Analysis and Control of the Dynamics of Economic Models

Chemical Engineering: An International Journal (CEIJ), Vol. 1, No.2, 2025
23
ANALYSIS AND CONTROL OF THE DYNAMICS OF
ECONOMIC MODELS
Lakshmi. N. Sridhar
Chemical Engineering Department University of Puerto Rico Mayaguez, PR 00681
ABSTRACT
The complex dynamics of various nations' economies have led to extensive research into economic models
to prevent market collapse and mitigate fluctuations, while developing strategies to control the gross
domestic product (GDP). Bifurcation analysis is a powerful mathematical tool for examining the nonlinear
dynamics of any process. Several factors must be considered, and multiple objectives need to be met
simultaneously. Bifurcation analysis and multiobjective nonlinear model predictive control (MNLMPC)
calculations are applied to two economic models. The MATLAB program MATCONT was used for the
bifurcation analysis. The MNLMPC calculations were carried out employing the optimization language
PYOMO in conjunction with advanced global optimization solvers IPOPT and BARON. The bifurcation
analysis identified Hopf bifurcation points that lead to unwanted limit cycles, which are mitigated using an
activation factor involving the tanh function. The MNLMPC calculations produced optimal control and
variable profiles. However, the control profiles exhibited a significant amount of noise, which was
addressed using the Savitzky-Golay Filter.
KEYWORDS
financial system, bifurcation, optimization, control
1. BACKGROUND
Lavoie and Godley(2001) [1] discussed Kaleckian models of growth in a coherent stock-flow
monetary framework. Szydłowsk et al (2001) [2] demonstrated the existence of nonlinear
oscillations in business cycle model with time lags. Gallegati et al (2003)[3] showed that the
Hicks’ trade cycle exhibited cycles and bifurcations. Torres and Vela(2003)[4] , researched trade
integration and synchronization between the business cycles of Mexico and the United States.
Puu and Sushko(2004)[5] developed a business cycle model with cubic nonlinearity. Dubois
(2004)[6] extended the Kaldor-Kalecki model of business cycle with a computationally
anticipated capital stock. Kyrtsou and Vorlow(2005)[7] demonstrated the existence of complex
dynamics in macroeconomics.
Rana(2007)[8] discussed the economic integration and synchronization of business cycles in East
Asia. De Paoli(2009)[9] investigated the monetary policy and welfare in an open economy.
Matsumoto A., and Szidarovszky(2010)[10] developed a continuous Hicksian trade cycle model
with consumption and investment time delays. Asada et al (2012)[11] , developed an analytical
expressions of periodic disequilibrium fluctuations generated by Hopf bifurcations in economic
dynamics. Volos et al (2012)[12] studied the synchronization phenomena in coupled nonlinear
systems applied in economic cycles. Bouali et al (2012)[13] emulated complex business cycles
by using an electronic analogue. He and Liao(2012)[14] , studied the asian business cycle
synchronization.
Calderon and Fuentes(2014)[15] conducted an empirical investigation of changes in business
cycles over the last two decades. Chu et al (2015)[16] discussed the inflation, R&D, and growth

24
in an open economy. Zhaoet al (2016)[17] demonstrated the existence of Hopf Bifurcations in
a nonlinear Business Cycle Model. Kufenko, and Geiger (2016)[18] provided an econometric
analysis of economic business cycles. Sella et al (2016)[19] investigated the economic cycles
and their synchronization in three European countries. Groth and Ghil(2017) [20] studied the
synchronization of world economic activity. Gardini et al (2018) [21] investigated the nonlinear
behavior of economic dynamics. Campos et al (2019)[22] reviewed the research on business
cycle synchronisation and currency unions. Hanias et al (2019)[23] describe the reverse
engineering in econo-physics. Camargo et al (2022)[24] demonstrated the Synchronization and
bifurcation in an economic model. Amaral et al (2022)[25] studied the interaction between
economies in a business cycle model.
Some of this work involved bifurcation analysis and single-objective optimal control calculation.
The main objective of this paper is to perform multiobjective nonlinear model predictive
control(MNLMPC) in conjunction with bifurcation analysis for two dynamic economic models.
The two models that will be used are those described in (Camargo et al (2022)[24]) and Zhao et
al (2016)[17]. The paper is organized as follows. First, the model equations are presented. The
numerical procedures (bifurcation analysis and multiobjective nonlinear model predictive
control(MNLMPC) are then described. The results, discussion, and conclusions are then
presented.
2. ECONOMY MODELS
The two models described here are those of (Camargo et al (2022)[24]);and Zhao et al
(2016)[17]
Model 1(Camargo et al (2022)[24])
The model equations are
 
2
( ) ( )( ( ))
( )
( )
w( ) c( )
( )
( )
( )
(
) ) 1 2
m yval pval xval d yval
v yval xv
d xva
al zval
s xval
l
dt
d yval
dt
d zval
dt
r yval u u zval
 
 


  


(1)
, ,
xval yval zval represent savings , gross domestic product (GDP) y, and foreign capital
inflow. The parameter values are
0.02; 0.4; 1.0; 50.0; 10; 0.1; 0.05; 0.1;
m pval d c s r v w
       
1
u is the bifurcation and control parameter, while 2
u is taken as 0.5 for the bifurcation analysis
and treated as a variable for the MNLMPC calculations. m is the marginal propensity to save,
pval represents the fraction of capitalized profit, d represents the GDP potential, and c is the
output-capital ratio. s is the inflow-saving ratio, r is the indebtedness factor, v is the marginal
propensity to consume, and w represents the proportion of savings. 1
u measures the effect of the
commercial trade and 2
u zval
 represents the commercial balance of the country.

25
Model 2 Zhao et al (2016)[17]
The model equations are
( )
* 0.66 15 0.3 108)
( )
15 0.3
(
( )
0.3 10 6 50
d yval
yval yval kval mval
dt
d kval
yval kval mval
dt
d mval
yval yval mval
dt

  
     
  
     
(2)
, ,
yval kval mval represent the actual gross domestic product, the physical capital stock and the
nominal currency supply, is the market adjustment coefficient, and is the degree of capital
movement. Both are used as bifurcation and control parameters.
3. NUMERICAL PROCEDURES
This section describes the numerical procedures for bifurcation analysis and multi-objective
nonlinear model predictive control.
Bifurcation analysis
The MATLAB software MATCONT is used to perform the bifurcation calculations. Bifurcation
analysis deals with multiple steady-states and limit cycles. Multiple steady states occur because
of the existence of branch and limit points. Hopf bifurcation points cause limit cycles . A
commonly used MATLAB program that locates limit points, branch points, and Hopf bifurcation
points is MATCONT(Dhooge Govearts, and Kuznetsov, 2003[26]; Dhooge Govearts, Kuznetsov,
Mestrom and Riet, 2004[27] ). This program detects Limit points(LP), branch points(BP), and
Hopf bifurcation points(H) for an ODE system
( , )
dx
f x
dt

 (3)
n
x R
 Let the bifurcation parameter be  Since the gradient is orthogonal to the tangent
vector,
The tangent plane at any point 1 2 3 4 1
[ , , , ,.... ]
n
w w w w w w 
 must satisfy
0
Aw  (4)
Where A is
[ / | / ]
A f x f 
     (5)
where /
f x
  is the Jacobian matrix. For both limit and branch points, the matrix [ / ]
f x
 
must be singular. The n+1 th
component of the tangent vector 1
n
w  = 0 for a limit point
(LP)and for a branch point (BP) the matrix T
A
w
 
 
 
must be singular. At a Hopf bifurcation point,

26
det(2 ( , )@ ) 0
x n
f x I
  (6)
@ indicates the bialternate product while n
I is the n-square identity matrix. Hopf bifurcations
cause limit cycles and should be eliminated because limit cycles make optimization and control
tasks very difficult. More details can be found in Kuznetsov ( 1998[28]; 2009[29]) and Govaerts
[2000] [30]
Hopf bifurcations cause unwanted oscillatory behavior and limit cycles. The tanh activation
function (where a control value u is replaced by ) ( tanh / )
u u  is commonly used in neural
nets (Dubey et al 2022[31]; Kamalov et al, 2021[32] and Szandała, 2020[33] )and optimal
control problems(Sridhar 2023[34] ) to eliminate spikes in the optimal control profile. Hopf
bifurcation points cause oscillatory behavior. Oscillations are similar to spikes, and the results in
Sridhar(2024)[35] demonstrate that the tanh factor also eliminates the Hopf bifurcation by
preventing the occurrence of oscillations. Sridhar (2024)[35] explained with several examples
how the activation factor involving the tanh function successfully eliminates the limit cycle
causing Hopf bifurcation points. This was because the tanh function increases the time period of
the oscillatory behavior, which occurs in the form of a limit cycle caused by Hopf bifurcations.
Multi-Objective Nonlinear Model Predictive Control (MNLMPC)
Flores Tlacuahuaz et al (2012)[36] developed a multiobjective nonlinear model predictive control
(MNLMPC) method that is rigorous and does not involve weighting functions or additional
constraints. This procedure is used for performing the MNLMPC calculations Here
0
( )
i f
i
t t
j i
t
q t


 (j=12..n) represents the variables that need to be minimized/maximized
simultaneously for a problem involving a set of ODE
( , )
dx
F x u
dt
 (7)
f
t being the final time value, and n the total number of objective variables and . u the control
parameter. This MNLMPC procedure first solves the single objective optimal control problem
independently optimizing each of the variables
0
( )
i f
i
t t
j i
t
q t


 individually. The
minimization/maximization of
0
( )
i f
i
t t
j i
t
q t


 will lead to the values
*
j
q . Then the optimization
problem that will be solved is
0
* 2
1
min( ( ( ) ))
( , );
i f
i
t t
n
j i j
j t
q t q
dx
subject to F x u
dt





 
(8)
This will provide the values of u at various times. The first obtained control value of u is
implemented and the rest are discarded. This procedure is repeated until the implemented and the

27
first obtained control values are the same or if the Utopia point where (
0
*
( )
i f
i
t t
j i j
t
q t q



 for all
j) is obtained. Pyomo (Hart et al, 2017)[37] is used for these calculations. Here, the differential
equations are converted to a Nonlinear Program (NLP) using the orthogonal collocation method
The NLP is solved using IPOPT (Wächter And Biegler, 2006)[38]and confirmed as a global
solution with BARON (Tawarmalani, M. and N. V. Sahinidis 2005)[39].
The steps of the algorithm are as follows
1. Optimize
0
( )
i f
i
t t
j i
t
q t


 and obtain
*
j
q at various time intervals ti. The subscript i is the
index for each time step.
2. Minimize
0
* 2
1
( ( ( ) ))
i f
i
t t
n
j i j
j t
q t q




  and get the control values for various times.
3. Implement the first obtained control values
4. Repeat steps 1 to 3 until there is an insignificant difference between the implemented
and the first obtained value of the control variables or if the Utopia point is achieved.
The Utopia point is when
0
*
( )
i f
i
t t
j i j
t
q t q



 for all j.
4. RESULTS AND DISCUSSION
For model 1, the bifurcation analysis revealed a Hopf bifurcation point of ( , , , 1)
xval yval zval u
of ( 0.014863 3.638778 -0.003669 0.427351 ). This is shown in Fig. 1a. The limit cycle
produced by this Hopf bifurcation point is shown in Fig. 1b. When u1 is modified to u1tanh(u1)
the Hopf bifurcation point and the limit cycle disappear (confirming the analysis of
Sridhar(2024).
In model 2, when  is the bifurcation parameter, the bifurcation analysis revealed a Hopf
bifurcation point of ( , , , )
yval kval mval  = ( 163.636364 1047.013683 122.223460 1.433291
)(Fig. 2a). The limit cycle produced by this Hopf bifurcation point is shown in Fig. 2b . When
 is modified to tanh( ) / 0.2
  the Hopf bifurcation point disappears. (Fig. 2c).When  is the
bifurcation parameter, the bifurcation analysis revealed a Hopf bifurcation point of
( , , , )
yval kval mval  = (( 163.636364 1047.009483 122.222200 3.012525 )(Fig. 3a). The limit
cycle produced by this Hopf bifurcation point is shown in Fig. 3b . When  is modified to
tanh( ) / 0.2
  the Hopf bifurcation point disappears. (Fig. 3c). Both models show the presence
of limit cycles causing Hopf bifurcations, which can be eliminated using the activation factor
involving the tanh function, confirming the analysis of Sridhar(2024)[35].
When the MNLMPC calculations were performed for model 1,
0 0 0
( ), ( ), ( )
i f i f i f
i i i
t t t t t t
j i j i j i
t t t
xval t yval t zval t
  
  
   were maximized individually, resulting in values of
7.1377e-03, 1000, and 1000, respectively. The overall optimal control problem will involve the

28
minimization of
0 0 0
2 2 2
7.1377 0
( ( ) ) ( ( ) 1000) ( ( ) 0 0
3 1 0 )
i f i f i f
i i i
t t t t t t
j i j i j i
t t t
xval t yval t zval t
e
  
  
    

  
was minimized subject to the equations governing the model. This led to a value of 5.02e-05.
The first of the control variables is implemented, and the rest are discarded. The process is
repeated until the difference between the first and second values of the control variables is the
same. This MNLMPC control value of 1
u obtained was 0.01063. The MNLMPC profiles of
xval, yval, and zval are shown in Figs 4a, 4b and 4c. . The obtained control profile of u1
exhibited noise (Fig. 4d). This was remedied using the Savitzky-Golay Filter. The smoothed-out
version of this profile is shown in Fig.4e.
When the MNLMPC calculations were performed for model 2,
0 0 0
( ), ( ), ( )
i f i f i f
i i i
t t t t t t
j i j i j i
t t t
yval t kval t mval t
  
  
   are maximized individually producing values of
11018.477, 20000, and 13221.553, respectively. The overall optimal control problem will
involve the minimization of
0 0 0
2 2 2
11018.477 1
( ( ) ) ( ( ) 20000) ( ( ) ) )1. 08
3221.553
i f i f i f
i i i
t t t t t t
j i j i j i
t t t
yval t kval t mval t e
  
  
     
  
was minimized subject to the equations governing the model. A scaling factor of 1.e-08 was
used. This minimization led to a value of 0.5679. The first of the control variables is
implemented, and the rest are discarded. The process is repeated until the difference between the
first and second values of the control variables is the same. The MNLMPC control values of
,
  obtained were 0.1394 and 0. The MNLMPC profiles of yval, kval, and mval are shown in
Figs 5a, 5b and 5c. . The obtained control profiles of ,
  exhibited noise (Fig. 5d). This was
remedied using the Savitzky-Golay Filter. The smoothed-out version of this profile is shown in
Fig.5e.
5. CONCLUSIONS
Multiobjective nonlinear model predictive control(MNLMPC) calculations were performed
along with bifurcation analysis on two dynamic financial engineering models. The bifurcation
analysis revealed the existence of Hopf bifurcation points. The Hopf bifurcation points cause
unwanted limit cycles and were eliminated using an activation factor involving the tanh
function.The MNLMPC calculations generated the optimal control and variable profiles. The
control profiles exhibited a considerable amount of noise. This was remedied using the Savitzky-
Golay Filter.
Data Availability Statement
All data used is presented in the paper
Conflict of interest
The author, Dr. Lakshmi N Sridhar has no conflict of interest.

29
ACKNOWLEDGEMENT
Dr. Sridhar thanks Dr. Carlos Ramirez and Dr. Suleiman for encouraging him to write single-
author papers.
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Fig. 1a (bifurcation analysis model 1 showing Hopf bifurcation

31
Fig. 1 b(limit cycle for Model 1)
Fig. 1c(Hopf bifurcation disappears with tanh activation factor in model 1)
Fig. 2a (hopf bifurcation model 2 alfa is bifurcation parameter)
Fig. 2b (limit cycle alfa is bifurcation parameter)

32
Fig. 2c Hopf bifurcation disappears with tanh activation factor alfa is bifurcation parameter
Fig. 3a(Hopf bifurcation model 2 beta is bifurcation parameter)
Fig. 3b(limit cycle beta is bifurcation parameter)
Fig. 3c Hopf bifurcation disappears with tanh activation factor alfa is bifurcation parameter

33
Fig. 4a (xval vs t MNLMPC model 1)
Fig. 4b (yval vs t MNLMPC model 1)
Fig. 4c (zval vs t MNLMPC model 1)

34
Fig. 4d (u1 vs t MNLMPC model 1)
Fig. 4e (u1 with Savitzky Golay filter MNLMPC model 1)
Fig. 5a (yval vs t MNLMPC model 2)

35
Fig. 5b (kval vs t MNLMPC model 2)
Fig. 5c (mval vs t MNLMPC model 2)
Fig. 5d (alfa beta vs t MNLMPC model 2)

36
Fig. 5e (alfa beta (with Savitzky Golay) vs t MNLMPC model 2)

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