![Notation
Z set of all integers.
Z+(Z++) set of all nonnegative (positive) integers.
R set of all real numbers.
R+(R++) set of all nonnegative (positive) real numbers.
R
set of all -vectors.
x = (xi ) = (x1, . . . , x) an element of R
.
x ≥ 0 xi ≥ 0 for i = 1, 2, . . . , ; [x is nonnegative].
x 0 xi ≥ 0 for all i; xi 0 for some i; [x is
positive].
x 0 xi 0 for all i; [x is strictly positive].
(S, S) a measurable space [when S is a metric space,
S = B(S) is the Borel sigmafield unless
otherwise specified].
xv](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-21-320.jpg)
![1.2 Basic Definitions: Fixed and Periodic Points 3
We have selected some descriptive models, some models of optimization
with a single decision maker, a dynamic game theoretic model, and an
example of intertemporal equilibrium with overlapping generations. An
interesting lesson that emerges is that variations of some well-known
models that generate monotone behavior lead to dynamical systems ex-
hibiting Li–Yorke chaos, or even to systems with the quadratic family as
possible laws of motion.
1.2 Basic Definitions: Fixed and Periodic Points
We begin with some formal definitions. A dynamical system is described
by a pair (S, α) where S is a nonempty set (called the state space) and α
is a function (called the law of motion) from S into S. Thus, if xt is the
state of the system in period t, then
xt+1 = α(xt ) (2.1)
is the state of the system in period t + 1.
In this chapter we always assume that the state space S is a (nonempty)
metric space (the metric is denoted by d). As examples of (2.1), take S
to be the set R of real numbers, and define
α(x) = ax + b, (2.2)
where a and b are real numbers.
Another example is provided by S = [0, 1] and
α(x) = 4x(1 − x). (2.3)
Here in (2.3), d(x, y) ≡ |x − y|.
The evolution of the dynamical system (R, α) where α is defined by
(2.2) is described by the difference equation
xt+1 = axt + b. (2.4)
Similarly, the dynamical system ([0, 1], α) where α is defined by (2.3)
is described by the difference equation
xt+1 = 4xt (1 − xt ). (2.5)
Once the initial state x (i.e., the state in period 0) is specified, we write
α0
(x) ≡ x, α1
(x) = α(x), and for every positive integer j ≥ 1,
α j+1
(x) = α(α j
(x)). (2.6)](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-25-320.jpg)
![4 Dynamical Systems
We refer to α j
as the jth iterate of α. For any initial x, the trajectory
from x is the sequence τ(x) = {α j
(x)∞
j=0}. The orbit from x is the set
γ (x) = {y: y = α j
(x) for some j ≥ 0}. The limit set w(x) of a trajectory
τ(x) is defined as
w(x) =
∞
j=1
[τ(α j (x)], (2.7)
where Ā is the closure of A.
Fixed and periodic points formally capture the intuitive idea of a sta-
tionary state or an equilibrium of a dynamical system. In his Foundations,
Samuelson (1947, p. 313) noted that “Stationary is a descriptive term
characterizing the behavior of an economic variable over time; it usually
implies constancy, but is occasionally generalized to include behavior
periodically repetitive over time.”
A point x ∈ S is a fixed point if x = α(x). A point x ∈ S is a periodic
point of period k ≥ 2 if αk
(x) = x and α j
(x) = x for 1 ≤ j k. Thus,
to prove that x is a periodic point of period, say, 3, one must prove that
x is a fixed point of α3
and that it is not a fixed point of α and α2
. Some
writers consider a fixed point as a periodic point of period 1.
Denote the set of all periodic points of S by ℘(S). We write ℵ(S) to
denote the set of nonperiodic points.
We now note some useful results on the existence of fixed points of α.
Proposition 2.1 Let S = R and α be continuous. If there is a (nondegen-
erate) closed interval I = [a, b] such that (i) α(I) ⊂ I or (ii) α(I) ⊃ I,
then there is a fixed point of α in I.
Proof.
(i) If α(I) ⊂ I, then α(a) ≥ a and α(b) ≤ b. If α(a) = a or α(b) = b,
the conclusion is immediate. Otherwise, α(a) a and α(b) b. This
means that the function β(x) = α(x) − x is positive at a and negative
at b. Using the intermediate value theorem, β(x∗
) = 0 for some x∗
in
(a, b). Then α(x∗
) = x∗
.
(ii) By the Weierstrass theorem, there are points xm and xM in I
such that α(xm) ≤ α(x) ≤ α(xM ) for all x in I. Write α(xm) = m and
α(xM ) = M. Then, by the intermediate value theorem, α(I) = [m, M].](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-26-320.jpg)
![1.2 Basic Definitions: Fixed and Periodic Points 5
Since α(I) ⊃ I, m ≤ a ≤ b ≤ M. In other words,
α(xm) = m ≤ a ≤ xm,
and
α(xM ) = M ≥ b ≥ xM .
The proof can now be completed by an argument similar to that in
case (i).
Remark 2.1 Let S = [a, b] and α be a continuous function from S into
S. Suppose that for all x in (a, b) the derivative α
(x) exists and |α
(x)|
1. Then α has a unique fixed point in S.
Proposition 2.2 Let S be a nonempty compact convex subset of R
, and
α be continuous. Then there is a fixed point of α.
A function α : S → S is a uniformly strict contraction if there is some
C, 0 C 1, such that for all x, y ∈ X, x = y, one has
d(α(x), α(y)) Cd(x, y). (2.8)
If d(α(x), α(y)) d(x, y) for x = y, we say that α is a strict contrac-
tion. If only
d(α(x), α(y)) d(x, y),
we say that α is a contraction.
If α is a contraction, α is continuous on S.
In this book, the following fundamental theorem is used many times:
Theorem 2.1 Let (S, d) be a nonempty complete metric space and
α : S → S be a uniformly strict contraction. Then α has a unique fixed
point x∗
∈ S. Moreover, for any x in S, the trajectory τ(x) = {α j
(x)∞
j=0}
converges to x∗
.
Proof. Choose an arbitrary x ∈ S. Consider the trajectory τ(x) = (xt )
from x, where
xt+1 = α(xt ). (2.9)](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-27-320.jpg)
![6 Dynamical Systems
Note that d(x2, x1) = d(α(x1), α(x)) Cd(x1, x) for some C ∈ (0, 1);
hence, for any t ≥ 1,
d(xt+1, xt ) Ct
d(x1, x). (2.10)
We note that
d(xt+2, xt ) ≤ d(xt+2, xt+1) + d(xt+1, xt )
Ct+1
d(x1, x) + Ct
d(x1, x)
= Ct
(1 + C)d(x1, x).
It follows that for any integer k ≥ 1,
d(xt+k, xt ) [Ct
/(1 − C)]d(x1, x),
and this implies that (xt ) is a Cauchy sequence. Since S is assumed to be
complete, limitt→∞ xt = x∗
exists. By continuity of α, and (2.9),
α(x∗
) = x∗
.
If there are two distinct fixed points x∗
and x∗∗
of α, we see that there is
a contradiction:
0 d(x∗
, x∗∗
) = d(α(x∗
), α(x∗∗
)) Cd(x∗
, x∗∗
), (2.11)
where 0 C 1.
Remark 2.2 For applications of this fundamental result, it is important
to reflect upon the following:
(i) for any x ∈ S, d(αn
(x), x∗
) ≤ Cn
(1 − C)−1
d(α(x), x)),
(ii) for any x ∈ S, d(x, x∗
) ≤ (1 − C)−1
d(α(x), x).
Theorem 2.2 Let S beanonemptycompletemetricspaceandα : S → S
be such that αk
is a uniformly strict contraction for some integer k 1.
Then α has a unique fixed point x∗
∈ S.
Proof. Let x∗
be the unique fixed point of αk
. Then
αk
(α(x∗
)) = α(αk
(x∗
)) = α(x∗
)
Hence α(x∗
) is also a fixed point of αk
. By uniqueness, α(x∗
) = x∗
.
This means that x∗
is a fixed point of α. But any fixed point of α is a
fixed point of αk
. Hence x∗
is the unique fixed point of α.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-28-320.jpg)
![1.2 Basic Definitions: Fixed and Periodic Points 7
Theorem 2.3 Let S be a nonempty compact metric space and α : S → S
be a strict contraction. Then α has a unique fixed point.
Proof. Since d(α(x), x) is continuous and S is compact, there is an
x∗
∈ S such that
d(α(x∗
), x∗
) = inf
x∈S
d(α(x), x). (2.12)
Then α(x∗
) = x∗
, otherwise
d(α2
(x∗
), α(x∗
)) d(α(x∗
), x∗
),
contradicting (2.12).
Exercise 2.1
(a) Let S = [0, 1], and consider the map α : S → S defined by
α(x) = x −
x2
2
.
Show that α is a strict contraction, but not a uniformly strict contraction.
Analyze the behavior of trajectories τ(x) from x ∈ S.
(b) Let S = R, and consider the map α : S → S defined by
α(x) = [x + (x2
+ 1)1/2
]/2.
Show that α(x) is a strict contraction, but does not have a fixed
point.
A fixed point x∗
of α is (locally) attracting or (locally) stable if there
is an open set U containing x∗
such that for all x ∈ U, the trajectory τ(x)
from x converges to x∗
.
Weshalloftendropthecaveat“local”:notethatlocalattractionorlocal
stability is to be distinguished from the property of global stability of a
dynamical system: (S, α) is globally stable if for all x ∈ S, the trajectory
τ(x) converges to the unique fixed point x∗
. Theorem 2.1 deals with
global stability.
A fixed point x∗
of α is repelling if there is an open set U containing
x∗
such that for any x ∈ U, x = x∗
, there is some k ≥ 1, αk
(x) /
∈ U.
Consider a dynamical system (S, α) where S is a (nondegenerate)
closed interval [a, b] and α is continuous on [a, b]. Suppose that α is](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-29-320.jpg)
![8 Dynamical Systems
also continuously differentiable on (a, b). A fixed point x∗
∈ (a, b) is
hyperbolic if |α
(x∗
)| = 1.
Proposition 2.3 Let S = [a, b] and α be continuous on [a, b] and con-
tinuously differentiable on (a, b). Let x∗
∈ (a, b) be a hyperbolic fixed
point of α.
(a) If |α
(x∗
)| 1, then x∗
is locally stable.
(b) If |α
(x∗
)| 1, then x∗
is repelling.
Proof.
(a) There is some u 0 such that |α
(x)| m 1 for all x in I =
[x∗
− u, x∗
+ u]. By the mean value theorem, if x ∈ I,
|α(x) − x∗
| = |α(x) − α(x∗
)| ≤ m|x − x∗
| mu u.
Hence, α maps I into I and, again, by the mean value theorem, is a
uniformly strict contraction on I. The result follows from Theorem 2.1.
(b) this is left as an exercise.
We can define “a hyperbolic periodic point of period k” and define
(locally) attracting and repelling periodic points accordingly.
Let x0 be a periodic point of period 2 and x1 = α(x0). By defini-
tion x0 = α(x1) = α2
(x0) and x1 = α(x0) = α2
(x1). Now if α is differ-
entiable, by the chain rule,
[α2
(x0)]
= α
(x1)α
(x0).
More generally, suppose that x0 is a periodic point of period k and its
orbit is denoted by {x0, x1, . . . , xk−1}. Then,
[αk
(x0)]
= α
(xk−1) · · · α
(x0).
It follows that
[αk
(x0)]
= [αk
(x1)]
· · · [αk
(xk−1)]
.
We can now extend Proposition 2.3 appropriately.
While the contraction property of α ensures that, independent of the
initial condition, the trajectories enter any neighborhood of the fixed
point, there are examples of simple nonlinear dynamical systems in
which trajectories “wander around” the state space. We shall examine
this feature more formally in Section 1.3.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-30-320.jpg)
![1.2 Basic Definitions: Fixed and Periodic Points 9
Example 2.1 Let S = R, α(x) = x2
. Clearly, the only fixed points of α
are 0, 1. More generally, keeping S = R, consider the family of dynam-
ical systems αθ (x) = x2
+ θ, where θ is a real number. For θ 1/4, αθ
does not have any fixed point; for θ = 1/4, αθ has a unique fixed point
x = 1/2; for θ 1/4, αθ has a pair of fixed points.
When θ = −1, the fixed points of the map α(−1)(x) = x2
− 1 are
[1 +
√
5]/2 and [1 −
√
5]/2. Now α(−1)(0) = −1; α(−1)(−1) = 0.
Hence, both 0 and −1 are periodic points of period 2 of α(−1). It fol-
lows that:
τ(0) = (0, −1, 0, −1, . . .), τ(−1) = (−1, 0, −1, 0, . . .),
γ (−1) = {−1, 0}, γ (0) = {0, −1}.
Since
α2
(−1)(x) = x4
− 2x2
,
we see that (i) α2
(−1) has four fixed points: the fixed points of α(−1), and
0, −1; (ii) the derivative of α2
(−1) with respect to x, denoted by [α2
(−1)(x)]
,
is given by
[α2
(−1)(x)]
= 4x3
− 4x.
Now, [α2
(−1)(x)]
x=0 = [α2
(−1)(x)]
x=−1 = 0. Hence, both 0 and −1 are
attracting fixed points of α2
.
Example 2.2 Let S = [0, 1]. Consider the “tent map” defined by
α(x) =
2x for x ∈ [0, 1/2]
2(1 − x) for x ∈ [1/2, 1].
Note that α has two fixed points “0” and “2/3.” It is tedious to write out
the functional form of α2
:
α2
(x) =
4x for x ∈ [0, 1/4]
2(1 − 2x) for x ∈ [1/4, 1/2]
2(2x − 1) for x ∈ [1/2, 3/4]
4(1 − x) for x ∈ [3/4, 1].
Verify the following:
(i) “2/5” and “4/5” are periodic points of period 2.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-31-320.jpg)
![10 Dynamical Systems
(ii) “2/9,” “4/9,” “8/9” are periodic points of period 3. It follows from
a well-known result (see Theorem 3.1) that there are periodic points of
all periods.
By using the graphs, if necessary, verify that the fixed and periodic
points of the tent map are repelling.
Example 2.3 In many applications to economics and biology, the state
space S is the set of all nonnegative reals, S = R+. The law of motion
α : S → S has the special form
α(x) = xβ(x), (2.11
)
where β(0) ≥ 0, β : R+ → R+ is continuous (and often has additional
properties). Now, the fixed points x̂ of α must satisfy
α(x̂) = x̂
or
x̂[1 − β(x̂)] = 0.
The fixed point x̂ = 0 may have a special significance in a particular
context (e.g., extinction of a natural resource). Some examples of α
satisfying (2.11
) are
(Verhulst 1845) α(x) =
θ1x
x + θ2
, θ1 0, θ2 0.
(Hassell 1975) α(x) = θ1x(1 + x)−θ2
, θ1 0, θ2 0.
(Ricker 1954) α(x) = θ1xe−θ2x
, θ1 0, θ2 0.
Here θ1, θ2 are interpreted as exogenous parameters that influence the
law of motion α.
Assume that β(x) is differentiable at x ≥ 0. Then,
α
(x) = β(x) + xβ
(x).
Hence,
α
(0) = β(0).
For each of the special maps, the existence of a fixed point x̂ = 0 and the
local stability properties depend on the values of the parameters θ1, θ2.
We shall now elaborate on this point.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-32-320.jpg)
![12 Dynamical Systems
A subinterval of an interval I is an interval contained in I. Since α is
continuous, α(I) is an interval. If I is a compact interval, so is α(I).
Suppose that a dynamical system (S, α) has a periodic point of pe-
riod k. Can we conclude that it also has a periodic point of some other
period k
= k? It is useful to look at a simple example first.
Example 3.1 Suppose that (S, α) has a periodic point of period k(≥2).
Then it has a fixed point (i.e., a periodic point of period one). To see this,
consider the orbit γ of the periodic point of period k, and let us write
γ = {x(1)
, . . . , x(k)
},
where x(1)
x(2)
· · · x(k)
. Both α(x(1)
) and α(x(k)
) must be in γ .
This means that
α(x(1)
) = x(i)
for some i 1
and
α(x(k)
) = x( j)
for some j k.
Hence, α(x(1)
) − x(1)
0 and α(x(k)
) − x(k)
0.
By the intermediate value theorem, there is some x in S such that
α(x) = x.
We shall now state the Li–Yorke theorem (Li and Yorke 1975) and
provide a brief sketch of the proof of one of the conclusions.
Theorem 3.1 Let I be an interval and α : I → I be continuous. Assume
that there is some point a in I for which there are points b = α(a),
c = α(b), and d = α(c) satisfying
d ≤ a b c (or d ≥ a b c). (3.1)
Then
[1] for every positive integer k = 1, 2, . . . there is a periodic point of
period k, x(k)
, in I,
[2] there is an uncountable set ℵ
⊂ ℵ(I) such that
(i) for all x, y in ℵ
, x = y,
lim sup
n→∞
|αn
(x) − αn
(y)| 0; (3.2)
lim inf
n→∞
|αn
(x) − αn
(y)| = 0. (3.3)](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-34-320.jpg)
![1.3 Complexity 13
(ii) If x ∈ ℵ
and y ∈ ℘(I)
lim sup
n→∞
|αn
(x) − αn
(y)| 0.
Proof of [1].
Step1. Let G beareal-valuedcontinuousfunctiononaninterval I.For
any compact subinterval I1 of G(I) there is a compact subinterval
Q of I such that G(Q) = I1.
Proof of Step 1. One can figure out the subinterval Q directly as fol-
lows. Let I1 = [G(x), G(y)] where x, y are in I. Assume that x y.
Let r be the last point of [x, y] such that G(r) = G(x); let s be the
first point after r such that G(s) = G(y). Then Q = [r, s] is mapped
onto I1 under G. The case x y is similar.
Step 2. Let I be an interval and α : I → I be continuous. Suppose
that (In)∞
n=0 is a sequence of compact subintervals of I, and for
all n,
In+1 ⊂ α(In). (3.4)
Then there is a sequence of compact subintervals (Qn) of I such
that for all n,
Qn+1 ⊂ Qn ⊂ Q0 = I0 (3.5)
and
αn
(Qn) = In. (3.6)
Hence, there is
x ∈
n
Qn such that αn
(x) ∈ In for all n. (3.7)
Proof of Step 2. The construction of the sequence Qn proceeds “in-
ductively” as follows: Define Q0 = I0. Recall that α0
is defined as
the identity mapping, so α0
(Q0) = I0 and I1 ⊂ α(I0). If Qn−1 is de-
fined as a compact subinterval such that αn−1
(Qn−1) = In−1, then
In ⊂ α(In−1) = αn
(Qn−1). Use Step 1, with G = αn
on Qn−1, in or-
der to get a compact subinterval Qn of Qn−1 such that αn
(Qn) = In.
This completes the induction argument (establishing (3.5) and
(3.6)). Compactness of Qn leads to (3.7).](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-35-320.jpg)
![14 Dynamical Systems
Now we prove [1]. Assume that d ≤ a b c (the other case
d ≥ a b c is treated similarly).
Write K = [a, b] and L = [b, c].
Let k be any positive integer.
For k 1, define a sequence of intervals (In) as follows:
In = L for n = 0, 1, 2, . . . , k − 2; Ik−1 = K; and In+k = In for
n = 0, 1, 2, . . . .
For k = 1, let In = L for all n.
Let Qn be the intervals in Step 2. Note that Qk ⊂ Q0 = I0 and
αk
(Qk) = Ik = I0. Hence, Proposition 2.1 applied to αk
gives us a fixed
point xk
of αk
in Qk. Now, xk
cannot have a period less than k; otherwise,
we need to have αk−1
(xk
) = b, contrary to αk+1
(xk
) ∈ L.
Proof of [2]. See Complements and Details.
We shall now state Sarkovskii’s theorem on periodic points. Consider
the following Sarkovskii ordering of the positive integers:
3 5 7 · · · 2.3 2.5 · · · 22
3 22
5 · · · (SO)
23
.3 23
.5 · · · 23
22
2 1
In other words, first list all the odd integers beginning with 3; next list 2
times the odds, 22
times the odds, etc. Finally, list all the powers of 2 in
decreasing order.
Theorem 3.2 Let S = R and α be a continuous function from S into S.
Suppose that α has a periodic point of period k. If k k
in the Sarkovskii
ordering (SO), then α has a periodic point of period k
.
Proof. See Devaney (1986).
It follows that if α has only finitely many periodic points, then they all
necessarily have periods that are powers of two.
1.3.2 A Remark on Robustness of Li–Yorke Complexity
Let S = [J, K], and suppose that a continuous function α satisfies the
Li–Yorke condition (3.1) with strict inequality throughout; i.e., suppose
that there are points a, b, c, d such that
d = α(c) a b = α(a) c = α(b) (3.8)](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-36-320.jpg)
![1.3 Complexity 15
or
d = α(c) a b = α(a) c = α(b). (3.9)
Consider the space C(S) of all continuous (hence, bounded) real-
valued functions on S = [J, K]. Let α = maxx∈S α(x). The conclu-
sions of Theorem 3.1 hold with respect to the dynamical system (S, α).
But the Li–Yorke complexity is now “robust” in a precise sense.
Proposition 3.1 Let S = [J, K], and let α satisfy (3.8). In addition,
assume that
J m(S, α) M(S, α) K, (3.10)
where m(S, α) and M(S, α) are respectively the minimum and maximum
of α on [J, K]. Then there is an open set N of C(S) containing α such
that β ∈ N implies that [1] and [2] of Theorem 3.1 hold with β in place
of α.
Proof. First, we show the following:
Fix x ∈ [J, K]. Given k ≥ 1, ε 0, there exists δ(k, ε) 0 such that
“β − α δ(k, ε)” implies |β j
(x) − α j
(x)| ε for all j = 1, . . . , k.
The proof is by induction on k. It is clearly true for k = 1, with
δ(1, ε) ≡ ε. Assume that the claim is true for k = m, but not for
k = m + 1. Then there exist some ε 0 and a sequence of functions
{βn} satisfying βn − α → 0 such that |βm+1
n (x) − αm+1
(x)| ≥ ε. Let
βm
n (x) = yn and αm
(x) = y. Then, by the induction hypothesis, yn → y.
From Rudin (1976, Chapter 7) we conclude that βn(yn) → α(y), which
yields a contraction.
Next, choose a real number ρ satisfying 0 ρ min[1/2(a − d),
1/2(b − a), 1/2(c − b)] and a positive number r such that β − α r
implies |β j
(a) − α j
(a)| ρ for j = 1, 2, 3, and also 0 r min{K −
M(S, α), m(S, α) − J}.
Define the open set N as
N = {β ∈ C(S) : β − α r}.
It follows that any β ∈ N maps S into S, since the maximum of β on
[J, K] is less than M(S, α) + r K. Similarly, the minimum of β on
[J, K] is likewise greater than J. It remains to show that the condition
(3.8) also holds for any β in N. Recall that α(a) = b, α(b) = c, α(c) = d.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-37-320.jpg)
![1.4 Linear Difference Equations 17
It is important to reflect upon the behavior of a topologically transitive
dynamical system and contrast it to one in which the law of motion
satisfies the strict contraction property. We now turn to another concept
that has profound implications for the long-run prediction of a dynamical
system. A dynamical system (S, α) has sensitive dependence on initial
condition if there is ∂ 0 such that for any x ∈ S and any neighborhood
N of x there exist y ∈ N and an integer j ≥ 0 with the property |α j
(x) −
α j
(y)| ∂.
Devaney (1986) asserted that if a dynamical system “possesses sen-
sitive dependence on initial condition, then for all practical purposes,
the dynamics defy numerical computation. Small errors in computation
which are introduced by round-off may become magnified upon itera-
tion. The results of numerical computation of an orbit, no matter how
accurate, may bear no resemblance whatsoever with the real orbit.”
Example 3.2 The map α(x) = 4x(1 − x) on [0, 1] is topologically tran-
sitive and has sensitive dependence on initial condition (see Devaney
1986).
1.4 Linear Difference Equations
Excellent coverage of this topic is available from many sources: we pro-
vide only a sketch. Consider
xt+1 = axt + b. (4.1)
When a = 1, xt = x0 + bt. When a = −1, xt = −x0 + b for t =
1, 3, . . . , and xt = x0 for t = 2, 4, . . . .
Nowassumea = 1.Themapα(x) = ax + b whena = 1,hasaunique
fixed point x∗
= b
1−a
. Given any initial x0 = x, the solution to (4.1) can
be verified as
xt = (x − x∗
)at
+ x∗
. (4.2)
The long-run behavior of trajectories from alternative initial x can
be analyzed from (4.2). In this context, the important fact is that
the sequence at
converges to 0 if |a| 1 and becomes unbounded if
|a| 1.
Example 4.1 Consider an economy where the output yt (≥0) in any pe-
riod is divided between consumption ct (≥0) and investment xt (≥0). The](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-39-320.jpg)
![18 Dynamical Systems
return function is given by
yt+1 = rxt , r 1, t ≥ 0. (4.3)
Given an initial stock y 0, a program x = (xt ) (from y) is a nonneg-
ative sequence satisfying x0 ≤ y, xt+1 ≤ rxt (for t ≥ 1). It generates a
corresponding consumption program c = (ct ) defined by c0 = y − x0;
ct+1 = yt+1 − xt+1 = rxt − xt+1 for t ≥ 0.
(a) Show that a consumption program c = (ct ) must satisfy
∞
t=0
ct /rt
≤ y.
(b) Call a program x from y [generating c] efficient if there does not
exist another program x
[generating (c
)] from y such that c
t ≥ ct for all
t ≥ 0 with strict inequality for some t. Show that a program x is efficient
if and only if
∞
t=0
ct /rt
= y.
(c) Suppose that for the economy to survive it must consume an
amount c 0 in every period. Informally, we say that a program x from
y that satisfies ct ≥ c for all t ≥ 0 survives at (or above) c. Note that the
law of motion of the economy that plans a consumption c ≥ 0 in every
period t ≥ 1, with an investment x0 initially, can be written as
xt+1 = rxt − c. (4.4)
This equation has a solution
xt = rt
(x0 − ξ) + ξ,
where
ξ = c/(r − 1). (4.5)
Hence,
(1) xt reaches 0 in finite time if x0 ξ;
(2) xt = ξ for all t if x0 = ξ;
(3) xt diverges to (plus) infinity if x0 ξ.
To summarize the implications for survival and sustainable develop-
ment we state the following:](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-40-320.jpg)
![1.4 Linear Difference Equations 19
Proposition 4.1 Survival at (or above) c 0 is possible if and only if
the initial stock y ≥ ξ + c, or equivalently,
y ≥
r
r − 1
c.
Of course if r ≤ 1, there is no program that can guarantee that ct ≥ c
for any c 0.
In this chapter, our exposition deals with processes that are generated
by an autonomous or time invariant law of motion [in (2.1), the function
α : S → S does not depend on time]. We shall digress briefly and study
a simple example which explicitly recognizes that the law of motion may
itself depend on time.
Example 4.2 Let xt (t ≥ 0) be the stock of a natural resource at the
beginning of period t, and assume that the evolution of xt is described
by the following (nonautonomous) difference equation
xt+1 = axt + bt+1, (4.6)
where 0 a 1 and (bt+1)t≥0 is a sequence of nonnegative num-
bers. During period t, (1 − a)xt is consumed or used up and bt+1
is the “new discovery” of the resource reported at the beginning of
period t + 1.
Given an initial x0 0, one can write
xt = at
x0 +
t−1
i=0
ai
bt−i . (4.7)
Now, if the sequence (bt+1)t≥0 is bounded above, i.e., if there is B 0
such that 0 ≤ bt+1 ≤ B for all t ≥ 0, then the sequence t−1
i=0 ai
bt−i
converges, so that the sequence (xt ) converges to a finite limit as
well.
For other examples of such nonautonomous systems, see Azariadis
(1993, Chapters 1–5). What happens if we want to introduce uncertainty
in the new discovery of the resource? Perhaps the natural first step is to
consider (bt+1)t≥0 as a sequence of independent, identically distributed
random variables (assuming values in R+). This leads us to the process
(2.1) studied in Chapter 4.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-41-320.jpg)
![22 Dynamical Systems
Now in Case I, x2 = α(x1) α(x0) = x1. Hence {xt } is an increasing
sequence. In Case II, xt = x0 for all t ≥ 0. Finally, in Case III {xt } is a
decreasing sequence. The appropriate rewording of Proposition 5.1 when
α is a continuous, increasing function is left as an exercise.
Proposition 5.2 (Uzawa–Inada condition) S = R+, α is continuous,
nondecreasing, and satisfies the Uzawa–Inada condition (UI):
A(x) ≡ [α(x)/x] is decreasing in x 0;
for some x̄ 0, A(x̄) 1,
and for some
=
x x̄ 0, A(
=
x) 1. (UI)
Then the condition (PI) holds.
Proof. Clearly, by the intermediate value theorem, there is some x∗
∈
(x̄,
=
x) such that A(x∗
) = 1, i.e., α(x∗
) = x∗
0. Since A(x) is decreas-
ing, A(x) 1 for x x∗
and A(x) 1 for x x∗
. In other words, for
all x ∈ (0, x∗
), α(x) x and for all x x∗
, α(x) x, i.e., the property
(PI) holds.
In the literature on economic growth, the (UI) condition is implied
by appropriate differentiability assumptions. We state a list of typical
assumptions from this literature.
Proposition 5.3 Let S = R+ and α : S → S be a function that is
(i) continuous on S,
(ii) twice continuously differentiable at x 0 satisfying:
[E.1] lim
x↓0
α
(x) = 1 + θ1, θ1 0,
[E.2] lim
x↑∞
α
(x) = 1 − θ2, θ2 0,
[E.3] α
(x) 0, α
(x) 0 at x 0.
Then the condition (UI) holds.
Proof. Take
=
x x̄ ≥ 0; by the mean value theorem
[α(
=
x) − α(x̄)] = (
=
x −x̄)α
(z) where x̄ z
=
x.
Since α
(z) 0, α(
=
x) α(x̄).
Thus α is increasing. Also, α
(x) 0 at x 0 means that α is strictly
concave. Take
=
x x̄ 0. Then x̄ ≡ t
=
x + (1 − t)0, 0 t 1.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-44-320.jpg)
![1.5 Increasing Laws of Motion 23
Now
α(x̄) = α(t
=
x + (1 − t)0)
tα(
=
x) + (1 − t)α(0)
≥ tα(
=
x)
or
α(x̄)
x̄
tα(
=
x)
t
=
x
=
α(
=
x)
=
x
.
Hence α(x)
x
is decreasing. Write B(x) ≡ α(x) − x. By [E.1]–[E.3]
there is some x̄ 0 such that α
(x) 1 for all x ∈ (0, x̄]. Hence,
B
(x) 0 for all x ∈ (0, x̄]. By the mean value theorem
α(x̄) = α(0) + x̄α
(z), 0 z x̄
≥ x̄α
(z)
or
α(x̄)
x̄
≥ α
(z) α
(x̄) 1.
Now, if α is bounded, i.e., if there is some N 0 such that α(x) ≤ N,
then [α(x)/x] ≤ [N/x]. Hence there is
=
x, sufficiently large, such that
α(
=
x)/
=
x ≤ [N/
=
x] 1.
If α is not bounded, we can find a sequence of points (xn) such that
α(xn) and xn go to infinity as n tends to infinity. Then
lim
n→∞
α(xn)
xn
= lim
n→∞
α
(xn) = 1 − θ2 1.
Hence, we can find some point
=
x sufficiently large, such that α(
=
x)/
=
x 1.
Thus, the Uzawa-Inada condition (UI) is satisfied.
Exercise 5.1 In his Economic Dynamics, Baumol (1970) presented a
simple model that captured some of the ideas of “classical” economists](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-45-320.jpg)
![24 Dynamical Systems
formally. Let Pt , the net total product in period t, be a function of the
working population Lt :
Pt = F(Lt ), t ≥ 0,
where F : R+ → R+ is continuous and increasing, F(0) = 0.
At any time, the working population tends to grow to a size where out-
put per worker is just enough to provide each worker with a “subsistence
level” of minimal consumption M 0. This is formally captured by the
relation
Lt+1 =
Pt
M
.
Hence,
Lt+1 =
F(Lt )
M
≡ α(Lt ).
Identify conditions on the average productivity function F(L)
L
(L 0)
that guarantee the following:
(i) There is a unique L∗
0 such that (F(L∗
)/L∗
) = M.
(ii) For any 0 L L∗
, the trajectory τ(L) is increasing and con-
verges to L∗
; for any L L∗
, the trajectory τ(L) decreases to L∗
.
Example 5.2 Consider an economy (or a fishery) which starts with an
initial stock y (the metaphorical corn of one-sector growth theory or the
stock of renewable resource, e.g., the stock of trouts, the population of
dodos, . . . ). In each period t, the economy is required to consume or
harvest a positive amount c out of the beginning of the period stock yt .
The remaining stock in that period xt = yt − c is “invested” and the
resulting output (principal plus return) is the beginning of the period
stock yt+1. The output yt+1 is related to the input xt by a “production”
function g. Assume that
[A.1] g : R → R is continuous, and increasing on R+, g(x) = 0
for x ≤ 0;
[A.2] there is some x∗
0 such that g(x) x for 0 x x∗
and
g(x) x for x x∗
;
[A.3] g is concave.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-46-320.jpg)
![1.5 Increasing Laws of Motion 25
The evolution of the system (given the initial y 0 and the planned
harvesting c 0) is described by
y0 = y,
xt = yt − c, t ≥ 0;
yt+1 = g(xt ) for t ≥ 0.
Let T be the first period t, if any, such that xt 0; if there is no
such t, then T = ∞. If T is finite we say that the agent (or the resource)
survives up to (but not including) period T . We say that the agent survives
(forever) if T = ∞ (i.e., if xt ≥ 0 for all t).
Define the net return function h(x) = g(x) − x. It follows that h
satisfies
h(x)
≥
0 as
0 x x∗
;
x = 0, x∗
;
x x∗
.
Since g(x) 0 for all x ≤ 0, all statements about g and h will be
understood to be for nonnegative arguments unless something explicit is
said to the contrary.
Actually, we are only interested in following the system up to the
“failure” or “extinction” time T .
The maximum sustainable harvest or consumption is
H = max
[0,x∗]
h(x). (5.3)
We write
xt+1 = xt + h(xt ) − c.
If c H, xt+1 − xt = h(xt ) − c H − c 0. Hence, xt will fall below
(extinction) 0 after a finite number of periods. On the other hand, if
0 c H, there will be two roots ξ
and ξ
of the equation
c = h(x),
which have the properties
0 ξ
ξ
x∗
.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-47-320.jpg)
![26 Dynamical Systems
and
h(x) − c
≥
0 as
ξ
x ξ
;
x = ξ
, ξ
;
x ξ
, x ξ
.
We can show that
(a) If x0 ξ
, then xt reaches or falls below 0 in finite time.
(b) If x0 = ξ
, then xt = ξ
for all t.
(c) If x0 ξ
, then xt converges monotonically to ξ
.
Note that if c = H, there are two possibilities: either ξ
= ξ
(i.e.,
h(x) attains the maximum H at a unique period ξ
) or for all x in a
nondegenerate interval [ξ
, ξ
], h(x) attains its maximum.
The implications of the foregoing discussion for survival and extinc-
tion are summarized as follows.
Proposition 5.4 Let c 0 be the planned consumption for every period
and H be the maximum sustainable consumption.
(1) If c H, there is no initial y from which survival (forever) is
possible.
(2) If 0 c H, then there is ξ
, with h(ξ
) = c, ξ
0
such that survival is possible if and only if the initial stock y ≥ ξ
+ c
(3) c = H implies h(ξ
) = H, and ξ
tends to 0 as c tends to 0.
1.6 Thresholds and Critical Stocks
We now consider some examples of dynamical systems that have been
of particular interest in various contexts in development economics and
in the literature on the management of a renewable resource. It has been
emphasized that the evolution of an economy may depend crucially on
the initial condition (hence, on the “history” that leads to it). It has also
been noted that if the stock of a renewable resource falls below a critical
level, the biological reproduction law may lead to its eventual extinction.
In sharp contrast with the dynamical systems identified in Proposition
5.1, in which trajectories from positive initial stocks all converge to
a positive fixed point, we sketch some examples where the long-run
behavior of trajectories changes remarkably as the initial condition goes
above a threshold (see Complements and Details).](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-48-320.jpg)
![1.6 Thresholds and Critical Stocks 27
Consider the “no-harvesting” case where S = R+, and the biological
reproduction law is described by a continuous function α : R+ → R+.
Given an initial state x ≥ 0, the state in period t is the stock of the
resources at the beginning of that period and (2.1) is assumed to hold.
When xt = 0, the resource is extinct. We state a general result.
Proposition 6.1 Let S = R+ and α : S → S be a continuous, increas-
ing function with the following properties [P2]:
[P2.1] α(0) = 0;
[P2.2] there are two positive fixed points x∗
(1), x∗
(2) (0 x∗
(1) x∗
(2))
such that
(i) α(x) x for x ∈ (0, x∗
(1));
(ii) α(x) x for x ∈ (x∗
(1), x∗
(2));
(iii) α(x) x for x x∗
(2).
For any x ∈ (0, x∗
(1)), the trajectory τ(x) from x is decreasing and
converges to 0; for any x ∈ (x∗
(1), x∗
(2)), the trajectory τ(x) from x is
increasing and converges to x∗
(2); for any x x∗
(2), the trajectory τ(x)
from x is decreasing and converges to x∗
(2).
Proof. Left as an exercise.
The striking feature of the trajectories from any x 0 is that these
are all convergent, but the limits depend on the initial condition. Also,
these are all monotone, but again depending on the initial condition –
some are increasing, others are decreasing. We interpret x∗
(1) 0 as the
critical level for survival of the resource.
Exercise 6.1 Consider S = R+ and α : S → S defined by
α(x) =
θ1x2
x2 + θ2
where θ1, θ2 are positive parameters. When θ1 2
√
θ2, compute the
fixed points of α and verify [P2].
Exercise 6.2 Consider the problem of survival with constant harvesting
discussed in Example 5.2. Work out the conditions for survival with a](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-49-320.jpg)
![28 Dynamical Systems
constant harvest c 0 when the “production” function g is continuous,
increasing on R+, and satisfies
[P2.1] g(x) = 0 for x ≤ 0;
[P2.2] there are two positive fixed points x∗
(1), x∗
(2) (0 x∗
(1) x∗
(2))
such that
(i) g(x) x for x ∈ (0, x∗
(1));
(ii) g(x) x for x ∈ (x∗
(1), x∗
(2));
(iii) g(x) x for x x∗
(2).
Exercise 6.3 Let S = R+ and α : S → S is defined by α(x) = xm
, where
m ≥ 1. Show that (i) the fixed points of α are “0” and “1”; (ii) for
all x ∈ [0, 1), the trajectory τ(x) is monotonically decreasing, and con-
verges to 0. For x 1, the trajectory α(x) is monotonically increasing
and unbounded. Thus, x = 1 is the “threshold” above which sustainable
growth is possible; if the initial x 1, the trajectory from x leads to
extinction.
Example 6.1 In parts of Section 1.9 we review dynamical systems that
arise out of “classical” optimization models (in which “convexity” as-
sumptions on the preferences and technology hold). Here we sketch an
example of a “nonclassical” optimization model in which a critical level
of initial stock has important policy implications.
Think of a competitive fishery (see Clark 1971, 1976, Chapter 7). Let
xt (≥0) be the stock or “input” of fish in period t, and f : R+ → R+ the
biological reproduction relationship. The stock x in any period gives rise
to output y = f (x) in the subsequent period. The following assumptions
on f are introduced:
[A.1] f (0) = 0;
[A.2] f (x) is twice continuously differentiable for x ≥ 0; f
(x) 0
for x 0.
[A.3] f satisfies the following end-point conditions: f
(∞) 1
f
(0) ∞; f
(x) 0 for x 0.
[A.4]Thereisa(finite)b1 0,suchthat (i) f
(b1) = 0;(ii) f
(x) 0
for 0 ≤ x b1; (iii) f
(x) 0 for x b1.
In contrast to the present (“nonclassical”) model, the traditional (or
“classical”) framework would replace [A.4] by](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-50-320.jpg)
![1.6 Thresholds and Critical Stocks 29
[A.4
]. f is strictly concave for x ≥ 0 ( f
(x) 0 for x 0), while
preserving [A.1]–[A.3].
In some versions, [A.2] and [A.3] are modified to allow f
(0) = ∞.
In the discussion to follow, we find it convenient to refer to a model
with assumptions [A.1]–[A.3] and [A.4
] as classical, and a model with
[A.1]–[A.4] as nonclassical.
We define a function h (representing the average product function) as
follows:
h(x) = [ f (x)/x] for x 0; h(0) = lim
x→0
[ f (x)/x]. (6.1)
Under [A.1]–[A.4], it is easily checked that h(0) = f
(0); furthermore,
there exist positive numbers k∗
, k̄, b2 satisfying (i) 0 b1 b2
k∗
k̄ ∞; (ii) f
(k∗
) = 1; (iii) f (k̄) = k̄; (iv) f
(b2) = h(b2). Also,
for 0 ≤ x k∗
, f
(x) 1 and for x k∗
, f
(x) 1; for 0 x k̄,
x f (x) k̄ and for x k̄, k̄ f (x) x; and for 0 x b2,
f
(x) h(x) and for x b2, f
(x) h(x). Also note that for 0 ≤ x
b2, h(x) is increasing, and for x b2, h(x) is decreasing; for 0 ≤ x b1,
f
(x) is increasing, and for x b1, f
(x) is decreasing.
A feasible production program from x
˜
0 is a sequence (x, y) =
(xt , yt+1) satisfying
x0 = x
˜
; 0 ≤ xt ≤ yt and yt = f (xt−1) for t ≥ 1. (6.2)
The sequence x = (xt )t≥0 is the input (or stock) program, while the cor-
responding y = (yt+1)t≥0 satisfying (6.2) is the output program. The har-
vest program c = (ct ) generated by (x, y) is defined by ct ≡ yt − xt for
t ≥ 1. We will refer to (x, y, c) briefly as a program from x
˜
, it being
understood that (x, y) is a feasible production program, and c is the
corresponding harvest program.
A slight abuse of notation: we shall often specify only the stock pro-
gram x = (xt )t≥0 from x
˜
0 to describe a program (x, y, c). It will
be understood that x0 = x
˜
; 0 ≤ xt ≤ f (xt−1) for all t ≥ 1, ct = yt − xt
for t ≥ 1.
Let the profit per unit of harvesting, denoted by q 0, and the rate
of interest γ 0 remain constant over time. Consider a firm that has an
objective of maximizing the discounted sum of profits from harvesting.](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-51-320.jpg)
![30 Dynamical Systems
A program x∗
= (x∗
t ) of stocks from x
˜
0 is optimal if
∞
t=1
q
(1 + γ )t−1
c∗
t ≥
∞
t=1
q
(1 + γ )(t−1)
ct
for every program x from x
˜
. Write δ = 1/(1 + γ ). Models of this type
have been used to discuss the possible conflict between profit maximiza-
tion and conservation of natural resources.
The program (x, y, c) from x
˜
0 defined as x0 = x
˜
, xt = 0 for t ≥ 1
is the extinction program. Here the entire output f (x
˜
) is harvested in
period 1, i.e., c1 = f (x
˜
), ct = 0 for t ≥ 2.
In the qualitative analysis of optimal programs, the roots of the equa-
tion δ f
(x) = 1 play an important role. This equation might not have a
nonnegative real root at all; if it has a pair of unique nonnegative real
roots, denote it by Z; if it has nonnegative real roots, the smaller one is
denoted by z and the larger one by Z.
The qualitative behavior of optimal programs depends on the value of
δ = 1/(1 + γ ). Three cases need to be distinguished. The first two were
analyzed and interpreted by Clark (1971).
Case 1. Strong discounting: δ f
(b2) ≤ 1
This is the case when δ is “sufficiently small,” i.e., 1 + γ ≥ f (x)/x
for all x 0.
Proposition 6.2 The extinction program is optimal from any x
˜
0, and
is the unique optimal program if δ f
(b̂2) 1.
Remark 6.1 First, if δ f
(b2) = 1, there are many optimal programs (see
Majumdar and Mitra 1983, p. 146). Second, if we consider the classical
model(satisfying[A.1]–[A.3]and[A.4
]),itisstilltruethatifδ f
(0) ≤ 1,
the extinction program is the unique optimal program from any x
˜
0.
Case 2. Mild discounting: δ f
(0) ≥ 1
This is the case where δ is “sufficiently close to 1” (δ 1/f
(0)) and
Z b2 exists (if z exists, z = 0).
Now, given x
˜
Z, let M be the smallest positive integer such that
x1
M ≥ Z; in other words, M is the first period in which the pure](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-52-320.jpg)
![1.6 Thresholds and Critical Stocks 31
accumulation program from x
˜
defined by x1
0 = x
˜
, x1
t+1 = f (x1
t ) for t ≥ 0
attains Z.
Proposition 6.3 If x
˜
≥ Z, then the program x∗
= (x∗
t )t≥0 from x
˜
defined
by x∗
0 = x
˜
, x∗
t = Z for t ≥ 1 is the unique optimal program from x
˜
.
Proposition 6.4 If x
˜
Z, the program x∗
= (x∗
t )t≥0 defined by x∗
0 = x
˜
,
x∗
t = x1
t for t = 1, . . . , M − 1, x∗
t = Z for t ≥ M is the unique optimal
program.
In the corresponding classical model for δ f
(0) 1, there is a unique
positive K∗
δ , solving δ f
(x) = 1. Propositions 6.3 and 6.4 continue to
hold with Z replaced by K∗
δ (also in the definition of M).
Case 3. Two turnpikes and the critical point of departure [δ f
(0)
1 δ f
(k2)]
In case (i) the extinction program (x, y, c) generated by xt = 0 for all
t ≥ 1 and in case (ii) the stationary program generated by xt = Z for all
t ≥ 0 (which is also the optimal program from Z) serve as the “turnpikes”
approached by the optimal programs. Both the classical and nonclassical
models share the feature that the long-run behavior of optimal programs
is independent of the positive initial stock. The “intermediate” case of
discounting, namely when
1/f
(b2) δ 1/f
(0),
turned out to be difficult and to offer a sharp contrast between the classical
and nonclassical models. In this case
0 z b1 b2 Z k∗
.
The qualitative properties of optimal programs are summarized in two
steps.
Proposition 6.5 x
˜
≥ Z, the program x∗
= (x∗
t )t≥0 defined by x∗
0 = x
˜
,
x∗
t = Z for t ≥ 1 is optimal.
A program x = (xt )t≥0 from x
˜
Z is a regeneration program if there
is some positive integer N ≥ 1 such that xt xt−1 for 1 ≤ t ≤ N, and
xt = Z for t ≥ N. It should be stressed that a regeneration program](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-53-320.jpg)
![32 Dynamical Systems
may allow for positive consumption in all periods, and need not specify
“pure accumulation” in the initial periods. For an interesting example
of a regeneration program that allows for positive consumption and is
optimal, the reader is referred to Clark (1971, p. 259).
Proposition 6.6 Let x
˜
Z. There is a critical stock Kc 0 such that
if 0 x
˜
Kc, the extinction program from x
˜
is an optimal program. If
Kc x
˜
Z, then any optimal program is a regeneration program.
In the literature on renewable resources, Kc is naturally called the
“minimum safe standard of conservation.” It has been argued that a policy
that prohibits harvesting of a fishery till the stock exceeds Kc will ensure
that the fishery will not become extinct, even under pure “economic
exploitation.”
Some conditions on x
˜
can be identified under which there is a unique
optimal program. But if x
˜
= Kc, then both the extinction program and
a regeneration program are optimal. For further details and proofs, see
Majumdar and Mitra (1982, 1983).
1.7 The Quadratic Family
Let S = [0, 1] and A = [0, 4]. The quadratic family of maps is then
defined by
αθ (x) = θx(1 − x) for (x, θ) ∈ S × A. (7.1)
We interpret x as the variable and θ as the parameter generating the
family.
We first describe some basic properties of this family of maps. First,
note that Fθ (0) = 0(= Fθ (1)) ∀θ ∈ [0, 4], so that 0 is a fixed point
of Fθ . By solving the quadratic equation Fθ (x) = x, one notes that
pθ = 1 − 1/θ is the only other fixed point that occurs if θ 1. For θ 3,
one can show (e.g., by numerical calculations) that the fourth-degree
polynomial equation F2
θ (x): = Fθ ◦ Fθ (x) has two other solutions (in ad-
dition to 0 and pθ ). This means that for 3 θ ≤ 4, Fθ has a period-two
orbit. For θ 1 +
√
6, a new period-four orbit appears. We refer to the
Li–Yorke and Sarkovskii theorems, stated in Section 1.3, for the succes-
sive appearance of periodic points of period 2k
(k ≥ 0), as θ increases to a
limit point of θc ≈ 3.57, which is followed by other cascades of periodic](https://image.slidesharecdn.com/1301079-250529020146-5b7c8c2e/85/Random-Dynamical-Systems-Bhattacharya-R-Majumdar-M-54-320.jpg)
Random Dynamical Systems Bhattacharya R Majumdar M
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- 7. RANDOM DYNAMICAL SYSTEMS This book provides an exposition of discrete time dynamic processes evolv- ing over an infinite horizon. Chapter 1 reviews some mathematical results from the theory of deterministic dynamical systems, with particular empha- sis on applications to economics. The theory of irreducible Markov pro- cesses, especially Markov chains, is surveyed in Chapter 2. Equilibrium and long-run stability of a dynamical system in which the law of motion is sub- ject to random perturbations are the central theme of Chapters 3–5. A unified account of relatively recent results, exploiting splitting and contractions, that have found applications in many contexts is presented in detail. Chapter 6 explains how a random dynamical system may emerge from a class of dy- namic programming problems. With examples and exercises, readers are guided from basic theory to the frontier of applied mathematical research. Rabi Bhattacharya is Professor of Mathematics at the University of Arizona. He has also taught at the University of California at Berkeley and Indiana University. Professor Bhattacharya has held visiting research professorships at the University of Goettingen, the University of Bielefeld, and the Indian Statistical Institute. He is a recipient of a Guggenheim Fellowship and an Alexander Von Humboldt Forschungspreis. He is a Fellow of the Institute of Mathematical Statistics and has served on the editorial boards of a number of international journals, including the Annals of Probability, Annals of Ap- pliedProbability,JournalofMultivariateAnalysis,EconometricTheory,and Statistica Sinica. He has co-authored Normal Approximations and Asymp- totic Expansions (with R. Ranga Rao), Stochastic Processes with Applica- tions (with E. C. Waymire), and Asymptotic Statistics (with M. Denker). Mukul Majumdar is H. T. and R. I. Warshow Professor of Economics at Cornell University. He has also taught at Stanford University and the London School of Economics. Professor Majumdar is a Fellow of the Economet- ric Society and has been a Guggenheim Fellow, a Ford Rotating Research Professor at the University of California at Berkeley, an Erskine Fellow at the University of Canterbury, an Oskar Morgenstern Visiting Professor at New York University, a Lecturer at the College de France, and an Overseas Fellow at Churchill College, Cambridge University. Professor Majumdar has served on the editorial boards of many leading journals, including The Review of Economic Studies, Journal of Economic Theory, Journal of Mathematical Economics, and Economic Theory, and he has edited the collection Organi- zations with Incomplete Information (Cambridge University Press, 1998).
- 8. To Urmi, Deepta, and Aveek
- 9. Random Dynamical Systems Theory and Applications RABI BHATTACHARYA University of Arizona MUKUL MAJUMDAR Cornell University
- 10. CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-82565-8 ISBN-13 978-0-521-53272-3 ISBN-13 978-0-511-27353-7 © Rabi Bhattacharya and Mukul Majumdar 2007 2007 Information on this title: www.cambridge.org/9780521825658 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. ISBN-10 0-511-27353-3 ISBN-10 0-521-82565-2 ISBN-10 0-521-53272-8 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback paperback paperback eBook (EBL) eBook (EBL) hardback
- 11. Contents Preface page ix Acknowledgment xiii Notation xv 1 Dynamical Systems 1 1.1 Introduction 1 1.2 Basic Definitions: Fixed and Periodic Points 3 1.3 Complexity 11 1.3.1 Li–Yorke Chaos and Sarkovskii Theorem 11 1.3.2 A Remark on Robustness of Li–Yorke Complexity 14 1.3.3 Complexity: Alternative Approaches 16 1.4 Linear Difference Equations 17 1.5 Increasing Laws of Motion 20 1.6 Thresholds and Critical Stocks 26 1.7 The Quadratic Family 32 1.7.1 Stable Periodic Orbits 33 1.8 Comparative Statics and Dynamics 38 1.8.1 Bifurcation Theory 39 1.9 Some Applications 46 1.9.1 The Harrod–Domar Model 46 1.9.2 The Solow Model 47 1.9.3 Balanced Growth and Multiplicative Processes 53 1.9.4 Models of Intertemporal Optimization with a Single Decision Maker 59 1.9.5 Optimization with Wealth Effects: Periodicity and Chaos 77 1.9.6 Dynamic Programming 83 1.9.7 Dynamic Games 95 1.9.8 Intertemporal Equilibrium 98 1.9.9 Chaos in Cobb–Douglas Economies 101 v
- 12. vi Contents 1.10 Complements and Details 104 1.11 Supplementary Exercises 113 2 Markov Processes 119 2.1 Introduction 119 2.2 Construction of Stochastic Processes 122 2.3 Markov Processes with a Countable Number of States 126 2.4 Essential, Inessential, and Periodic States of a Markov Chain 131 2.5 Convergence to Steady States for Markov Processes on Finite State Spaces 133 2.6 Stopping Times and the Strong Markov Property of Markov Chains 143 2.7 Transient and Recurrent Chains 150 2.8 Positive Recurrence and Steady State Distributions of Markov Chains 159 2.9 Markov Processes on Measurable State Spaces: Existence of and Convergence to Unique Steady States 176 2.10 Strong Law of Large Numbers and Central Limit Theorem 185 2.11 Markov Processes on Metric Spaces: Existence of Steady States 191 2.12 Asymptotic Stationarity 196 2.13 Complements and Details 201 2.13.1 Irreducibility and Harris Recurrent Markov Processes 210 2.14 Supplementary Exercises 239 3 Random Dynamical Systems 245 3.1 Introduction 245 3.2 Random Dynamical Systems 246 3.3 Evolution 247 3.4 The Role of Uncertainty: Two Examples 248 3.5 Splitting 250 3.5.1 Splitting and Monotone Maps 250 3.5.2 Splitting: A Generalization 255 3.5.3 The Doeblin Minorization Theorem Once Again 260 3.6 Applications 262 3.6.1 First-Order Nonlinear Autoregressive Processes (NLAR(1)) 262
- 13. Contents vii 3.6.2 Stability of Invariant Distributions in Models of Economic Growth 263 3.6.3 Interaction of Growth and Cycles 267 3.6.4 Comparative Dynamics 273 3.7 Contractions 275 3.7.1 Iteration of Random Lipschitz Maps 275 3.7.2 A Variant Due to Dubins and Freedman 281 3.8 Complements and Details 284 3.9 Supplementary Exercises 294 4 Random Dynamical Systems: Special Structures 296 4.1 Introduction 296 4.2 Iterates of Real-Valued Affine Maps (AR(1) Models) 297 4.3 Linear Autoregressive (LAR(k)) and Other Linear Time Series Models 304 4.4 Iterates of Quadratic Maps 310 4.5 NLAR (k) and NLARCH (k) Models 317 4.6 Random Continued Fractions 323 4.6.1 Continued Fractions: Euclid’s Algorithm and the Dynamical System of Gauss 324 4.6.2 General Continued Fractions and Random Continued Fractions 325 4.6.3 Bernoulli Innovation 330 4.7 Nonnegativity Constraints 336 4.8 A Model with Multiplicative Shocks, and the Survival Probability of an Economic Agent 338 4.9 Complements and Details 342 5 Invariant Distributions: Estimation and Computation 349 5.1 Introduction 349 5.2 Estimating the Invariant Distribution 350 5.3 A Sufficient Condition for √ n-Consistency 351 5.3.1 √ n-Consistency 352 5.4 Central Limit Theorems 360 5.5 The Nature of the Invariant Distribution 365 5.5.1 Random Iterations of Two Quadratic Maps 367 5.6 Complements and Details 369 5.7 Supplementary Exercises 375 6 Discounted Dynamic Programming Under Uncertainty 379 6.1 Introduction 379 6.2 The Model 380 6.2.1 Optimality and the Functional Equation of Dynamic Programming 381
- 14. viii Contents 6.3 The Maximum Theorem: A Digression 385 6.3.1 Continuous Correspondences 385 6.3.2 The Maximum Theorem and the Existence of a Measurable Selection 386 6.4 Dynamic Programming with a Compact Action Space 388 6.5 Applications 390 6.5.1 The Aggregative Model of Optimal Growth Under Uncertainty: The Discounted Case 390 6.5.2 Interior Optimal Processes 397 6.5.3 The Random Dynamical System of Optimal Inputs 402 6.5.4 Accumulation of Risky Capital 407 6.6 Complements and Details 409 6.6.1 Upper Semicontinuous Model 409 6.6.2 The Controlled Semi-Markov Model 410 6.6.3 State-Dependent Actions 415 A Appendix 419 A1. Metric Spaces: Separability, Completeness, and Compactness 419 A1.1. Separability 420 A1.2. Completeness 420 A1.3. Compactness 422 A2. Infinite Products of Metric Spaces and the Diagonalization Argument 423 A3. Measurability 425 A3.1. Subspaces 426 A3.2. Product Spaces: Separability Once Again 426 A3.3. The Support of a Measure 428 A3.4. Change of Variable 428 A4. Borel-Cantelli Lemma 430 A5. Convergence 431 Bibliography 435 Author Index 453 Subject Index 457
- 15. Preface The scope of this book is limited to the study of discrete time dynamic processes evolving over an infinite horizon. Its primary focus is on mod- els with a one-period lag: “tomorrow” is determined by “today” through an exogenously given rule that is itself stationary or time-independent. A finite lag of arbitrary length may sometimes be incorporated in this scheme. In the deterministic case, the models belong to the broad math- ematical class, known as dynamical systems, discussed in Chapter 1, with particular emphasis on those arising in economics. In the presence of random perturbations, the processes are random dynamical systems whose long-term stability is our main quest. These occupy a central place in the theory of discrete time stochastic processes. Aside from the appearance of many examples from economics, there is a significant distinction between the presentation in this book and that found in standard texts on Markov processes. Following the exposition in Chapter 2 of the basic theory of irreducible processes, especially Markov chains, much of Chapters 3–5 deals with the problem of stability of random dynamical systems which may not, in general, be irreducible. The latter models arise, for example, if the random perturbation is limited to a finite or countable number of choices. Quite a bit of this theory is of relatively recent origin and appears especially relevant to economics because of underlying structures of monotonicity or contraction. But it is useful in other contexts as well. In view of our restriction to discrete time frameworks, we have not touched upon powerful techniques involving deterministic and stochastic differential equations or calculus of variations that have led to significant advances in many disciplines, including economics and finance. It is not possible to rely on the economic data to sift through vari- ous possibilities and to compute estimates with the degrees of precision ix
- 16. x Preface that natural or biological scientists can often achieve through controlled experiments. We duly recognize that there are obvious limits to the lessons that formal models with exogenously specified laws of motion can offer. The first chapter of the book presents a treatment of deterministic dynamical systems. It has been used in a course on dynamic models in economics, addressed to advanced undergraduate students at Cornell. Supplemented by appropriate references, it can also be part of a graduate course on dynamic economics. It requires a good background in calculus and real analysis. Chapters 2–6 have been used as the core material in a graduate course at Cornell on Markov processes and their applications to economics. An alternative is to use Chapters 1–3 and 5 to introduce models of intertem- poral optimization/equilibrium and the role of uncertainty. Complements and Details make it easier for the researchers to follow up on some of the themes in the text. In addition to numerous examples illustrating the theory, many ex- ercises are included for pedagogic purposes. Some of the exercises are numbered and set aside in paragraphs, and a few appear at the end of some chapters. But quite a few exercises are simply marked as (Exercise), in the body of a proof or an argument, indicating that a relatively minor step in reasoning needs to be formally completed. Given the extensive use of the techniques that we review, we are unable to provide a bibliography that can do justice to researchers in many disciplines. We have cited several well-known monographs, texts, and reviewarticleswhich,inturn,haveextendedlistsofreferencesforcurious readers. The quote attributed to Toni Morrison in Chapter 1 is available on the Internet from Simpson’s Contemporary Quotations, compiled by J. B. Simpson. The quote from Shizuo Kakutani in Chapter 2 is available on the In- ternet at www.uml.edu/Dept/Math/alumni/tangents/tangents Fall2004/ MathInTheNews.htm. Endnote 1 of the document describes it as “a joke by Shizuo Kakutani at a UCLA colloquium talk as attributed in Rick Dur- rett’s book Probability: Theory and Examples.” The other quote in this chapter is adapted from Bibhuti Bandyopadhyay’s original masterpiece in Bengali. The quote from Gerard Debreu in Chapter 4 appeared in his article in American Economic Review (Vol. 81, 1991, pp. 1–7).
- 17. Preface xi The quote from Patrick Henry in Chapter 5 is from Bartlett’s Quota- tions (no. 4598), available on the Internet. The quote attributed to Freeman J. Dyson in the same chapter appeared in the circulated abstract of his Nordlander Lecture (“The Predictable and the Unpredictable: How to Tell the Difference”) at Cornell University on October 21, 2004. The quote from Kenneth Arrow at the beginning of Chapter 6 appears in Chapter 2 of his classic Essays in the Theory of Risk-Bearing. Other quotes are from sources cited in the text.
- 19. Acknowledgment We would like to thank Vidya Atal, Kuntal Banerjee, Seung Han Yoo, Benjarong Suwankiri, Jayant Ganguli, Souvik Ghosh, Chao Gu, and Wee Lee Loh for research assistance. In addition, for help in locating ref- erences, we would like to thank Vidhi Chhaochharia and Kameshwari Shankar. For direct and indirect contributions we are thankful to many col- leagues: Professors Krishna Athreya, Robert Becker, Venkatesh Bala, Jess Benhabib, William Brock, Partha Dasgupta, Richard Day, Prajit Dutta, David Easley, Ani Guerdjikova, Nigar Hashimzade, Ali Khan, Nicholas Kiefer, Kaushik Mitra, Tapan Mitra, Kazuo Nishimura, Man- fred Nermuth, Yaw Nyarko, Bezalel Peleg, Uri Possen, Debraj Ray, Roy Radner, Rangarajan Sundaram, Edward Waymire, Makoto Yano, and Ithzak Zilcha. Professor Santanu Roy was always willing to help out, with comments on stylistic and substantive matters. We are most appreciative of the efforts of Ms. Amy Moesch: her pa- tience and skills transformed our poorly scribbled notes into a presentable manuscript. Mukul Majumdar is grateful for the support from the Warshow endow- ment and the Department of Economics at Cornell, as well as from the Institute of Economic Research at Kyoto University. Rabi Bhattacharya gratefully acknowledges support from the National Science Foundation with grants DMS CO-73865, 04-06143. Two collections of published articles have played an important role in our exposition: a symposium on Chaotic Dynamical Systems (edited by Mukul Majumdar) and a symposium on Dynamical Systems Subject to Random Shocks (edited by Rabi Bhattacharya and Mukul Majumdar) that xiii
- 20. xiv Acknowledgment appeared in Economic Theory (the first in Vol. 4, 1995, and the second in Vol. 23, 2004). We acknowledge the enthusiastic support of Professor C. D. Aliprantis in this context. Finally, thanks are due to Scott Parris, who initiated the project.
- 21. Notation Z set of all integers. Z+(Z++) set of all nonnegative (positive) integers. R set of all real numbers. R+(R++) set of all nonnegative (positive) real numbers. R set of all -vectors. x = (xi ) = (x1, . . . , x) an element of R . x ≥ 0 xi ≥ 0 for i = 1, 2, . . . , ; [x is nonnegative]. x 0 xi ≥ 0 for all i; xi 0 for some i; [x is positive]. x 0 xi 0 for all i; [x is strictly positive]. (S, S) a measurable space [when S is a metric space, S = B(S) is the Borel sigmafield unless otherwise specified]. xv
- 23. 1 Dynamical Systems Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties. Robert M. May There is nothing more to say – except why. But since why is difficult to handle, one must take refuge in how. Toni Morrison 1.1 Introduction There is a rich literature on discrete time models in many disciplines – including economics – in which dynamic processes are described for- mally by first-order difference equations (see (2.1)). Studies of dynamic properties of such equations usually involve an appropriate definition of a steady state (viewed as a dynamic equilibrium) and conditions that guarantee its existence and local or global stability. Also of importance, particularly in economics following the lead of Samuelson (1947), have been the problems of comparative statics and dynamics: a systematic analysis of how the steady states or trajectories respond to changes in some parameter that affects the law of motion. While the dynamic prop- erties of linear systems (see (4.1)) have long been well understood, rela- tively recent studies have emphasized that “the very simplest” nonlinear difference equations can exhibit “a wide spectrum of qualitative behav- ior,” from stable steady states, “through cascades of stable cycles, to a regime in which the behavior (although fully deterministic) is in many respects chaotic or indistinguishable from the sample functions of a ran- dom process” (May 1976, p. 459). This chapter is not intended to be a 1
- 24. 2 Dynamical Systems comprehensive review of the properties of complex dynamical systems, the study of which has benefited from a collaboration between the more “abstract” qualitative analysis of difference and differential equations, and a careful exploration of “concrete” examples through increasingly sophisticated computer experiments. It does recall some of the basic re- sults on dynamical systems, and draws upon a variety of examples from economics (see Complements and Details). There is by now a plethora of definitions of “chaotic” or “complex” behavior, and we touch upon a few properties of chaotic systems in Sections 1.2 and 1.3. However, the map (2.3) and, more generally, the quadratic family discussed in Section 1.7 provide a convenient frame- work for understanding many of the definitions, developing intuition and achieving generalizations (see Complements and Details). It has been stressed that the qualitative behavior of the solution to Equation (2.5) depends crucially on the initial condition. Trajectories emanating from initial points that are very close may display radically different proper- ties. This may mean that small changes in the initial condition “lead to predictions so different, after a while, that prediction becomes in effect useless” (Ruelle 1991, p. 47). Even within the quadratic family, com- plexities are not “knife-edge,” “abnormal,” or “rare” possibilities. These observations are particularly relevant for models in social sciences, in which there are obvious limits to gathering data to identify the initial condition, and avoiding computational errors at various stages. In Section 1.2 we collect some basic results on the existence of fixed points and their stability properties. Of fundamental importance is the contraction mapping theorem (Theorem 2.1) used repeatedly in subse- quent chapters. Section 1.3 introduces complex dynamical systems, and the central result is the Li–Yorke theorem (Theorem 3.1). In Section 1.4 we briefly touch upon linear difference equations. In Section 1.5 we ex- plore in detail dynamical systems in which the state space is R+, the set of nonnegative reals, and the law of motion α is an increasing function. Proposition 5.1 is widely used in economics and biology: it identifies a class of dynamical systems in which all trajectories (emanating from initial x in R++) converge to a unique fixed point. In contrast, Sec- tion 1.6 provides examples in which the long-run behavior depends on initial conditions. In the development of complex dynamical systems, the “quadratic family” of laws of motion (see (7.11)) has played a distin- guished role. After a review of some results on this family in Section 1.7, we turn to examples of dynamical systems from economics and biology.
- 25. 1.2 Basic Definitions: Fixed and Periodic Points 3 We have selected some descriptive models, some models of optimization with a single decision maker, a dynamic game theoretic model, and an example of intertemporal equilibrium with overlapping generations. An interesting lesson that emerges is that variations of some well-known models that generate monotone behavior lead to dynamical systems ex- hibiting Li–Yorke chaos, or even to systems with the quadratic family as possible laws of motion. 1.2 Basic Definitions: Fixed and Periodic Points We begin with some formal definitions. A dynamical system is described by a pair (S, α) where S is a nonempty set (called the state space) and α is a function (called the law of motion) from S into S. Thus, if xt is the state of the system in period t, then xt+1 = α(xt ) (2.1) is the state of the system in period t + 1. In this chapter we always assume that the state space S is a (nonempty) metric space (the metric is denoted by d). As examples of (2.1), take S to be the set R of real numbers, and define α(x) = ax + b, (2.2) where a and b are real numbers. Another example is provided by S = [0, 1] and α(x) = 4x(1 − x). (2.3) Here in (2.3), d(x, y) ≡ |x − y|. The evolution of the dynamical system (R, α) where α is defined by (2.2) is described by the difference equation xt+1 = axt + b. (2.4) Similarly, the dynamical system ([0, 1], α) where α is defined by (2.3) is described by the difference equation xt+1 = 4xt (1 − xt ). (2.5) Once the initial state x (i.e., the state in period 0) is specified, we write α0 (x) ≡ x, α1 (x) = α(x), and for every positive integer j ≥ 1, α j+1 (x) = α(α j (x)). (2.6)
- 26. 4 Dynamical Systems We refer to α j as the jth iterate of α. For any initial x, the trajectory from x is the sequence τ(x) = {α j (x)∞ j=0}. The orbit from x is the set γ (x) = {y: y = α j (x) for some j ≥ 0}. The limit set w(x) of a trajectory τ(x) is defined as w(x) = ∞ j=1 [τ(α j (x)], (2.7) where Ā is the closure of A. Fixed and periodic points formally capture the intuitive idea of a sta- tionary state or an equilibrium of a dynamical system. In his Foundations, Samuelson (1947, p. 313) noted that “Stationary is a descriptive term characterizing the behavior of an economic variable over time; it usually implies constancy, but is occasionally generalized to include behavior periodically repetitive over time.” A point x ∈ S is a fixed point if x = α(x). A point x ∈ S is a periodic point of period k ≥ 2 if αk (x) = x and α j (x) = x for 1 ≤ j k. Thus, to prove that x is a periodic point of period, say, 3, one must prove that x is a fixed point of α3 and that it is not a fixed point of α and α2 . Some writers consider a fixed point as a periodic point of period 1. Denote the set of all periodic points of S by ℘(S). We write ℵ(S) to denote the set of nonperiodic points. We now note some useful results on the existence of fixed points of α. Proposition 2.1 Let S = R and α be continuous. If there is a (nondegen- erate) closed interval I = [a, b] such that (i) α(I) ⊂ I or (ii) α(I) ⊃ I, then there is a fixed point of α in I. Proof. (i) If α(I) ⊂ I, then α(a) ≥ a and α(b) ≤ b. If α(a) = a or α(b) = b, the conclusion is immediate. Otherwise, α(a) a and α(b) b. This means that the function β(x) = α(x) − x is positive at a and negative at b. Using the intermediate value theorem, β(x∗ ) = 0 for some x∗ in (a, b). Then α(x∗ ) = x∗ . (ii) By the Weierstrass theorem, there are points xm and xM in I such that α(xm) ≤ α(x) ≤ α(xM ) for all x in I. Write α(xm) = m and α(xM ) = M. Then, by the intermediate value theorem, α(I) = [m, M].
- 27. 1.2 Basic Definitions: Fixed and Periodic Points 5 Since α(I) ⊃ I, m ≤ a ≤ b ≤ M. In other words, α(xm) = m ≤ a ≤ xm, and α(xM ) = M ≥ b ≥ xM . The proof can now be completed by an argument similar to that in case (i). Remark 2.1 Let S = [a, b] and α be a continuous function from S into S. Suppose that for all x in (a, b) the derivative α (x) exists and |α (x)| 1. Then α has a unique fixed point in S. Proposition 2.2 Let S be a nonempty compact convex subset of R , and α be continuous. Then there is a fixed point of α. A function α : S → S is a uniformly strict contraction if there is some C, 0 C 1, such that for all x, y ∈ X, x = y, one has d(α(x), α(y)) Cd(x, y). (2.8) If d(α(x), α(y)) d(x, y) for x = y, we say that α is a strict contrac- tion. If only d(α(x), α(y)) d(x, y), we say that α is a contraction. If α is a contraction, α is continuous on S. In this book, the following fundamental theorem is used many times: Theorem 2.1 Let (S, d) be a nonempty complete metric space and α : S → S be a uniformly strict contraction. Then α has a unique fixed point x∗ ∈ S. Moreover, for any x in S, the trajectory τ(x) = {α j (x)∞ j=0} converges to x∗ . Proof. Choose an arbitrary x ∈ S. Consider the trajectory τ(x) = (xt ) from x, where xt+1 = α(xt ). (2.9)
- 28. 6 Dynamical Systems Note that d(x2, x1) = d(α(x1), α(x)) Cd(x1, x) for some C ∈ (0, 1); hence, for any t ≥ 1, d(xt+1, xt ) Ct d(x1, x). (2.10) We note that d(xt+2, xt ) ≤ d(xt+2, xt+1) + d(xt+1, xt ) Ct+1 d(x1, x) + Ct d(x1, x) = Ct (1 + C)d(x1, x). It follows that for any integer k ≥ 1, d(xt+k, xt ) [Ct /(1 − C)]d(x1, x), and this implies that (xt ) is a Cauchy sequence. Since S is assumed to be complete, limitt→∞ xt = x∗ exists. By continuity of α, and (2.9), α(x∗ ) = x∗ . If there are two distinct fixed points x∗ and x∗∗ of α, we see that there is a contradiction: 0 d(x∗ , x∗∗ ) = d(α(x∗ ), α(x∗∗ )) Cd(x∗ , x∗∗ ), (2.11) where 0 C 1. Remark 2.2 For applications of this fundamental result, it is important to reflect upon the following: (i) for any x ∈ S, d(αn (x), x∗ ) ≤ Cn (1 − C)−1 d(α(x), x)), (ii) for any x ∈ S, d(x, x∗ ) ≤ (1 − C)−1 d(α(x), x). Theorem 2.2 Let S beanonemptycompletemetricspaceandα : S → S be such that αk is a uniformly strict contraction for some integer k 1. Then α has a unique fixed point x∗ ∈ S. Proof. Let x∗ be the unique fixed point of αk . Then αk (α(x∗ )) = α(αk (x∗ )) = α(x∗ ) Hence α(x∗ ) is also a fixed point of αk . By uniqueness, α(x∗ ) = x∗ . This means that x∗ is a fixed point of α. But any fixed point of α is a fixed point of αk . Hence x∗ is the unique fixed point of α.
- 29. 1.2 Basic Definitions: Fixed and Periodic Points 7 Theorem 2.3 Let S be a nonempty compact metric space and α : S → S be a strict contraction. Then α has a unique fixed point. Proof. Since d(α(x), x) is continuous and S is compact, there is an x∗ ∈ S such that d(α(x∗ ), x∗ ) = inf x∈S d(α(x), x). (2.12) Then α(x∗ ) = x∗ , otherwise d(α2 (x∗ ), α(x∗ )) d(α(x∗ ), x∗ ), contradicting (2.12). Exercise 2.1 (a) Let S = [0, 1], and consider the map α : S → S defined by α(x) = x − x2 2 . Show that α is a strict contraction, but not a uniformly strict contraction. Analyze the behavior of trajectories τ(x) from x ∈ S. (b) Let S = R, and consider the map α : S → S defined by α(x) = [x + (x2 + 1)1/2 ]/2. Show that α(x) is a strict contraction, but does not have a fixed point. A fixed point x∗ of α is (locally) attracting or (locally) stable if there is an open set U containing x∗ such that for all x ∈ U, the trajectory τ(x) from x converges to x∗ . Weshalloftendropthecaveat“local”:notethatlocalattractionorlocal stability is to be distinguished from the property of global stability of a dynamical system: (S, α) is globally stable if for all x ∈ S, the trajectory τ(x) converges to the unique fixed point x∗ . Theorem 2.1 deals with global stability. A fixed point x∗ of α is repelling if there is an open set U containing x∗ such that for any x ∈ U, x = x∗ , there is some k ≥ 1, αk (x) / ∈ U. Consider a dynamical system (S, α) where S is a (nondegenerate) closed interval [a, b] and α is continuous on [a, b]. Suppose that α is
- 30. 8 Dynamical Systems also continuously differentiable on (a, b). A fixed point x∗ ∈ (a, b) is hyperbolic if |α (x∗ )| = 1. Proposition 2.3 Let S = [a, b] and α be continuous on [a, b] and con- tinuously differentiable on (a, b). Let x∗ ∈ (a, b) be a hyperbolic fixed point of α. (a) If |α (x∗ )| 1, then x∗ is locally stable. (b) If |α (x∗ )| 1, then x∗ is repelling. Proof. (a) There is some u 0 such that |α (x)| m 1 for all x in I = [x∗ − u, x∗ + u]. By the mean value theorem, if x ∈ I, |α(x) − x∗ | = |α(x) − α(x∗ )| ≤ m|x − x∗ | mu u. Hence, α maps I into I and, again, by the mean value theorem, is a uniformly strict contraction on I. The result follows from Theorem 2.1. (b) this is left as an exercise. We can define “a hyperbolic periodic point of period k” and define (locally) attracting and repelling periodic points accordingly. Let x0 be a periodic point of period 2 and x1 = α(x0). By defini- tion x0 = α(x1) = α2 (x0) and x1 = α(x0) = α2 (x1). Now if α is differ- entiable, by the chain rule, [α2 (x0)] = α (x1)α (x0). More generally, suppose that x0 is a periodic point of period k and its orbit is denoted by {x0, x1, . . . , xk−1}. Then, [αk (x0)] = α (xk−1) · · · α (x0). It follows that [αk (x0)] = [αk (x1)] · · · [αk (xk−1)] . We can now extend Proposition 2.3 appropriately. While the contraction property of α ensures that, independent of the initial condition, the trajectories enter any neighborhood of the fixed point, there are examples of simple nonlinear dynamical systems in which trajectories “wander around” the state space. We shall examine this feature more formally in Section 1.3.
- 31. 1.2 Basic Definitions: Fixed and Periodic Points 9 Example 2.1 Let S = R, α(x) = x2 . Clearly, the only fixed points of α are 0, 1. More generally, keeping S = R, consider the family of dynam- ical systems αθ (x) = x2 + θ, where θ is a real number. For θ 1/4, αθ does not have any fixed point; for θ = 1/4, αθ has a unique fixed point x = 1/2; for θ 1/4, αθ has a pair of fixed points. When θ = −1, the fixed points of the map α(−1)(x) = x2 − 1 are [1 + √ 5]/2 and [1 − √ 5]/2. Now α(−1)(0) = −1; α(−1)(−1) = 0. Hence, both 0 and −1 are periodic points of period 2 of α(−1). It fol- lows that: τ(0) = (0, −1, 0, −1, . . .), τ(−1) = (−1, 0, −1, 0, . . .), γ (−1) = {−1, 0}, γ (0) = {0, −1}. Since α2 (−1)(x) = x4 − 2x2 , we see that (i) α2 (−1) has four fixed points: the fixed points of α(−1), and 0, −1; (ii) the derivative of α2 (−1) with respect to x, denoted by [α2 (−1)(x)] , is given by [α2 (−1)(x)] = 4x3 − 4x. Now, [α2 (−1)(x)] x=0 = [α2 (−1)(x)] x=−1 = 0. Hence, both 0 and −1 are attracting fixed points of α2 . Example 2.2 Let S = [0, 1]. Consider the “tent map” defined by α(x) = 2x for x ∈ [0, 1/2] 2(1 − x) for x ∈ [1/2, 1]. Note that α has two fixed points “0” and “2/3.” It is tedious to write out the functional form of α2 : α2 (x) = 4x for x ∈ [0, 1/4] 2(1 − 2x) for x ∈ [1/4, 1/2] 2(2x − 1) for x ∈ [1/2, 3/4] 4(1 − x) for x ∈ [3/4, 1]. Verify the following: (i) “2/5” and “4/5” are periodic points of period 2.
- 32. 10 Dynamical Systems (ii) “2/9,” “4/9,” “8/9” are periodic points of period 3. It follows from a well-known result (see Theorem 3.1) that there are periodic points of all periods. By using the graphs, if necessary, verify that the fixed and periodic points of the tent map are repelling. Example 2.3 In many applications to economics and biology, the state space S is the set of all nonnegative reals, S = R+. The law of motion α : S → S has the special form α(x) = xβ(x), (2.11 ) where β(0) ≥ 0, β : R+ → R+ is continuous (and often has additional properties). Now, the fixed points x̂ of α must satisfy α(x̂) = x̂ or x̂[1 − β(x̂)] = 0. The fixed point x̂ = 0 may have a special significance in a particular context (e.g., extinction of a natural resource). Some examples of α satisfying (2.11 ) are (Verhulst 1845) α(x) = θ1x x + θ2 , θ1 0, θ2 0. (Hassell 1975) α(x) = θ1x(1 + x)−θ2 , θ1 0, θ2 0. (Ricker 1954) α(x) = θ1xe−θ2x , θ1 0, θ2 0. Here θ1, θ2 are interpreted as exogenous parameters that influence the law of motion α. Assume that β(x) is differentiable at x ≥ 0. Then, α (x) = β(x) + xβ (x). Hence, α (0) = β(0). For each of the special maps, the existence of a fixed point x̂ = 0 and the local stability properties depend on the values of the parameters θ1, θ2. We shall now elaborate on this point.
- 33. 1.3 Complexity 11 For the Verhulst map α(x) = θ1x/(x + θ2), where x ≥ 0, θ1 0, and θ2 0, there are two cases: Case I: θ1 ≤ θ2. Here x∗ = 0 is the unique fixed point; Case II: θ1 θ2. Here there are two fixed points x∗ (1) = 0 and x∗ (2) = θ1 − θ2. Verify that α (0) = (θ1/θ2). Hence, in Case I, x∗ = 0 is locally attract- ing if (θ1/θ2) 1. In Case II, however, α (x∗ 1 ) = α (0) 1, so x∗ 1 = 0 is repelling, whereas x∗ 2 is locally attracting, since α (x∗ 2 ) ≡ α (θ1 − θ2) = (θ2/θ1) 1. For the Hassell map, there are two cases: Case I: θ1 ≤ 1. Here x∗ = 0 is the unique fixed point. Case II: θ1 1. Here there are two fixed points x∗ (1) = 0, x∗ (2) = (θ1)1/θ2 − 1. In Case I, if θ1 1, x∗ = 0 is locally attracting. In Case II, x∗ (1) = 0 is repelling. Some calculations are needed to show that the fixed point x∗ (2) = (θ1)1/θ2 − 1 is locally stable if θ1 θ2 θ2 − 2 θ2 and θ2 2. For the Ricker map, there are two cases. Case I: θ1 ≤ 1. Here x∗ = 0 is the unique fixed point. Case II: θ1 1. Here x∗ (1) = 0 and x∗ (2) = (log θ1)/θ2 both are fixed points. Note that for all 0 θ1 1, x∗ = 0 is locally attracting. For θ1 1, x∗ (1) = 0 is repelling. The fixed point x∗ (2) = (log θ1)/θ2 is locally attracting if |1 − log θ1| 1 (which holds when 1 θ1 e2 ). 1.3 Complexity 1.3.1 Li–Yorke Chaos and Sarkovskii Theorem In this section we take the state space S to be a (nondegenerate) interval I in the real line, and α a continuous function from I into I.
- 34. 12 Dynamical Systems A subinterval of an interval I is an interval contained in I. Since α is continuous, α(I) is an interval. If I is a compact interval, so is α(I). Suppose that a dynamical system (S, α) has a periodic point of pe- riod k. Can we conclude that it also has a periodic point of some other period k = k? It is useful to look at a simple example first. Example 3.1 Suppose that (S, α) has a periodic point of period k(≥2). Then it has a fixed point (i.e., a periodic point of period one). To see this, consider the orbit γ of the periodic point of period k, and let us write γ = {x(1) , . . . , x(k) }, where x(1) x(2) · · · x(k) . Both α(x(1) ) and α(x(k) ) must be in γ . This means that α(x(1) ) = x(i) for some i 1 and α(x(k) ) = x( j) for some j k. Hence, α(x(1) ) − x(1) 0 and α(x(k) ) − x(k) 0. By the intermediate value theorem, there is some x in S such that α(x) = x. We shall now state the Li–Yorke theorem (Li and Yorke 1975) and provide a brief sketch of the proof of one of the conclusions. Theorem 3.1 Let I be an interval and α : I → I be continuous. Assume that there is some point a in I for which there are points b = α(a), c = α(b), and d = α(c) satisfying d ≤ a b c (or d ≥ a b c). (3.1) Then [1] for every positive integer k = 1, 2, . . . there is a periodic point of period k, x(k) , in I, [2] there is an uncountable set ℵ ⊂ ℵ(I) such that (i) for all x, y in ℵ , x = y, lim sup n→∞ |αn (x) − αn (y)| 0; (3.2) lim inf n→∞ |αn (x) − αn (y)| = 0. (3.3)
- 35. 1.3 Complexity 13 (ii) If x ∈ ℵ and y ∈ ℘(I) lim sup n→∞ |αn (x) − αn (y)| 0. Proof of [1]. Step1. Let G beareal-valuedcontinuousfunctiononaninterval I.For any compact subinterval I1 of G(I) there is a compact subinterval Q of I such that G(Q) = I1. Proof of Step 1. One can figure out the subinterval Q directly as fol- lows. Let I1 = [G(x), G(y)] where x, y are in I. Assume that x y. Let r be the last point of [x, y] such that G(r) = G(x); let s be the first point after r such that G(s) = G(y). Then Q = [r, s] is mapped onto I1 under G. The case x y is similar. Step 2. Let I be an interval and α : I → I be continuous. Suppose that (In)∞ n=0 is a sequence of compact subintervals of I, and for all n, In+1 ⊂ α(In). (3.4) Then there is a sequence of compact subintervals (Qn) of I such that for all n, Qn+1 ⊂ Qn ⊂ Q0 = I0 (3.5) and αn (Qn) = In. (3.6) Hence, there is x ∈ n Qn such that αn (x) ∈ In for all n. (3.7) Proof of Step 2. The construction of the sequence Qn proceeds “in- ductively” as follows: Define Q0 = I0. Recall that α0 is defined as the identity mapping, so α0 (Q0) = I0 and I1 ⊂ α(I0). If Qn−1 is de- fined as a compact subinterval such that αn−1 (Qn−1) = In−1, then In ⊂ α(In−1) = αn (Qn−1). Use Step 1, with G = αn on Qn−1, in or- der to get a compact subinterval Qn of Qn−1 such that αn (Qn) = In. This completes the induction argument (establishing (3.5) and (3.6)). Compactness of Qn leads to (3.7).
- 36. 14 Dynamical Systems Now we prove [1]. Assume that d ≤ a b c (the other case d ≥ a b c is treated similarly). Write K = [a, b] and L = [b, c]. Let k be any positive integer. For k 1, define a sequence of intervals (In) as follows: In = L for n = 0, 1, 2, . . . , k − 2; Ik−1 = K; and In+k = In for n = 0, 1, 2, . . . . For k = 1, let In = L for all n. Let Qn be the intervals in Step 2. Note that Qk ⊂ Q0 = I0 and αk (Qk) = Ik = I0. Hence, Proposition 2.1 applied to αk gives us a fixed point xk of αk in Qk. Now, xk cannot have a period less than k; otherwise, we need to have αk−1 (xk ) = b, contrary to αk+1 (xk ) ∈ L. Proof of [2]. See Complements and Details. We shall now state Sarkovskii’s theorem on periodic points. Consider the following Sarkovskii ordering of the positive integers: 3 5 7 · · · 2.3 2.5 · · · 22 3 22 5 · · · (SO) 23 .3 23 .5 · · · 23 22 2 1 In other words, first list all the odd integers beginning with 3; next list 2 times the odds, 22 times the odds, etc. Finally, list all the powers of 2 in decreasing order. Theorem 3.2 Let S = R and α be a continuous function from S into S. Suppose that α has a periodic point of period k. If k k in the Sarkovskii ordering (SO), then α has a periodic point of period k . Proof. See Devaney (1986). It follows that if α has only finitely many periodic points, then they all necessarily have periods that are powers of two. 1.3.2 A Remark on Robustness of Li–Yorke Complexity Let S = [J, K], and suppose that a continuous function α satisfies the Li–Yorke condition (3.1) with strict inequality throughout; i.e., suppose that there are points a, b, c, d such that d = α(c) a b = α(a) c = α(b) (3.8)
- 37. 1.3 Complexity 15 or d = α(c) a b = α(a) c = α(b). (3.9) Consider the space C(S) of all continuous (hence, bounded) real- valued functions on S = [J, K]. Let α = maxx∈S α(x). The conclu- sions of Theorem 3.1 hold with respect to the dynamical system (S, α). But the Li–Yorke complexity is now “robust” in a precise sense. Proposition 3.1 Let S = [J, K], and let α satisfy (3.8). In addition, assume that J m(S, α) M(S, α) K, (3.10) where m(S, α) and M(S, α) are respectively the minimum and maximum of α on [J, K]. Then there is an open set N of C(S) containing α such that β ∈ N implies that [1] and [2] of Theorem 3.1 hold with β in place of α. Proof. First, we show the following: Fix x ∈ [J, K]. Given k ≥ 1, ε 0, there exists δ(k, ε) 0 such that “β − α δ(k, ε)” implies |β j (x) − α j (x)| ε for all j = 1, . . . , k. The proof is by induction on k. It is clearly true for k = 1, with δ(1, ε) ≡ ε. Assume that the claim is true for k = m, but not for k = m + 1. Then there exist some ε 0 and a sequence of functions {βn} satisfying βn − α → 0 such that |βm+1 n (x) − αm+1 (x)| ≥ ε. Let βm n (x) = yn and αm (x) = y. Then, by the induction hypothesis, yn → y. From Rudin (1976, Chapter 7) we conclude that βn(yn) → α(y), which yields a contraction. Next, choose a real number ρ satisfying 0 ρ min[1/2(a − d), 1/2(b − a), 1/2(c − b)] and a positive number r such that β − α r implies |β j (a) − α j (a)| ρ for j = 1, 2, 3, and also 0 r min{K − M(S, α), m(S, α) − J}. Define the open set N as N = {β ∈ C(S) : β − α r}. It follows that any β ∈ N maps S into S, since the maximum of β on [J, K] is less than M(S, α) + r K. Similarly, the minimum of β on [J, K] is likewise greater than J. It remains to show that the condition (3.8) also holds for any β in N. Recall that α(a) = b, α(b) = c, α(c) = d.
- 38. 16 Dynamical Systems Since |β(a) − α(a)| ≡ |β(a) − b| ρ, we have (i) β(a) b − ρ a + ρ a. Likewise, since β(a) b + ρ and |β2 (a) − c| ρ, we get (ii) β2 (a) c − ρ b + ρ β(a). Finally, since |β3 (a) − d| ρ, we get (iii) β3 (a) d + ρ a − ρ a. 1.3.3 Complexity: Alternative Approaches Attempts to capture the complexity of dynamical systems have led to alternative definitions of chaos that capture particular properties. Here, we briefly introduce two interesting properties: topological transitivity and sensitive dependence on initial condition. A dynamical system (S, α) is topologically transitive if for any pair of nonempty open sets U and V , there exists k ≥ 1 such that αk (U) ∩ V = φ. Of interest are the following two results. Proposition 3.2 If there is some x such that γ (x), the orbit from x, is dense in S, then (S, α) is topologically transitive. Proof. Left as an exercise. Proposition 3.3 Let S be a (nonempty) compact metric space. Assume that (S, α) is topologically transitive. Then there is some x ∈ S such that the orbit γ (x) from x is dense in S. Proof. Since S is compact, it has a countable base of open sets; i.e., there is a family {Vn} of open sets in S with the property that if M is any open subset of S, there is some Vn ⊂ M. Corresponding to each Vn, define the set On as follows: On = {x ∈ S : α j (x) ∈ Vn, for some j ≥ 0}. On is open, by continuity of α. By topological transitivity it is also dense in S. By the Baire category theorem (see Appendix), the intersection O = ∩n On is nonempty (in fact, dense in S). Take any x ∈ O, and con- sider the orbit γ (x) from x. Take any y in S and any open M containing y. Then M contains some Vn. Since x belongs to the corresponding On, there is some element of γ (x) in Vn. Hence, γ (x) is dense in S.
- 39. 1.4 Linear Difference Equations 17 It is important to reflect upon the behavior of a topologically transitive dynamical system and contrast it to one in which the law of motion satisfies the strict contraction property. We now turn to another concept that has profound implications for the long-run prediction of a dynamical system. A dynamical system (S, α) has sensitive dependence on initial condition if there is ∂ 0 such that for any x ∈ S and any neighborhood N of x there exist y ∈ N and an integer j ≥ 0 with the property |α j (x) − α j (y)| ∂. Devaney (1986) asserted that if a dynamical system “possesses sen- sitive dependence on initial condition, then for all practical purposes, the dynamics defy numerical computation. Small errors in computation which are introduced by round-off may become magnified upon itera- tion. The results of numerical computation of an orbit, no matter how accurate, may bear no resemblance whatsoever with the real orbit.” Example 3.2 The map α(x) = 4x(1 − x) on [0, 1] is topologically tran- sitive and has sensitive dependence on initial condition (see Devaney 1986). 1.4 Linear Difference Equations Excellent coverage of this topic is available from many sources: we pro- vide only a sketch. Consider xt+1 = axt + b. (4.1) When a = 1, xt = x0 + bt. When a = −1, xt = −x0 + b for t = 1, 3, . . . , and xt = x0 for t = 2, 4, . . . . Nowassumea = 1.Themapα(x) = ax + b whena = 1,hasaunique fixed point x∗ = b 1−a . Given any initial x0 = x, the solution to (4.1) can be verified as xt = (x − x∗ )at + x∗ . (4.2) The long-run behavior of trajectories from alternative initial x can be analyzed from (4.2). In this context, the important fact is that the sequence at converges to 0 if |a| 1 and becomes unbounded if |a| 1. Example 4.1 Consider an economy where the output yt (≥0) in any pe- riod is divided between consumption ct (≥0) and investment xt (≥0). The
- 40. 18 Dynamical Systems return function is given by yt+1 = rxt , r 1, t ≥ 0. (4.3) Given an initial stock y 0, a program x = (xt ) (from y) is a nonneg- ative sequence satisfying x0 ≤ y, xt+1 ≤ rxt (for t ≥ 1). It generates a corresponding consumption program c = (ct ) defined by c0 = y − x0; ct+1 = yt+1 − xt+1 = rxt − xt+1 for t ≥ 0. (a) Show that a consumption program c = (ct ) must satisfy ∞ t=0 ct /rt ≤ y. (b) Call a program x from y [generating c] efficient if there does not exist another program x [generating (c )] from y such that c t ≥ ct for all t ≥ 0 with strict inequality for some t. Show that a program x is efficient if and only if ∞ t=0 ct /rt = y. (c) Suppose that for the economy to survive it must consume an amount c 0 in every period. Informally, we say that a program x from y that satisfies ct ≥ c for all t ≥ 0 survives at (or above) c. Note that the law of motion of the economy that plans a consumption c ≥ 0 in every period t ≥ 1, with an investment x0 initially, can be written as xt+1 = rxt − c. (4.4) This equation has a solution xt = rt (x0 − ξ) + ξ, where ξ = c/(r − 1). (4.5) Hence, (1) xt reaches 0 in finite time if x0 ξ; (2) xt = ξ for all t if x0 = ξ; (3) xt diverges to (plus) infinity if x0 ξ. To summarize the implications for survival and sustainable develop- ment we state the following:
- 41. 1.4 Linear Difference Equations 19 Proposition 4.1 Survival at (or above) c 0 is possible if and only if the initial stock y ≥ ξ + c, or equivalently, y ≥ r r − 1 c. Of course if r ≤ 1, there is no program that can guarantee that ct ≥ c for any c 0. In this chapter, our exposition deals with processes that are generated by an autonomous or time invariant law of motion [in (2.1), the function α : S → S does not depend on time]. We shall digress briefly and study a simple example which explicitly recognizes that the law of motion may itself depend on time. Example 4.2 Let xt (t ≥ 0) be the stock of a natural resource at the beginning of period t, and assume that the evolution of xt is described by the following (nonautonomous) difference equation xt+1 = axt + bt+1, (4.6) where 0 a 1 and (bt+1)t≥0 is a sequence of nonnegative num- bers. During period t, (1 − a)xt is consumed or used up and bt+1 is the “new discovery” of the resource reported at the beginning of period t + 1. Given an initial x0 0, one can write xt = at x0 + t−1 i=0 ai bt−i . (4.7) Now, if the sequence (bt+1)t≥0 is bounded above, i.e., if there is B 0 such that 0 ≤ bt+1 ≤ B for all t ≥ 0, then the sequence t−1 i=0 ai bt−i converges, so that the sequence (xt ) converges to a finite limit as well. For other examples of such nonautonomous systems, see Azariadis (1993, Chapters 1–5). What happens if we want to introduce uncertainty in the new discovery of the resource? Perhaps the natural first step is to consider (bt+1)t≥0 as a sequence of independent, identically distributed random variables (assuming values in R+). This leads us to the process (2.1) studied in Chapter 4.
- 42. 20 Dynamical Systems 1.5 Increasing Laws of Motion Consider first the dynamical system (S, α) where S ≡ R (the set of reals) and α : S → S is continuous and nondecreasing (i.e., if x, x ∈ S and x ≥ x , then α(x) ≥ α(x )). Consider the trajectory τ(x0) from any initial x0. Since xt+1 = α(xt ) for t ≥ 0, (5.1) there are three possibilities: Case I: x1 x0. Case II: x1 = x0. Case III: x1 x0. In Case I, x2 = α(x1) ≥ α(x0) = x1. It follows that {xt } is a non- decreasing sequence. In Case II, it is clear that xt = x0 for all t ≥ 0. In Case III, x2 = α(x1) ≤ α(x0) = x1. It follows that {xt } is a non- increasing sequence. In Case I, if {xt } is bounded above, lim t→∞ xt = x∗ exists. From (5.1) by taking limits and using the continuity of α, we have x∗ = α(x∗ ). Similarly, in Case III, if {xt } is bounded below, lim t→∞ xt = x∗ exists, and, again by the continuity of α, we have, by taking limits in (5.1), x∗ = α(x∗ ). We shall now identify some well-known conditions under which the long-run behavior of all the trajectories can be precisely characterized. Example 5.1 Let S = R+ and α : S → S be a continuous, nondecreas- ing function that satisfies the following condition (PI): there is a unique x∗ 0 such that α(x) x for all 0 x x∗ , α(x) x for all x x∗ . (PI)
- 43. 1.5 Increasing Laws of Motion 21 In this case, if the initial x0 ∈ (0, x∗ ), then x1 = α(x0) x0, and we are in Case I. But note that x0 x∗ implies that x1 = α(x0) ≤ α(x∗ ) = x∗ . Repeating the argument, we get x∗ ≥ xt+1 ≥ xt ≥ x1 x0 0. (5.2) Thus, the sequence {xt } is nondecreasing, bounded above by x∗ . Hence, limt→∞xt = x̂ exists and, using the continuity of α, we conclude that x̂ = α(x̂) 0. Now, the uniqueness of x∗ implies that x̂ = x∗ . If x0 = x∗ , then xt = x∗ for all t ≥ 0 and we are in Case II. If x0 x∗ , then x1 = α(x0) x0, and we are in Case III. Now, x1 ≥ x∗ , and, repeating the argument, x∗ ≤ xt+1 ≤ xt · · · ≤ x1 ≤ x0. Thus, the sequence {xt } is nonincreasing and bounded below by x∗ . Hence, limt→∞xt = x̂ exists. Again, by continuity of α, x̂ = α(x̂), so that the uniqueness of x∗ implies that x∗ = x̂. To summarize: Proposition 5.1 Let S = R+ and α : S → S be a continuous, nonde- creasing function that satisfies (PI). Then for any x 0 the trajectory τ(x) from x converges to x∗ . If x x∗ , τ(x) is a nondecreasing sequence. If x x∗ , τ(x) is a nonincreasing sequence. Here, the long-run behavior of the dynamical system is independent of the initial condition on x 0. Remark 5.1 Suppose α is continuous and increasing, i.e., “x x ” im- plies “α(x) α(x ).” Again, there are three possibilities: Case I: x1 x0. Case II: x1 = x0. Case III: x x0.
- 44. 22 Dynamical Systems Now in Case I, x2 = α(x1) α(x0) = x1. Hence {xt } is an increasing sequence. In Case II, xt = x0 for all t ≥ 0. Finally, in Case III {xt } is a decreasing sequence. The appropriate rewording of Proposition 5.1 when α is a continuous, increasing function is left as an exercise. Proposition 5.2 (Uzawa–Inada condition) S = R+, α is continuous, nondecreasing, and satisfies the Uzawa–Inada condition (UI): A(x) ≡ [α(x)/x] is decreasing in x 0; for some x̄ 0, A(x̄) 1, and for some = x x̄ 0, A( = x) 1. (UI) Then the condition (PI) holds. Proof. Clearly, by the intermediate value theorem, there is some x∗ ∈ (x̄, = x) such that A(x∗ ) = 1, i.e., α(x∗ ) = x∗ 0. Since A(x) is decreas- ing, A(x) 1 for x x∗ and A(x) 1 for x x∗ . In other words, for all x ∈ (0, x∗ ), α(x) x and for all x x∗ , α(x) x, i.e., the property (PI) holds. In the literature on economic growth, the (UI) condition is implied by appropriate differentiability assumptions. We state a list of typical assumptions from this literature. Proposition 5.3 Let S = R+ and α : S → S be a function that is (i) continuous on S, (ii) twice continuously differentiable at x 0 satisfying: [E.1] lim x↓0 α (x) = 1 + θ1, θ1 0, [E.2] lim x↑∞ α (x) = 1 − θ2, θ2 0, [E.3] α (x) 0, α (x) 0 at x 0. Then the condition (UI) holds. Proof. Take = x x̄ ≥ 0; by the mean value theorem [α( = x) − α(x̄)] = ( = x −x̄)α (z) where x̄ z = x. Since α (z) 0, α( = x) α(x̄). Thus α is increasing. Also, α (x) 0 at x 0 means that α is strictly concave. Take = x x̄ 0. Then x̄ ≡ t = x + (1 − t)0, 0 t 1.
- 45. 1.5 Increasing Laws of Motion 23 Now α(x̄) = α(t = x + (1 − t)0) tα( = x) + (1 − t)α(0) ≥ tα( = x) or α(x̄) x̄ tα( = x) t = x = α( = x) = x . Hence α(x) x is decreasing. Write B(x) ≡ α(x) − x. By [E.1]–[E.3] there is some x̄ 0 such that α (x) 1 for all x ∈ (0, x̄]. Hence, B (x) 0 for all x ∈ (0, x̄]. By the mean value theorem α(x̄) = α(0) + x̄α (z), 0 z x̄ ≥ x̄α (z) or α(x̄) x̄ ≥ α (z) α (x̄) 1. Now, if α is bounded, i.e., if there is some N 0 such that α(x) ≤ N, then [α(x)/x] ≤ [N/x]. Hence there is = x, sufficiently large, such that α( = x)/ = x ≤ [N/ = x] 1. If α is not bounded, we can find a sequence of points (xn) such that α(xn) and xn go to infinity as n tends to infinity. Then lim n→∞ α(xn) xn = lim n→∞ α (xn) = 1 − θ2 1. Hence, we can find some point = x sufficiently large, such that α( = x)/ = x 1. Thus, the Uzawa-Inada condition (UI) is satisfied. Exercise 5.1 In his Economic Dynamics, Baumol (1970) presented a simple model that captured some of the ideas of “classical” economists
- 46. 24 Dynamical Systems formally. Let Pt , the net total product in period t, be a function of the working population Lt : Pt = F(Lt ), t ≥ 0, where F : R+ → R+ is continuous and increasing, F(0) = 0. At any time, the working population tends to grow to a size where out- put per worker is just enough to provide each worker with a “subsistence level” of minimal consumption M 0. This is formally captured by the relation Lt+1 = Pt M . Hence, Lt+1 = F(Lt ) M ≡ α(Lt ). Identify conditions on the average productivity function F(L) L (L 0) that guarantee the following: (i) There is a unique L∗ 0 such that (F(L∗ )/L∗ ) = M. (ii) For any 0 L L∗ , the trajectory τ(L) is increasing and con- verges to L∗ ; for any L L∗ , the trajectory τ(L) decreases to L∗ . Example 5.2 Consider an economy (or a fishery) which starts with an initial stock y (the metaphorical corn of one-sector growth theory or the stock of renewable resource, e.g., the stock of trouts, the population of dodos, . . . ). In each period t, the economy is required to consume or harvest a positive amount c out of the beginning of the period stock yt . The remaining stock in that period xt = yt − c is “invested” and the resulting output (principal plus return) is the beginning of the period stock yt+1. The output yt+1 is related to the input xt by a “production” function g. Assume that [A.1] g : R → R is continuous, and increasing on R+, g(x) = 0 for x ≤ 0; [A.2] there is some x∗ 0 such that g(x) x for 0 x x∗ and g(x) x for x x∗ ; [A.3] g is concave.
- 47. 1.5 Increasing Laws of Motion 25 The evolution of the system (given the initial y 0 and the planned harvesting c 0) is described by y0 = y, xt = yt − c, t ≥ 0; yt+1 = g(xt ) for t ≥ 0. Let T be the first period t, if any, such that xt 0; if there is no such t, then T = ∞. If T is finite we say that the agent (or the resource) survives up to (but not including) period T . We say that the agent survives (forever) if T = ∞ (i.e., if xt ≥ 0 for all t). Define the net return function h(x) = g(x) − x. It follows that h satisfies h(x) ≥ 0 as 0 x x∗ ; x = 0, x∗ ; x x∗ . Since g(x) 0 for all x ≤ 0, all statements about g and h will be understood to be for nonnegative arguments unless something explicit is said to the contrary. Actually, we are only interested in following the system up to the “failure” or “extinction” time T . The maximum sustainable harvest or consumption is H = max [0,x∗] h(x). (5.3) We write xt+1 = xt + h(xt ) − c. If c H, xt+1 − xt = h(xt ) − c H − c 0. Hence, xt will fall below (extinction) 0 after a finite number of periods. On the other hand, if 0 c H, there will be two roots ξ and ξ of the equation c = h(x), which have the properties 0 ξ ξ x∗ .
- 48. 26 Dynamical Systems and h(x) − c ≥ 0 as ξ x ξ ; x = ξ , ξ ; x ξ , x ξ . We can show that (a) If x0 ξ , then xt reaches or falls below 0 in finite time. (b) If x0 = ξ , then xt = ξ for all t. (c) If x0 ξ , then xt converges monotonically to ξ . Note that if c = H, there are two possibilities: either ξ = ξ (i.e., h(x) attains the maximum H at a unique period ξ ) or for all x in a nondegenerate interval [ξ , ξ ], h(x) attains its maximum. The implications of the foregoing discussion for survival and extinc- tion are summarized as follows. Proposition 5.4 Let c 0 be the planned consumption for every period and H be the maximum sustainable consumption. (1) If c H, there is no initial y from which survival (forever) is possible. (2) If 0 c H, then there is ξ , with h(ξ ) = c, ξ 0 such that survival is possible if and only if the initial stock y ≥ ξ + c (3) c = H implies h(ξ ) = H, and ξ tends to 0 as c tends to 0. 1.6 Thresholds and Critical Stocks We now consider some examples of dynamical systems that have been of particular interest in various contexts in development economics and in the literature on the management of a renewable resource. It has been emphasized that the evolution of an economy may depend crucially on the initial condition (hence, on the “history” that leads to it). It has also been noted that if the stock of a renewable resource falls below a critical level, the biological reproduction law may lead to its eventual extinction. In sharp contrast with the dynamical systems identified in Proposition 5.1, in which trajectories from positive initial stocks all converge to a positive fixed point, we sketch some examples where the long-run behavior of trajectories changes remarkably as the initial condition goes above a threshold (see Complements and Details).
- 49. 1.6 Thresholds and Critical Stocks 27 Consider the “no-harvesting” case where S = R+, and the biological reproduction law is described by a continuous function α : R+ → R+. Given an initial state x ≥ 0, the state in period t is the stock of the resources at the beginning of that period and (2.1) is assumed to hold. When xt = 0, the resource is extinct. We state a general result. Proposition 6.1 Let S = R+ and α : S → S be a continuous, increas- ing function with the following properties [P2]: [P2.1] α(0) = 0; [P2.2] there are two positive fixed points x∗ (1), x∗ (2) (0 x∗ (1) x∗ (2)) such that (i) α(x) x for x ∈ (0, x∗ (1)); (ii) α(x) x for x ∈ (x∗ (1), x∗ (2)); (iii) α(x) x for x x∗ (2). For any x ∈ (0, x∗ (1)), the trajectory τ(x) from x is decreasing and converges to 0; for any x ∈ (x∗ (1), x∗ (2)), the trajectory τ(x) from x is increasing and converges to x∗ (2); for any x x∗ (2), the trajectory τ(x) from x is decreasing and converges to x∗ (2). Proof. Left as an exercise. The striking feature of the trajectories from any x 0 is that these are all convergent, but the limits depend on the initial condition. Also, these are all monotone, but again depending on the initial condition – some are increasing, others are decreasing. We interpret x∗ (1) 0 as the critical level for survival of the resource. Exercise 6.1 Consider S = R+ and α : S → S defined by α(x) = θ1x2 x2 + θ2 where θ1, θ2 are positive parameters. When θ1 2 √ θ2, compute the fixed points of α and verify [P2]. Exercise 6.2 Consider the problem of survival with constant harvesting discussed in Example 5.2. Work out the conditions for survival with a
- 50. 28 Dynamical Systems constant harvest c 0 when the “production” function g is continuous, increasing on R+, and satisfies [P2.1] g(x) = 0 for x ≤ 0; [P2.2] there are two positive fixed points x∗ (1), x∗ (2) (0 x∗ (1) x∗ (2)) such that (i) g(x) x for x ∈ (0, x∗ (1)); (ii) g(x) x for x ∈ (x∗ (1), x∗ (2)); (iii) g(x) x for x x∗ (2). Exercise 6.3 Let S = R+ and α : S → S is defined by α(x) = xm , where m ≥ 1. Show that (i) the fixed points of α are “0” and “1”; (ii) for all x ∈ [0, 1), the trajectory τ(x) is monotonically decreasing, and con- verges to 0. For x 1, the trajectory α(x) is monotonically increasing and unbounded. Thus, x = 1 is the “threshold” above which sustainable growth is possible; if the initial x 1, the trajectory from x leads to extinction. Example 6.1 In parts of Section 1.9 we review dynamical systems that arise out of “classical” optimization models (in which “convexity” as- sumptions on the preferences and technology hold). Here we sketch an example of a “nonclassical” optimization model in which a critical level of initial stock has important policy implications. Think of a competitive fishery (see Clark 1971, 1976, Chapter 7). Let xt (≥0) be the stock or “input” of fish in period t, and f : R+ → R+ the biological reproduction relationship. The stock x in any period gives rise to output y = f (x) in the subsequent period. The following assumptions on f are introduced: [A.1] f (0) = 0; [A.2] f (x) is twice continuously differentiable for x ≥ 0; f (x) 0 for x 0. [A.3] f satisfies the following end-point conditions: f (∞) 1 f (0) ∞; f (x) 0 for x 0. [A.4]Thereisa(finite)b1 0,suchthat (i) f (b1) = 0;(ii) f (x) 0 for 0 ≤ x b1; (iii) f (x) 0 for x b1. In contrast to the present (“nonclassical”) model, the traditional (or “classical”) framework would replace [A.4] by
- 51. 1.6 Thresholds and Critical Stocks 29 [A.4 ]. f is strictly concave for x ≥ 0 ( f (x) 0 for x 0), while preserving [A.1]–[A.3]. In some versions, [A.2] and [A.3] are modified to allow f (0) = ∞. In the discussion to follow, we find it convenient to refer to a model with assumptions [A.1]–[A.3] and [A.4 ] as classical, and a model with [A.1]–[A.4] as nonclassical. We define a function h (representing the average product function) as follows: h(x) = [ f (x)/x] for x 0; h(0) = lim x→0 [ f (x)/x]. (6.1) Under [A.1]–[A.4], it is easily checked that h(0) = f (0); furthermore, there exist positive numbers k∗ , k̄, b2 satisfying (i) 0 b1 b2 k∗ k̄ ∞; (ii) f (k∗ ) = 1; (iii) f (k̄) = k̄; (iv) f (b2) = h(b2). Also, for 0 ≤ x k∗ , f (x) 1 and for x k∗ , f (x) 1; for 0 x k̄, x f (x) k̄ and for x k̄, k̄ f (x) x; and for 0 x b2, f (x) h(x) and for x b2, f (x) h(x). Also note that for 0 ≤ x b2, h(x) is increasing, and for x b2, h(x) is decreasing; for 0 ≤ x b1, f (x) is increasing, and for x b1, f (x) is decreasing. A feasible production program from x ˜ 0 is a sequence (x, y) = (xt , yt+1) satisfying x0 = x ˜ ; 0 ≤ xt ≤ yt and yt = f (xt−1) for t ≥ 1. (6.2) The sequence x = (xt )t≥0 is the input (or stock) program, while the cor- responding y = (yt+1)t≥0 satisfying (6.2) is the output program. The har- vest program c = (ct ) generated by (x, y) is defined by ct ≡ yt − xt for t ≥ 1. We will refer to (x, y, c) briefly as a program from x ˜ , it being understood that (x, y) is a feasible production program, and c is the corresponding harvest program. A slight abuse of notation: we shall often specify only the stock pro- gram x = (xt )t≥0 from x ˜ 0 to describe a program (x, y, c). It will be understood that x0 = x ˜ ; 0 ≤ xt ≤ f (xt−1) for all t ≥ 1, ct = yt − xt for t ≥ 1. Let the profit per unit of harvesting, denoted by q 0, and the rate of interest γ 0 remain constant over time. Consider a firm that has an objective of maximizing the discounted sum of profits from harvesting.
- 52. 30 Dynamical Systems A program x∗ = (x∗ t ) of stocks from x ˜ 0 is optimal if ∞ t=1 q (1 + γ )t−1 c∗ t ≥ ∞ t=1 q (1 + γ )(t−1) ct for every program x from x ˜ . Write δ = 1/(1 + γ ). Models of this type have been used to discuss the possible conflict between profit maximiza- tion and conservation of natural resources. The program (x, y, c) from x ˜ 0 defined as x0 = x ˜ , xt = 0 for t ≥ 1 is the extinction program. Here the entire output f (x ˜ ) is harvested in period 1, i.e., c1 = f (x ˜ ), ct = 0 for t ≥ 2. In the qualitative analysis of optimal programs, the roots of the equa- tion δ f (x) = 1 play an important role. This equation might not have a nonnegative real root at all; if it has a pair of unique nonnegative real roots, denote it by Z; if it has nonnegative real roots, the smaller one is denoted by z and the larger one by Z. The qualitative behavior of optimal programs depends on the value of δ = 1/(1 + γ ). Three cases need to be distinguished. The first two were analyzed and interpreted by Clark (1971). Case 1. Strong discounting: δ f (b2) ≤ 1 This is the case when δ is “sufficiently small,” i.e., 1 + γ ≥ f (x)/x for all x 0. Proposition 6.2 The extinction program is optimal from any x ˜ 0, and is the unique optimal program if δ f (b̂2) 1. Remark 6.1 First, if δ f (b2) = 1, there are many optimal programs (see Majumdar and Mitra 1983, p. 146). Second, if we consider the classical model(satisfying[A.1]–[A.3]and[A.4 ]),itisstilltruethatifδ f (0) ≤ 1, the extinction program is the unique optimal program from any x ˜ 0. Case 2. Mild discounting: δ f (0) ≥ 1 This is the case where δ is “sufficiently close to 1” (δ 1/f (0)) and Z b2 exists (if z exists, z = 0). Now, given x ˜ Z, let M be the smallest positive integer such that x1 M ≥ Z; in other words, M is the first period in which the pure
- 53. 1.6 Thresholds and Critical Stocks 31 accumulation program from x ˜ defined by x1 0 = x ˜ , x1 t+1 = f (x1 t ) for t ≥ 0 attains Z. Proposition 6.3 If x ˜ ≥ Z, then the program x∗ = (x∗ t )t≥0 from x ˜ defined by x∗ 0 = x ˜ , x∗ t = Z for t ≥ 1 is the unique optimal program from x ˜ . Proposition 6.4 If x ˜ Z, the program x∗ = (x∗ t )t≥0 defined by x∗ 0 = x ˜ , x∗ t = x1 t for t = 1, . . . , M − 1, x∗ t = Z for t ≥ M is the unique optimal program. In the corresponding classical model for δ f (0) 1, there is a unique positive K∗ δ , solving δ f (x) = 1. Propositions 6.3 and 6.4 continue to hold with Z replaced by K∗ δ (also in the definition of M). Case 3. Two turnpikes and the critical point of departure [δ f (0) 1 δ f (k2)] In case (i) the extinction program (x, y, c) generated by xt = 0 for all t ≥ 1 and in case (ii) the stationary program generated by xt = Z for all t ≥ 0 (which is also the optimal program from Z) serve as the “turnpikes” approached by the optimal programs. Both the classical and nonclassical models share the feature that the long-run behavior of optimal programs is independent of the positive initial stock. The “intermediate” case of discounting, namely when 1/f (b2) δ 1/f (0), turned out to be difficult and to offer a sharp contrast between the classical and nonclassical models. In this case 0 z b1 b2 Z k∗ . The qualitative properties of optimal programs are summarized in two steps. Proposition 6.5 x ˜ ≥ Z, the program x∗ = (x∗ t )t≥0 defined by x∗ 0 = x ˜ , x∗ t = Z for t ≥ 1 is optimal. A program x = (xt )t≥0 from x ˜ Z is a regeneration program if there is some positive integer N ≥ 1 such that xt xt−1 for 1 ≤ t ≤ N, and xt = Z for t ≥ N. It should be stressed that a regeneration program
- 54. 32 Dynamical Systems may allow for positive consumption in all periods, and need not specify “pure accumulation” in the initial periods. For an interesting example of a regeneration program that allows for positive consumption and is optimal, the reader is referred to Clark (1971, p. 259). Proposition 6.6 Let x ˜ Z. There is a critical stock Kc 0 such that if 0 x ˜ Kc, the extinction program from x ˜ is an optimal program. If Kc x ˜ Z, then any optimal program is a regeneration program. In the literature on renewable resources, Kc is naturally called the “minimum safe standard of conservation.” It has been argued that a policy that prohibits harvesting of a fishery till the stock exceeds Kc will ensure that the fishery will not become extinct, even under pure “economic exploitation.” Some conditions on x ˜ can be identified under which there is a unique optimal program. But if x ˜ = Kc, then both the extinction program and a regeneration program are optimal. For further details and proofs, see Majumdar and Mitra (1982, 1983). 1.7 The Quadratic Family Let S = [0, 1] and A = [0, 4]. The quadratic family of maps is then defined by αθ (x) = θx(1 − x) for (x, θ) ∈ S × A. (7.1) We interpret x as the variable and θ as the parameter generating the family. We first describe some basic properties of this family of maps. First, note that Fθ (0) = 0(= Fθ (1)) ∀θ ∈ [0, 4], so that 0 is a fixed point of Fθ . By solving the quadratic equation Fθ (x) = x, one notes that pθ = 1 − 1/θ is the only other fixed point that occurs if θ 1. For θ 3, one can show (e.g., by numerical calculations) that the fourth-degree polynomial equation F2 θ (x): = Fθ ◦ Fθ (x) has two other solutions (in ad- dition to 0 and pθ ). This means that for 3 θ ≤ 4, Fθ has a period-two orbit. For θ 1 + √ 6, a new period-four orbit appears. We refer to the Li–Yorke and Sarkovskii theorems, stated in Section 1.3, for the succes- sive appearance of periodic points of period 2k (k ≥ 0), as θ increases to a limit point of θc ≈ 3.57, which is followed by other cascades of periodic
- 55. Other documents randomly have different content
- 56. A Bibliographical Melody, printed in Richard Thomson. 1820 at the press of John Johnson, as a gift to the members of the Roxburghe Club. That Life is a Comedy oft hath been shown, By all who Mortality's changes have known; But more like a Volume its actions appear, Where each Day is a Page and each Chapter a year. 'Tis a Manuscript Time shall full surely unfold, Though with Black-Letter shaded, or shining with gold; The Initial, like Youth, glitters bright on its Page, But its Text is as dark—as the gloom of Old Age. Then Life's Counsels of Wisdom engrave on thy breast, And deep on thine Heart be her lessons imprest. Though the Title stands first it can little declare The Contents which the Pages ensuing shall bear; As little the first day of Life can explain The succeeding events which shall glide in its train, The Book follows next, and, delighted, we trace An Elzevir's beauty, a Guttemberg's grace; Thus on pleasure we gaze with as raptured an eye, Till, cut off like a Volume imperfect, we die! Then Life's Counsels of Wisdom engrave on thy breast, And deep on thine Heart be her lessons imprest. Yet e'en thus imperfect, complete, or defaced, The skill of the Printer is still to be traced; And though death bend us early in life to his will, The wise hand of our Author is visible still. Like the Colophon lines is the Epitaph's lay, Which tells of what age and what nation our day, And, like the Device of the Printer, we bear The form of the Founder, whose Image we wear.
- 57. Then Life's Counsels of Wisdom engrave on thy breast, And deep on thine Heart be her lessons imprest. The work thus completed its Boards shall inclose, Till a Binding more bright and more beauteous it shows; And who can deny, when Life's Vision hath past, That the dark Boards of Death shall surround us at last. Yet our Volume illumed with fresh splendors shall rise, To be gazed at by Angels, and read to the skies, Reviewed by its Author, revised by his Pen, In a fair new Edition to flourish again. Then Life's Counsels of Wisdom engrave on thy breast, And deep on thine Heart be her lessons imprest. ON CERTAIN BOOKS. Charles Tennyson Turner. From 'Sonnets.' 1864. Faith and fixt hope these pages may peruse, And still be faith and hope; but, O ye winds! Blow them far off from all unstable minds, And foolish grasping hands of youth! Ye dews Of heaven! be pleased to rot them where they fall, Lest loitering boys their fancies should abuse, And they get harm by chance, that cannot choose; So be they stain'd and sodden, each and all! And if, perforce, on dry and gusty days, Upon the breeze some truant leaf should rise, Brittle with many weathers, to the skies, Or flit and dodge about the public ways— Man's choral shout, or organ's peal of praise Shall shake it into dust, like older lies.
- 58. TO HIS BOOKS. Henry Vaughan. From 'Silex Scintillans: Sacred Poems and Pious Ejaculations.' 1678. Bright books: perspectives on our weak sights, The clear projections of discerning lights, Burning in shining thoughts, man's posthume day, The track of fled souls in their milkie way, The dead alive and busy, the still voice Of enlarged spirits, kind heaven's white decoys! Who lives with you lives like those knowing flowers Which in commerce with light spend all their hours; Which shut to clouds, and shadows nicely shun, But with glad haste unveil to kiss the sun. Beneath you all is dark and a dead night, Which whoso lives in wants both health and sight. By sucking you, the wise, like bees, do grow Healing and rich, though this they do most slow, Because most choicely; for as great a store Have we of books as bees, of herbs, or more; And the great task to try, then know, the good, To discern weeds, and judge of wholesome food, Is a rare scant performance. For man dies Oft ere 'tis done, while the bee feeds and flies. But you were all choice flowers; all set and drest By old sage florists, who well knew the best; And I amidst you all am turned to weed! Not wanting knowledge, but for want of heed. Then thank thyself, wild fool, that would'st not be Content to know what was too much for thee! LITERATURE AND NATURE.
- 59. Samuel Waddington. Written for the present collection. 'Mid Cambrian heights around Dolgelly vale, What time we scaled great Cader's rugged pile, Or loitered idly where still meadows smile Beside the Mawddach-stream, or far Cynfael— Nor tome, nor rhythmic page, nor pastoral tale, Our summer-sated senses would beguile; Or lull our ears to melody, the while The voiceful rill ran lilting down the dale. In London town once more—behold, once more The old delight returns! 'Mid heights how vast, In Milton's verse, through what dim paths we wind; How Keats's canvas glows, and Wordsworth's lore, As tarn or torrent pure, by none surpass'd, Sheds light and love—unfathomed, undefined. THE LIBRARY. John Greenleaf Whittier. Sung at the opening of the Library at Haverhill, Mass. Let there be Light! God spake of old, And over chaos dark and cold, And through the dead and formless frame Of nature, life and order came. Faint was the light at first that shone On giant fern and mastodon, On half-formed plant and beast of prey, And man as rude and wild as they. Age after age, like waves o'erran The earth, uplifting brute and man;
- 60. And mind, at length, in symbols dark Its meanings traced on stone and bark. On leaf of palm, on sedge-wrought roll, On plastic clay and leathern scroll, Man wrote his thoughts; the ages passed, And lo! the Press was found at last! Then dead souls woke; the thoughts of men Whose bones were dust revived again; The cloister's silence found a tongue, Old prophets spake, old poets sung. And here, to-day, the dead look down, The kings of mind again we crown; We hear the voices lost so long, The sage's word, the sibyl's song. Here Greek and Roman find themselves Alive along these crowded shelves; And Shakspere treads again his stage, And Chaucer paints anew his age. As if some Pantheon's marbles broke Their stony trance, and lived and spoke, Life thrills along the alcoved hall, The lords of thought awake our call. THE COUNTRY SQUIRE. Tomas Yriarte. An anonymous translation of one of the 'Literary Fables.' A country squire, of greater wealth than wit (For fools are often blessed with fortune's smile),
- 61. Had built a splendid house, and furnished it In splendid style. One thing is wanting, said a friend; for, though The rooms are fine, the furniture profuse, You lack a library, dear sir, for show, If not for use. 'Tis true; but 'zounds! replied the squire with glee, The lumber-room in yonder northern wing (I wonder I ne'er thought of it) will be The very thing. I'll have it fitted up without delay With shelves and presses of the newest mode And rarest wood, befitting every way A squire's abode. And when the whole is ready, I'll dispatch My coachman—a most knowing fellow—down To buy me, by admeasurement, a batch Of books in town. But ere the library was half supplied With all its pomps of cabinet and shelf, The booby squire repented him, and cried Unto himself:— This room is much more roomy than I thought; Ten thousand volumes hardly would suffice To fill it, and would cost, however bought, A plaguy price. Now as I only want them for their looks, It might, on second thoughts, be just as good, And cost me next to nothing, if the books Were made of wood.
- 62. It shall be so, I'll give the shaven deal A coat of paint—a colorable dress, To look like calf or vellum, and conceal Its nakedness. And, gilt and lettered with the author's name, Whatever is most excellent and rare Shall be, or seem to be ('tis all the same), Assembled there. The work was done; the simulated hoards Of wit and wisdom round the chamber stood, In binding some; and some, of course, in boards, Where all were wood. From bulky folios down to slender twelves The choicest tomes, in many an even row Displayed their lettered backs upon the shelves, A goodly show. With such a stock as seemingly surpassed The best collection ever formed in Spain, What wonder if the owner grew at last Supremely vain? What wonder, as he paced from shelf to shelf, And conned their titles, that the squire began, Despite his ignorance, to think himself A learned man? Let every amateur, who merely looks To backs and binding, take the hint, and sell His costly library—for painted books Would serve as well. OLD BOOKS.
- 63. From the appendix of 'How to Read Anon. a Book in the Best Way.' New York, n. d. I must confess I love old books! The dearest, too, perhaps most dearly; Thick, clumpy tomes, of antique looks, In pigskin covers fashioned queerly. Clasped, chained, or thonged, stamped quaintly too, With figures wondrous strange, or holy Men and women, and cherubs, few Might well from owls distinguish duly. I love black-letter books that saw The light of day at least three hundred Long years ago; and look with awe On works that live, so often plundered. I love the sacred dust the more It clings to ancient lore, enshrining Thoughts of the dead, renowned of yore, Embalmed in books, for age declining. Fit solace, food, and friends more sure To have around one, always handy, When sinking spirits find no cure In news, election brawls, or brandy. In these old books, more soothing far Than balm of Gilead or Nepenthè, I seek an antidote for care— Of which most men indeed have plenty. Five hundred times at least, I've said— My wife assures me—I would never Buy more old books; yet lists are made,
- 64. And shelves are lumbered more than ever. Ah! that our wives could only see How well the money is invested In these old books, which seem to be By them, alas! so much detested. There's nothing hath enduring youth, Eternal newness, strength unfailing, Except old books, old friends, old truth, That's ever battling—still prevailing. 'T is better in the past to live Than grovel in the present vilely, In clubs, and cliques, where placemen hive, And faction hums, and dolts rank highly. To be enlightened, counselled, led, By master minds of former ages, Come to old books—consult the dead— Commune with silent saints and sages. Leave me, ye gods! to my old books— Polemics yield to sects that wrangle— Vile parish politics to folks Who love to squabble, scheme, and jangle. Dearly beloved old pigskin tomes! Of dingy hue—old bookish darlings! Oh, cluster ever round my rooms, And banish strifes, disputes, and snarlings.
- 65. Appendix THE LIBRARY BY GEORGE CRABBE THE LIBRARY. In want and danger, the unknown poet sent this poem to Edmund George Crabbe. Burke, who saw its merit, befriended its author, and procured its publication. When the sad soul, by care and grief oppressed, Looks round the world, but looks in vain for rest, When every object that appears in view Partakes her gloom and seems dejected too; Where shall affliction from itself retire? Where fade away and placidly expire? Alas! we fly to silent scenes in vain; Care blasts the honors of the flowery plain; Care veils in clouds the sun's meridian beam, Sighs through the grove, and murmurs in the stream; For when the soul is laboring in despair, In vain the body breathes a purer air: No storm-tost sailor sighs for slumbering seas—
- 66. He dreads the tempest, but invokes the breeze; On the smooth mirror of the deep resides Reflected woe, and o'er unruffled tides The ghost of every former danger glides. Thus, in the calms of life, we only see A steadier image of our misery; But lively gales and gently clouded skies Disperse the sad reflections as they rise; And busy thoughts and little cares avail To ease the mind, when rest and reason fail. When the dull thought, by no designs employed, Dwells on the past, or suffered or enjoyed, We bleed anew in every former grief, And joys departed furnish no relief. Not Hope herself, with all her flattering art, Can cure this stubborn sickness of the heart: The soul disdains each comfort she prepares, And anxious searches for congenial cares; Those lenient cares, which, with our own combined, By mixed sensations ease th' afflicted mind, And steal our grief away, and leave their own behind; A lighter grief! which feeling hearts endure Without regret, nor e'en demand a cure. But what strange art, what magic can dispose The troubled mind to change its native woes? Or lead us, willing from ourselves, to see Others more wretched, more undone than we? This Books can do;—nor this alone; they give New views to life, and teach us how to live; They soothe the grieved, the stubborn they chastise, Fools they admonish and confirm the wise: Their aid they yield to all: they never shun The man of sorrow, nor the wretch undone: Unlike the hard, the selfish, and the proud, They fly not sullen from the suppliant crowd; Nor tell to various people various things,
- 67. But show to subjects what they show to kings. Come, Child of Care! to make thy soul serene, Approach the treasures of this tranquil scene; Survey the dome, and, as the doors unfold, The soul's best cure, in all her cares behold! Where mental wealth the poor in thought may find, And mental physic the diseased in mind; See here the balms that passion's wounds assuage; See coolers here, that damp the fire of rage; Here alteratives, by slow degrees control The chronic habits of the sickly soul; And round the heart, and o'er the aching head, Mild opiates here their sober influence shed. Now bid thy soul man's busy scenes exclude, And view composed this silent multitude:— Silent they are—but though deprived of sound, Here all the living languages abound; Here all that live no more; preserved they lie, In tombs that open to the curious eye. Blest be the gracious Power, who taught mankind To stamp a lasting image of the mind! Beasts may convey, and tuneful birds may sing, Their mutual feelings, in the opening spring; But Man alone has skill and power to send The heart's warm dictates to the distant friend; 'Tis his alone to please, instruct, advise Ages remote, and nations yet to rise. In sweet repose, when Labor's children sleep, When Joy forgets to smile and Care to weep, When Passion slumbers in the lover's breast, And Fear and Guilt partake the balm of rest, Why then denies the studious man to share Man's common good, who feels his common care? Because the hope is his that bids him fly Night's soft repose, and sleep's mild power defy, That after-ages may repeat his praise,
- 68. And fame's fair meed be his, for length of days. Delightful prospect! when we leave behind A worthy offspring of the fruitful mind! Which, born and nursed through many an anxious day, Shall all our labor, all our care repay. Yet all are not these births of noble kind, Not all the children of a vigorous mind; But where the wisest should alone preside, The weak would rule us, and the blind would guide; Nay, man's best efforts taste of man, and show The poor and troubled source from which they flow; Where most he triumphs we his wants perceive, And for his weakness in his wisdom grieve. But though imperfect all; yet wisdom loves This seat serene, and virtue's self approves:— Here come the grieved, a change of thought to find; The curious here to feed a craving mind; Here the devout their peaceful temple choose; And here the poet meets his favoring Muse. With awe, around these silent walks I tread; These are the lasting mansions of the dead:— The dead! methinks a thousand tongues reply; These are the tombs of such as cannot die! Crowned with eternal fame, they sit sublime, And laugh at all the little strife of time. Hail, then, immortals! ye who shine above, Each, in his sphere, the literary Jove; And ye, the common people of these skies, A humbler crowd of nameless deities; Whether 't is yours to lead the willing mind Through History's mazes, and the turnings find; Or, whether led by Science, ye retire, Lost and bewildered in the vast desire, Whether the Muse invites you to her bowers, And crowns your placid brows with living flowers! Or godlike Wisdom teaches you to show
- 69. The noblest road to happiness below; Or men and manners prompt the easy page To mark the flying follies of the age; Whatever good ye boast, that good impart; Inform the head and rectify the heart. Lo, all in silence, all in order stand, And mighty folios, first a lordly band; Then quartos their well-ordered ranks maintain, And light octavos fill a spacious plain: See yonder, ranged in more frequented rows, A humbler band of duodecimos; While undistinguish'd trifles swell the scene, The last new play and frittered magazine. Thus 't is in life, where first the proud, the great, In leagued assembly keep their cumbrous state: Heavy and huge, they fill the world with dread, Are much admired, and are but little read: The commons next, a middle rank, are found; Professions fruitful pour their offspring round; Reasoners and wits are next their place allowed, And last, of vulgar tribes a countless crowd. First, let us view the form, the size, the dress: For these the manners, nay the mind, express: That weight of wood, with leathern coat o'erlaid; Those ample clasps of solid metal made; The close-pressed leaves, unclosed for many an age; The dull red edging of the well-filled page; On the broad back the stubborn ridges rolled, Where yet the title stands in tarnished gold; These all a sage and labored work proclaim, A painful candidate for lasting fame: No idle wit, no trifling verse can lurk In the deep bosom of that weighty work; No playful thoughts degrade the solemn style, Nor one light sentence claims a transient smile. Hence, in these times, untouched the pages lie,
- 70. And slumber out their immortality: They had their day, when, after all his toil, His morning study, and his midnight oil, At length an author's one great work appeared, By patient hope, and length of days endeared: Expecting nations haled it from the press; Poetic friends prefixed each kind address; Princes and kings received the pond'rous gift, And ladies read the work they could not lift. Fashion, though Folly's child, and guide of fools, Rules e'en the wisest, and in learning rules; From crowds and courts to Wisdom's seat she goes, And reigns triumphant o'er her mother's foes. For lo! these favorites of the ancient mode Lie all neglected like the Birthday Ode. Ah! needless now this weight of massy chain, Safe in themselves, the once-loved works remain; No readers now invade their still retreat, None try to steal them from their parent seat; Like ancient beauties, they may now discard Chains, bolts, and locks, and lie without a guard. Our patient fathers trifling themes laid by, And rolled, o'er labored works, th' attentive eye: Page after page the much enduring men Explored the deeps and shallows of the pen: Till, every former note and comment known, They marked the spacious margin with their own; Minute corrections proved their studious care; The little index, pointing, told us where; And many an emendation showed the age Looked far beyond the rubric title-page. Our nicer palates lighter labors seek, Cloyed with a folio-Number once a week; Bibles, with cuts and comments, thus go down: E'en light Voltaire is numbered through the town: Thus physic flies abroad, and thus the law,
- 71. From men of study, and from men of straw; Abstracts, abridgments, please the fickle times, Pamphlets and plays, and politics and rhymes: But though to write be now a task of ease, The task is hard by manly arts to please, When all our weakness is exposed to view, And half our judges are our rivals too. Amid these works, on which the eager eye Delights to fix, or glides reluctant by, When all combined, their decent pomp display, Where shall we first our early offering pay?— To thee, Divinity! to thee, the light And guide of mortals, through their mental night; By whom we learn our hopes and fears to guide; To bear with pain, and to contend with pride; When grieved, to pray; when injured, to forgive; And with the world in charity to live. Not truths like these inspired that numerous race, Whose pious labors fill this ample space; But questions nice, where doubt on doubt arose, Awaked to war the long-contending foes. For dubious meanings, learned polemics strove, And wars on faith prevented works of love; The brands of discord far around were hurled, And holy wrath inflamed a sinful world:— Dull though impatient, peevish though devout, With wit, disgusting and despised without; Saints in design, in execution men, Peace in their looks, and vengeance in their pen. Methinks I see, and sicken at the sight, Spirits of spleen from yonder pile alight; Spirits who prompted every damning page, With pontiff pride, and still increasing rage: Lo! how they stretch their gloomy wings around, And lash with furious strokes the trembling ground! They pray, they fight, they murder, and they weep,
- 72. Wolves in their vengeance, in their manners sheep; Too well they act the prophet's fatal part, Denouncing evil with a zealous heart; And each, like Jonah, is displeased if God Repent his anger, or withold his rod. But here the dormant fury rests unsought, And Zeal sleeps soundly by the foes she fought; Here all the rage of controversy ends, And rival zealots rest like bosom friends: An Athanasian here, in deep repose, Sleeps with the fiercest of his Arian foes; Socinians here with Calvinists abide, And thin partitions angry chiefs divide; Here wily Jesuits simple Quakers meet, And Bellarmine has rest at Luther's feet. Great authors, for the church's glory fired, Are for the church's peace to rest retired; And close beside, a mystic, maudlin race, Lie Crumbs of Comfort for the Babes of Grace. Against her foes Religion well defends Her sacred truths, but often fears her friends; If learned, their pride, if weak, their zeal she dreads, And their hearts' weakness, who have soundest heads. But most she fears the controversial pen, The holy strife of disputatious men; Who the blest Gospel's peaceful page explore, Only to fight against its precepts more. Near to these seats behold yon slender frames, All closely filled and marked with modern names; Where no fair science ever shows her face, Few sparks of genius, and no spark of grace; There sceptics rest, a still increasing throng, And stretch their widening wings ten thousand strong; Some in close fight their dubious claims maintain; Some skirmish lightly, fly, and fight again; Coldly profane, and impiously gay,
- 73. Their end the same, though various in their way. When first Religion came to bless the land, Her friends were then a firm believing band; To doubt was then to plunge in guilt extreme, And all was gospel that a monk could dream; Insulted Reason fled the grov'lling soul, For Fear to guide and visions to control: But now, when Reason has assumed her throne, She, in her turn demands to reign alone; Rejecting all that lies beyond her view, And, being judge, will be a witness too: Insulted Faith then leaves the doubtful mind, To seek for truth, without a power to find: Ah! when will both in friendly beams unite, And pour on erring man resistless light! Next to the seats, well stored with works divine, An ample space, Philosophy! is thine; Our reason's guide, by whose assisting light We trace the moral bounds of wrong and right; Our guide through nature, from the sterile clay, To the bright orbs of yon celestial way! 'T is thine, the great, the golden chain to trace, Which runs through all, connecting race with race Save where those puzzling, stubborn links remain, Which thy inferior light pursues in vain:— How vice and virtue in the soul contend; How widely differ, yet how nearly blend; What various passions war on either part, And now confirm, now melt the yielding heart: How Fancy loves around the world to stray, While Judgment slowly picks his sober way; The stores of memory and the flights sublime Of genius, bound by neither space nor time;— All these divine Philosophy explores, Till, lost in awe, she wonders and adores. From these, descending to the earth, she turns,
- 74. And matter, in its various forms, discerns; She parts the beamy light with skill profound, Metes the thin air, and weighs the flying sound; 'T is hers the lightning from the clouds to call, And teach the fiery mischief where to fall. Yet more her volumes teach—on these we look Abstracts drawn from Nature's larger book; Here, first described, the torpid earth appears, And next, the vegetable robe it wears; Where flowery tribes in valleys, fields, and groves, Nurse the still flame, and feed the silent loves; Loves where no grief, nor joy, nor bliss, nor pain, Warm the glad heart or vex the laboring brain; But as the green blood moves along the blade, The bed of Flora on the branch is made; Where, without passion, love instinctive lives, And gives new life, unconscious that it gives. Advancing still in Nature's maze, we trace, In dens and burning plains, her savage race With those tame tribes who on their lord attend, And find in man a master and a friend; Man crowns the scene, a world of wonders new, A moral world, that well demands our view. This world is here; for, of more lofty kind, These neighboring volumes reason on the mind; They paint the state of man ere yet endued With knowledge;—man, poor, ignorant, and rude; Then, as his state improves, their pages swell, And all its cares, and all its comforts tell: Here we behold how inexperience buys, At little price, the wisdom of the wise; Without the troubles of an active state, Without the cares and dangers of the great, Without the miseries of the poor, we know What wisdom, wealth, and poverty bestow; We see how reason calms the raging mind,
- 75. And how contending passions urge mankind: Some, won by virtue, glow with sacred fire; Some, lured by vice, indulge the low desire; Whilst others, won by either, now pursue The guilty chase, now keep the good in view; Forever wretched, with themselves at strife, They lead a puzzled, vexed, uncertain life; For transient vice bequeaths a lingering pain, Which transient virtue seeks to cure in vain. Whilst thus engaged, high views enlarge the soul, New interest draws, new principles control: Nor thus the soul alone resigns her grief, But here the tortured body finds relief; For see where yonder sage Arachnè shapes Her subtle gin, that not a fly escapes! There Physic fills the space, and far around, Pile above pile her learned works abound: Glorious their aim—to ease the laboring heart; To war with death, and stop his flying dart; To trace the source whence the fierce contest grew; And life's short lease on easier terms renew; To calm the frenzy of the burning brain; To heal the tortures of imploring pain; Or, when more powerful ills all efforts brave, To ease the victim no device can save, And smooth the stormy passage to the grave. But man, who knows no good unmixed and pure, Oft finds a poison where he sought a cure; For grave deceivers lodge their labors here, And cloud the science they pretend to clear; Scourges for sin, the solemn tribe are sent; Like fire and storms, they call us to repent; But storms subside, and fires forget to rage. These are eternal scourges of the age: 'T is not enough that each terrific hand Spreads desolation round a guilty land;
- 76. But trained to ill, and hardened by its crimes, Their pen relentless kills through future times, Say, ye, who search these records of the dead— Who read huge works, to boast what ye have read, Can all the real knowledge ye possess, Or those—if such there are—who more than guess, Atone for each impostor's wild mistakes, And mend the blunders pride or folly makes? What thought so wild, what airy dream so light, That will not prompt a theorist to write? What art so prevalent, what proofs so strong, That will convince him his attempt is wrong? One in the solids finds each lurking ill, Nor grants the passive fluids power to kill; A learned friend some subtler reason brings, Absolves the channels, but condemns their spring; The subtile nerves, that shun the doctor's eye, Escape no more his subtler theory; The vital heat, that warms the laboring heart, Lends a fair system to these sons of art; The vital air, a pure and subtile stream, Serves a foundation for an airy scheme, Assists the doctor and supports his dream. Some have their favorite ills, and each disease Is but a younger branch that kills from these; One to the gout contracts all human pain; He views it raging in the frantic brain; Finds it in fevers all his efforts mar, And sees it lurking in the cold catarrh; Bilious by some, by others nervous seen, Rage the fantastic demons of the spleen; And every symptom of the strange disease With every system of the sage agrees. Ye frigid tribe, on whom I wasted long The tedious hours, and ne'er indulged in song; Ye first seducers of my easy heart,
- 77. Who promised knowledge ye could not impart; Ye dull deluders, truth's destructive foes; Ye sons of fiction, clad in stupid prose; Ye treacherous leaders, who, yourselves in doubt, Light up false fires, and send us far about;— Still may yon spider round your pages spin, Subtile and slow, her emblematic gin! Buried in dust and lost in silence, dwell, Most potent, grave, and reverend friends—farewell! Near these, and where the setting sun displays, Through the dim window, his departing rays, And gilds yon columns, there, on either side, The huge Abridgments of the Law abide; Fruitful as vice, the dread correctors stand, And spread their guardian terrors round the land; Yet, as the best that human care can do Is mixed with error, oft with evil too, Skilled in deceit, and practised to evade, Knaves stand secure, for whom these laws were made, And justice vainly each expedient tries, While art eludes it, or while power defies. Ah! happy age, the youthful poet sings, When the free nations knew not laws nor kings, When all were blest to share a common store, And none were proud of wealth, for none were poor, No wars nor tumults vexed each still domain, No thirst of empire, no desire of gain; No proud great man, nor one who would be great, Drove modest merit from its proper state; Nor into distant climes would Avarice roam, To fetch delights for Luxury at home: Bound by no ties which kept the soul in awe, They dwelt at liberty, and love was law! Mistaken youth! each nation first was rude, Each man a cheerless son of solitude, To whom no joys of social life were known,
- 78. None felt a care that was not all his own; Or in some languid clime his abject soul Bowed to a little tyrant's stern control; A slave, with slaves his monarch's throne he raised, And in rude song his ruder idol praised; The meaner cares of life were all he knew; Bounded his pleasures, and his wishes few; But when by slow degrees the Arts arose, And Science wakened from her long repose; When Commerce, rising from the bed of ease, Ran round the land, and pointed to the seas; When Emulation, born with jealous eye, And Avarice, lent their spurs to industry; Then one by one the numerous laws were made, Those to control, and these to succor trade; To curb the insolence of rude command, To snatch the victim from the usurer's hand; To awe the bold, to yield the wronged redress, And feed the poor with Luxury's excess. Like some vast flood, unbounded, fierce, and strong, His nature leads ungoverned man along; Like mighty bulwarks made to stem that tide, The laws are formed and placed on every side; Whene'er it breaks the bounds by these decreed, New statutes rise, and stronger laws succeed; More and more gentle grows the dying stream, More and more strong the rising bulwarks seem; Till, like a miner working sure and slow, Luxury creeps on, and ruins all below; The basis sinks, the ample piles decay; The stately fabric shakes and falls away; Primeval want and ignorance come on, But Freedom, that exalts the savage state, is gone. Next History ranks;—there full in front she lies, And every nation her dread tale supplies; Yet History has her doubts, and every age
- 79. With sceptic queries marks the passing page; Records of old nor later date are clear, Too distant those, and these are placed too near; There time conceals the objects from our view, Here our own passions and a writer's too: Yet, in these volumes, see how states arose! Guarded by virtue from surrounding foes; Their virtue lost, and of their triumphs vain, Lo! how they sunk to slavery again! Satiate with power, of fame and wealth possessed, A nation grows too glorious to be blest; Conspicuous made, she stands the mark of all, And foes join foes to triumph in her fall. Thus speaks the page that paints ambition's race, The monarch's pride, his glory, his disgrace; The headlong course that maddening heroes run, How soon triumphant, and how soon undone; How slaves, turned tyrants, offer crowns to sale, And each fallen nation's melancholy tale. Lo! where of late the Book of Martyrs stood, Old pious tracts, and Bibles bound in wood; There, such the taste of our degenerate age, Stand the profane delusions of the Stage: Yet virtue owns the Tragic Muse a friend, Fable her means, morality her end; For this she rules all passions in their turns, And now the bosom bleeds, and now it burns; Pity with weeping eye surveys her bowl, Her anger swells, her terror chills the soul; She makes the vile to virtue yield applause, And own her sceptre while they break her laws; For vice in others is abhorred of all, And villains triumph when the worthless fall. Not thus her sister Comedy prevails, Who shoots at Folly, for her arrow fails; Folly, by Dulness armed, eludes the wound,
- 80. And harmless sees the feathered shafts rebound; Unhurt she stands, applauds the archer's skill, Laughs at her malice, and is Folly still. Yet well the Muse portrays, in fancied scenes, What pride will stoop to, what profession means; How formal fools the farce of state applaud; How caution watches at the lips of fraud; The wordy variance of domestic life; The tyrant husband, the retorting wife; The snares for innocence, the lie of trade, And the smooth tongue's habitual masquerade. With her the Virtues to obtain a place, Each gentle passion, each becoming grace; The social joy in life's securer road, Its easy pleasure, its substantial good; The happy thought that conscious virtue gives, And all that ought to live, and all that lives. But who are these? Methinks a noble mien And awful grandeur in their form are seen, Now in disgrace: what though by time is spread Polluting dust o'er every reverend head; What though beneath yon gilded tribe they lie, And dull observers pass insulting by: Forbid it shame, forbid it decent awe, What seems so grave, should no attention draw! Come, let us then with reverend step advance, And greet—the ancient worthies of Romance. Hence, ye profane! I feel a former dread, A thousand visions float around my head: Hark! hollow blasts through empty courts resound, And shadowy forms with staring eyes stalk round; See! moats and bridges, walls and castles rise, Ghosts, fairies, demons, dance before our eyes; Lo! magic verse inscribed on golden gate; And bloody hand that beckons on to fate:— And who art thou, thou little page, unfold?
- 81. Say, doth thy lord my Claribel withhold? Go tell him straight, Sir Knight, thou must resign The captive queen;—for Claribel is mine. Away he flies; and now for bloody deeds, Black suits of armor, masks, and foaming steeds; The giant falls; his recreant throat I seize, And from his corselet take the massy keys:— Dukes, lords, and knights in long procession move, Released from bondage with my virgin love:— She comes! she comes! in all the charms of youth, Unequalled love, and unsuspected truth! Ah! happy he who thus, in magic themes, O'er worlds bewitched, in early rapture dreams, Where wild Enchantment waves her potent wand, And Fancy's beauties fill her fairy land; Where doubtful objects strange desires excite, And Fear and Ignorance afford delight. But lost, for ever lost, to me these joys, Which Reason scatters, and which Time destroys; Too dearly bought: maturer judgment calls My busied mind from tales and madrigals; My doughty giants all are slain or fled And all my knights—blue, green, and yellow—dead! No more the midnight fairy tribe I view, All in the merry moonshine tippling dew; E'en the last lingering fiction of the brain, The churchyard ghost is now at rest again; And all these wayward wanderings of my youth Fly Reason's power, and shun the light of Truth. With Fiction then does real joy reside, And is our reason the delusive guide? Is it then right to dream the sirens sing? Or mount enraptured on the dragon's wing? No; 't is the infant mind, to care unknown, That makes th' imagined paradise its own; Soon as reflections in the bosom rise,
- 82. Light slumbers vanish from the clouded eyes: The tear and smile, that once together rose, Are then divorced; the head and heart are foes: Enchantment bows to Wisdom's serious plan, And Pain and Prudence make and mar the man. While thus, of power and fancied empire vain, With various thoughts my mind I entertain; While books, my slaves, with tyrant hand I seize, Pleased with the pride that will not let them please, Sudden I find terrific thoughts arise, And sympathetic sorrow fills my eyes; For, lo! while yet my heart admits the wound, I see the Critic army ranged around. Foes to our race! if ever ye have known A father's fears for offspring of your own; If ever, smiling o'er a lucky line, Ye thought the sudden sentiment divine, Then paused and doubted, and then, tired of doubt, With rage as sudden dashed the stanza out;— If, after fearing much and pausing long, Ye ventured on the world your labored song, And from the crusty critics of those days Implored the feeble tribute of their praise; Remember now the fears that moved you then, And, spite of truth, let mercy guide your pen. What vent'rous race are ours! what mighty foes Lie waiting all around them to oppose! What treacherous friends betray them to the fight! What dangers threaten them:—yet still they write: A hapless tribe! to every evil born, Whom villains hate, and fools affect to scorn: Strangers they come, amid a world of woe, And taste the largest portion ere they go. Pensive I spoke, and cast mine eyes around; The roof, methought, returned a solemn sound; Each column seemed to shake, and clouds, like smoke,
- 83. From dusty piles and ancient volumes broke; Gathering above, like mists condensed they seem, Exhaled in summer from the rushy stream; Like flowing robes they now appear, and twine Round the large members of a form divine; His silver beard, that swept his aged breast, His piercing eye, that inward light expressed, Were seen—but clouds and darkness veiled the rest. Fear chilled my heart: to one of mortal race, How awful seemed the Genius of the place! So in Cimmerian shores, Ulysses saw His parent-shade, and shrunk in pious awe; Like him I stood, and wrapped in thought profound, When from the pitying power broke forth a solemn sound:— Care lives with all; no rules, no precepts save The wise from woe, no fortitude the brave; Grief is to man as certain as the grave: Tempests and storms in life's whole progress rise, And hope shines dimly through o'erclouded skies. Some drops of comfort on the favored fall, But showers of sorrow are the lot of all: Partial to talents, then, shall Heaven withdraw Th' afflicting rod, or break the general law? Shall he who soars, inspired by loftier views, Life's little cares and little pains refuse? Shall he not rather feel a double share Of mortal woe, when doubly armed to bear? Hard is his fate who builds his peace of mind On the precarious mercy of mankind; Who hopes for wild and visionary things, And mounts o'er unknown seas with vent'rous wings; But as, of various evils that befall The human race, some portion goes to all; To him perhaps the milder lot's assigned Who feels his consolation in his mind. And, locked within his bosom, bears about
- 84. A mental charm for every care without. E'en in the pangs of each domestic grief, Or health or vigorous hope affords relief; And every wound the tortured bosom feels, Or virtue bears, or some preserver heals; Some generous friend of ample power possessed; Some feeling heart, that bleeds for the distressed; Some breast that glows with virtues all divine; Some noble Rutland, misery's friend and thine. Nor say, the Muse's song, the Poet's pen, Merit the scorn they meet from little men. With cautious freedom if the numbers flow, Not wildly high, nor pitifully low; If vice alone their honest aims oppose, Why so ashamed their friends, so loud their foes? Happy for men in every age and clime, If all the sons of vision dealt in rhyme. Go on, then, Son of Vision! still pursue Thy airy dreams; the world is dreaming too. Ambition's lofty views, the pomp of state, The pride of wealth, the splendor of the great, Stripped of their mask, their cares and troubles known, Are visions far less happy than thy own: Go on! and, while the sons of care complain, Be wisely gay and innocently vain; While serious souls are by their fears undone, Blow sportive bladders in the beamy sun, And call them worlds! and bid the greatest show More radiant colors in their worlds below: Then, as they break, the slaves of care reprove, And tell them, Such are all the toys they love.
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