Investigation of Dependent Rikitake System to Initiation Point

International Journal of Applied Power Engineering (IJAPE)
Vol. 4, No. 2, August 2015, pp. 70~76
ISSN: 2252-8792  70
Journal homepage: http://iaesjournal.com/online/index.php/IJAPE
Investigation of Dependent Rikitake System to Initiation Point
*Yousof Gholipour, **Amin Ramezani, ***Mahmood Mola
*Department of Electrical Eng., Mehriz Branch. Islamic Azad University. Mehriz.Iran.
**Department of Electrical Eng., Faculty of Eng., Tarbiat Modarres University. Tehran, Iran
***Department of Electrical Eng., University of A. A. Boroujerdi, Boroujerd, Iran.
Article Info ABSTRACT
Article history:
Received Mei 2, 2015
Revised Jun 10, 2015
Accepted Jul 23, 2015
In this paper we investigate depending of the Rikitake system to initiation
point, and monitor changing behavior of this system. We will have 4
initiation points in Cartesian system. We at 4 positions, will monitor behavior
of this system, while holding constant other values, and after per position,
will draw operation of system on axes of x, y, z and 3-D plot. We want to
know, what is the effect of initiation point on Rikitake system? Numerical
simulations to illustrate the effect of initiation point are presented, and at the
end conclusions and comparing the states together are obtained.Keyword:
Chaos
Dependent.
Initiation point
Rikitake system
Sinusoid
Stability Copyright © 2015 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Yousof Gholipour,
Department of Electrical Engineering,
Islamic Azad University,
Mehriz Branch, Mehriz.Iran.
Email: y.gholipour@yahoo.com
1. INTRODUCTION
At the first, I want to explain, the physics rules behind of Rikitake system for more and better
understanding behavior of Rikitake system. The Rikitake system is a mathematical model that obtained from
a simple mechanical system that it used by Rikitake (Denis de Carvalho Braga et al., 2010) to study the
reversals of the Earth’s magnetic field.
This system made from two magnetic disks (D1,D2) and two shafts and two coils (L1, L2) see
Figure1.This system run by an external power supply on two brushes (terminals) for a moment (for start after
starting the system power supply will disconnect) and the system will be run. There is a G for both
conducting disks (D1,D2) Both disks rotate in the same sense, one with an angular velocity of ω1 around the
axis of rotation A1and the other with angular velocity ω2 around the axis of rotation A2. And there is too two
I1 and I2 that these made from L1 and L2 the coils the cross connect to D2 and D1 the currents in coils cause a
magnetic field that it is opposite the magnetic field of disks (D1, D2) The opposition was so continued until
the currents in two loops L1 and L2 be reverse this reversal in currents means that the magnetic field (emf) of
earth be reversal. This procedure happens for magnetic field of the earth but without control and any
prediction this procedure called chaos. Chaos Is very sensitive to initiation point. This system under
particular initial conditions this process becomes chaotic. (Ahmad Harb, Nabil Ayoub, 2013).
Many works have described the dynamics of Rikitake system. Historically E. C. Bullard has
described the magnetic field of the earth, and too described the behavior of earth’s magnetic field and its
simulations. He then in 1955 the similar behavior between a set of homogeneous dynamos and terrestrial
magnetism and described the stability of a homopolar dynamo (Liu Xiao-Jun, et.al, 1999), analyzed the

IJAPE ISSN: 2252-8792 
Investigation of Dependent Rikitake System to Initiation Point (Yousof Gholipour)
71
dynamics of the Rikitake system to describe the reversals of the Earth’s magnetic field. They concluded that
the chaotic behavior of the system can be used for simulation the geomagnetic field reversal.. The Rikitake
chaotic attractor was studied by several authors (T. McMillen, 1999) and (Mohammad Javidi,et al., 2013) has
studied the shape and dynamics of the Rikitake attractor. (J. Llibre .et al., 2009) used the Poincare
compactification to study the dynamics of the Rikitake system at infinity. Chien- Chih Chen et al. have
studied the stochastic resonance in the periodically forced Rikitake dynamo.( Chien- Chih Chen et al., 2007)
In the past years, many scientists are working on control the chaotic behaviors. Harb and Harb have designed
a nonlinear controller to control the chaotic behavior in the phase-locked loop by means of nonlinear control
(Harb and Harb, 2007). Ahmad Harb have designed a controller to control the unstable chaotic oscillations by
means of back stepping method (Ahmad Harb, Bassam Harb, 2004). The modern nonlinear theory for
bifurcation has been discussed and chaos theory was used to investigated dynamics of the Rikitake system
and an equation was found that was the same as the mathematical model of the Lorenz system(Ahmad Harb,
Nabil Ayoub, 2013)The synchronization for chaotic of Rikitake system was studied by several authors.
(Mohamad Ali khan, 2012) and (Carlos Aguilar-Ibañez, 2010) and (U.E. Vincent, 2011), (Yousof Gholipour
and Mahmood moula, 2014) Investigated stability of Rikitake system with changing the resistance wires of
system.
In this paper we suppose that the all situations and values are constant, and we change initiation
point E(x0,y0,z0), and compare states of system in different points.
2. MODELING AND ANALYSIS
Figure 1. The Rikitake dynamo is composed of two disk dynamos coupled to one another.
The mathematical model of system is as follows:
 















yxz
xazuyy
zyuxx
1'
'
'
(1)
Where (x, y, z) R3
are the state variables and a, u > 0 parameters are positive. Note that the
mathematical model of Rikitake system is a quadratic system in R3
. The choice value of the parameters a and
u reflects a physical meaning in the Rikitake model.
It is well known that system (1) has two equilibrium points E+ = (x0, y0, z0) ; E - = (−x0,−y0, z0) In
order to study the stability of E±, it is only sufficient to study the stability of the equilibrium point E+.

 ISSN: 2252-8792
IJAPE Vol. 4, No. 2, August 2015 : 70 – 76
72

















u
uaa
Z
uaa
u
Y
u
uaa
X
2
242
242
2
2
242
(2)
Where:
  GL
CM
u
GM
LC
Ra  21

(3)
3. SIMULATION RESULT AT 4 POINTS
In this section we will investigate dependent Rikitake system to initiation point. While all the
situations and values are constant and suppose chaos situation for Rikitake system (Yousof Gholipour and
Mahood Moula, 2014). When a=3 and u=1.2, change initiation point E(x0,y0,z0) and observe amount of
dependent and change of Rikitake system.
We consider 4 positions:
1- E+(x0,y0,z0)=(1,1,1)
2- E+(x0,y0,z0)=(0.5,0.5,0.5)
3- E+(x0,y0,z0)=(3,2,1)
4- E-(x0,y0,z0)=(-1,-2,-3)
Now display the numerical solution of Rikitake system. For all positions the situations and values
are constant, and set u=1.2, a=3 and steps h=0.01. We plot the system behavior for four positions around X,
Y, Z axes and 3-D plot.
1. E+(x0,y0,z0)=(1,1,1)
Figure 2. Rikitake System Behavior for E = (1, 1, 1)
0 5 10 15 20 25 30 35 40 45 50
-5
0
5
X
0 5 10 15 20 25 30 35 40 45 50
-2
0
2
4
Y
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
Z
Time(s)

IJAPE ISSN: 2252-8792 
73
Figure 3. The Limit Cycle. Amount of Stability System When E = (1,1,1)
2. E+(x0,y0,z0)=(0.5,0.5,0.5)
Figure 4. Rikitake System Behavior for E = (0.5,0.5,0.5)
Figure 5. The Limit Cycle Amount of Stability System When E = (0.5,0.5,0.5)
-4
-3
-2
-1
0
1
2
3
4
5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Z
Limit Cycle
X
Y
0 5 10 15 20 25 30 35 40 45 50
-5
0
5
X
0 5 10 15 20 25 30 35 40 45 50
-4
-2
0
2
4
Y
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
Z
Time(s)
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
-5
-4
-3
-2
-1
0
1
2
3
4
Z
Limit Cycle
X
Y

 ISSN: 2252-8792
74
3. E+(x0,y0,z0)=(3,2,1)
Figure 6. Rikitake System Behavior for E = (3,2,1)
Figure 7. The Limit Cycle Amount of Stability System When E = (3,2,1)
4. E-(x0,y0,z0)=(-1,-2,-3)
0 5 10 15 20 25 30 35 40 45 50
-5
0
5
X
0 5 10 15 20 25 30 35 40 45 50
-4
-2
0
2
4
Y
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
Z
Time(s)
0
1
2
3
4
5
6
7
-3
-2
-1
0
1
2
3
-5
-4
-3
-2
-1
0
1
2
3
4
Z
Limit Cycle
X
Y
0 5 10 15 20 25 30 35 40 45 50
-10
-5
0
5
X
0 5 10 15 20 25 30 35 40 45 50
-4
-2
0
2
Y
0 5 10 15 20 25 30 35 40 45 50
-5
0
5
10
Z
Time(s)

IJAPE ISSN: 2252-8792 
75
Figure 8. Rikitake System Behavior for E = (-1,-2,-3)
Figure 9. The Limit Cycle Amount of Stability System When E = (-1,-2,-3)
In the numerical simulations we see that the initial point has effect on behavior of system, the effect
of initial point on this system is too deep, and all behavior of the system changed when initial point changed.
4. CONCLUSION
In this paper we, Investigated of dependent Rikitake system to initiation point. Firstly, discussed the
physics rules behind Rikitake system and, last works about this system, after that presented the Rikitake
model and at the behavior of system at four positions, investigated amount of dependent system to initiation
point, and plotted axes x, y, z and 3-D plot, while all the situations and values was constant and suppose
chaos situation for Rikitake system and a=3, u=1.2. We concluded from numerical simulations that, the
Rikitake system is sharply dependent to initiation point. By note that, the studies showed that this system has
intrinsic chaotic behavior, change in initiation point, will change behavior of system. Because of this Intrinsic
properties we should adjust a & u (Equation 3) of this system carefully for have a system with fix behavior.
ACKNOWLEDGEMENTS
We acknowledge our friend, Mehdi fatemi and the Associate Editor and anonymous reviewers for
their valuable comments and suggestions that have helped us to improving our paper.
REFERENCES
[1] Ahmad Harb, Bassam Harb, 2004. Chaos control of third-order phase-locked loops using backstepping nonlinear
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[3] Carlos Aguilar-Ibañez, Rafael Martinez-Guerra, Ricardo Aguilar-López, Juan L. Mata-Machuca, 2010.
Synchronization and parameter estimations of an uncertain Rikitake system, Physics Letters A 374, 3625–3628.
[4] C.-C. Chen, C.-Y. Tseng, 2007. A study of stochastic resonance in the periodically forced Rikitake dynamo. Terr.
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-4
-2
0
2
4
6
8
-4
-3
-2
-1
0
1
2
-6
-5
-4
-3
-2
-1
0
1
2
3
4
Z
Limit Cycle
X
Y

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Investigation of Dependent Rikitake System to Initiation Point