Chaos in a generalized van der Pol system
and in its fractional order system
Zheng-Ming Ge
*, Mao-Yuan Hsu
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Accepted 7 March 2006
Abstract
In this paper, chaos of a generalized van der Pol system with fractional orders is studied. Both nonautonomous and
autonomous systems are considered in detail. Chaos in the nonautonomous generalized van der Pol system excited by a
sinusoidal time function with fractional orders is studied. Next, chaos in the autonomous generalized van der Pol
sys-tem with fractional orders is considered. By numerical analyses, such as phase portraits, Poincare´ maps and bifurcation
diagrams, periodic, and chaotic motions are observed. Finally, it is found that chaos exists in the fractional order
sys-tem with the order both less than and more than the number of the states of the integer order generalized van der Pol
system.
2006 Elsevier Ltd. All rights reserved.
1. Introduction
A van der Pol system, which is a typical nonlinear chaotic system has many interesting features, and numerous
appli-cations. It has been used for the design of various systems including biological ones, such as the heartbeats
[1]
or the
generation of action potentials by neurons, acoustic models, the radiation of mobile phones, and as a model of electrical
systems. Chaos, chaos control and chaos synchronization of nonlinear systems, including the van der Pol system have
been subjected to extensive studies
[2–18]
.
Fractional calculus is an old mathematical topic from 17th century. Although it has a long history, applications are
only recent focus of interest. Many systems are known to display fractional order dynamics, such as viscoelastic systems
[19]
, dielectric polarization, electrode–electrolyte polarization, and electromagnetic waves. Furthermore, researchers
have found some systems for which the chaotic motions exist in the fractional orders
[20–24]
.
This paper is organized as follows. In Section
2
, a review and the approximation of the fractional order operator is
presented. In Section
3
, chaos of the generalized van der Pol system and the nonautonomous and autonomous
fractional order generalized van der Pol systems is given. In Section
4
, numerical simulations, such as phase portraits,
Poincare´ maps and bifurcation diagrams, of various nonautonomous and autonomous fractional order generalized van
der Pol systems are presented. In Section
5
, some conclusions are drawn.
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.028
*Corresponding author. Fax: +886 35720634.
E-mail address:zmg@cc.nctu.edu.tw(Z.-M. Ge).
Chaos, Solitons and Fractals 33 (2007) 1711–1745
2. The review and the approximation of fractional order operators
There are many ways to define a fractional differential operator
[25–27]
. The commonly used definition for general
fractional derivative is the Riemann–Liouville definition. The Riemann–Liouville definition of the fractional order
derivative is
D
ayðtÞ ¼
d
ndt
nD
anyðtÞ ¼
1
Cðn aÞ
d
ndt
nZ
t 0yðsÞ
ðt sÞ
anþ1ds;
ð1Þ
where C(Æ) is a gamma function and n is an integer such that n
1 6 a < n. This definition is different from the usual
intuitive definition of derivative.
Thus, it is necessary to develop approximations to the fractional operators using the standard integer order
opera-tors. Fortunately, the Laplace transform which is basic engineering tool for analysing linear systems is still applicable
and works:
L
d
af
ðtÞ
dt
a¼ s
aLff ðtÞg
X
n1 k¼0s
kd
a1kf
ðtÞ
dt
a1k t¼0for all a;
ð2Þ
where n is an integer such that n
1 6 a < n. Upon considering the initial conditions to be zero, this formula reduces to
the more expected form
L
d
af
ðtÞ
dt
a¼ s
aLff ðtÞg.
ð3Þ
Using the algorithm in
[21,28]
, linear transfer function of approximations of the fractional integrator is adopted.
Basically the idea is to approximate the system behavior based on frequency domain arguments. From
[29]
, we get
the table of approximating transfer functions for 1/s
awith different fractional orders, a = 0.1
0.9, in steps of 0.1,
which give the maximum error 2 dB in calculations. These approximations will be used in the following study.
3. The chaos of the generalized van der Pol system and its fractional order form
The van der Pol system
[30]
was first given to study the oscillations in vacuum tube circuits. The equivalent state
space formulation has the form of an autonomous system
dx
1dt
¼ x
2;
dx
2dt
¼ x
1eðx
2 11Þx
2;
ð4Þ
where e is a parameter. The generalized van der Pol system
[31–35]
has the form of a nonautonomous system which is
written as
dx
1dt
¼ x
2;
dx
2dt
¼ x
1eð1 x
2 1Þðc ax
2 1Þx
2þ b sin xt;
ð5Þ
where e, a, b, c are parameters, and x is the circular frequency of the external excitation bsin xt.
Figs. 1 and 2
show the
bifurcation diagram and Lyapunov exponent diagram for the nonautonomous generalized van der Pol system
(5)
, and
Figs. 3–6
are the phase portraits and Poincare´ maps, while Poincare´ maps are taken by period
2px
. The corresponding
nonautonomous fractional order system is
d
ax
1dt
a¼ x
2;
d
bx
2dt
b¼ x
1eð1 x
2 1Þðc ax
2 1Þx
2þ b sin xt;
ð6Þ
where a, b are fractional numbers.
A modified version of Eq.
(6)
is now proposed. The nonautonomous generalized fractional order van der Pol system
(6)
with two states is transformed into an autonomous generalized fractional order van der Pol system with three states
Fig. 1. The bifurcation diagram of the nonautonomous generalized van der Pol system.
Fig. 2. Lyapunov exponent diagram of the nonautonomous generalized van der Pol system for x between 0.1295 and 0.1308.
Fig. 3. The phase portrait and the Poincare´ map of the nonautonomous generalized van der Pol system.
Fig. 4. The phase portrait and the Poincare´ map of the nonautonomous generalized van der Pol system.
Fig. 5. The phase portrait and the Poincare´ map of the nonautonomous generalized van der Pol system.
Fig. 6. The phase portrait and the Poincare´ map of the nonautonomous generalized van der Pol system.
d
ax
1dt
a¼ x
2;
d
bx
2dt
b¼ x
1eð1 x
2 1Þðc ax
2 1Þx
2þ b sin xx
3;
d
cx
3dt
c¼ 1;
ð7Þ
where a, b, c are fractional numbers, in which the original time t in Eq.
(6)
is changed to a new state x
3. When c = 1,
x
3= t, Eq.
(7)
reduces to Eq.
(6)
.
In this paper, we analyse and present simulation results of the chaotic dynamics produced from a new generalized
fractional van der Pol system as the state of fractional order derivatives in Eq.
(6)
are varied from 0.1 to 1.3. It is
observed that the different orders have large influences upon the overall system dynamics.
4. Numerical simulations for the fractional order generalized van der Pol systems
Our study of system
(7)
consists of three parts:
Part 1: c equals one, and a, b take the same fractional numbers. The system is equivalent to a nonautonomous
frac-tional order system with two states, Eq.
(6)
.
Part 2: a, b, c take the same fractional numbers.
Part 3: c equals 1.1 or 0.9, and a, b take the same fractional numbers.
In Part 1, c = 1, Eq.
(7)
reduces to Eq.
(6)
with two states x
1and x
2. a, b take the same fractional numbers and vary
from 1.1 to 0.7 in steps of 0.1.
Fig. 7
shows the bifurcation diagram and
Figs. 8–11
are the phase portraits and Poincare´
maps with a = b = 1.1, while Poincare´ maps are taken by period
2px
.
Fig. 12
shows the bifurcation diagram and
Figs. 13–
16
are the phase portraits and Poincare´ maps with a = b = 0.9.
Fig. 17
shows the bifurcation diagram and
Figs. 18–21
are the phase portraits and Poincare´ maps with a = b = 0.8.
Fig. 22
shows the bifurcation diagram and
Figs. 23–26
are
Fig. 7. The bifurcation diagram of the nonautonomous fractional order system with order a = b = 1.1, c = 1.
Fig. 8. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 1.1, c = 1, x = 0.435.
Fig. 9. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 1.1, c = 1, x= 0.4732.
Fig. 10. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 1.1, c = 1, x= 0.4462.
Fig. 11. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 1.1, c = 1, x= 0.445.
Fig. 13. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.9, c = 1, x= 0.127.
Fig. 12. The bifurcation diagram of the nonautonomous fractional order system with order a = b = 0.9, c = 1.
Fig. 15. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.9, c = 1, x= 0.12624.
Fig. 14. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.9, c = 1, x= 0.1263.
Fig. 16. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.9, c = 1, x= 0.1275.
Fig. 17. The bifurcation diagram of the nonautonomous fractional order system with order a = b = 0.8, c = 1.
Fig. 18. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.8, c = 1, x= 0.135.
Fig. 19. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.8, c = 1, x= 0.133.
Fig. 20. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.8, c = 1, x= 0.13295.
Fig. 21. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.8, c = 1, x= 0.1315.
Fig. 22. The bifurcation diagram of the nonautonomous fractional order system with order a = b = 0.7, c = 1.
Fig. 23. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.7, c = 1, x= 0.315.
Fig. 24. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.7, c = 1, x = 0.32.
Fig. 25. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.7, c = 1, x= 0.31758.
Fig. 26. The phase portrait and Poincare´ maps of the nonautonomous fractional order system with order a = b = 0.7, c = 1, x= 0.31812.
Fig. 27. The bifurcation diagram of the autonomous fractional order system with order a = b = c = 1.1.
Fig. 28. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = c = 1.1, x = 0.37.
Fig. 29. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = c = 1.1, x = 0.36418.
Fig. 30. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = c = 1.1, x = 0.36417.
Fig. 31. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = c = 1.1, x = 0.34.
Fig. 32. The bifurcation diagram of the autonomous fractional order system with order a = b = 0.9, c = 1.1
Fig. 33. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.9, c = 1.1, x = 0.5498.
Fig. 34. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.9, c = 1.1, x = 0.5531.
Fig. 35. The bifurcation diagram of the autonomous fractional order system with order a = b = 0.8, c = 1.1.
Fig. 36. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.8, c = 1.1, x = 0.2851.
Fig. 37. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.8, c = 1.1, x = 0.2807.
Fig. 38. The bifurcation diagram of the autonomous fractional order system with order a = b = 0.7, c = 1.1.
Fig. 39. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.7, c = 1.1, x = 0.141.
Fig. 40. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.7, c = 1.1, x = 0.1408.
Fig. 41. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.7, c = 1.1, x = 0.1404.
Fig. 42. The bifurcation diagram of the autonomous fractional order system with order a = b = 0.6, c = 1.1.
Fig. 43. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.6, c = 1.1, x = 0.13.
Fig. 44. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.6, c = 1.1, x = 0.11.
Fig. 45. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.6, c = 1.1, x = 0.0107.
Fig. 46. The bifurcation diagram of the autonomous fractional order system with order a = b = 0.5, c = 1.1.
Fig. 47. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.5, c = 1.1, x = 0.075.
Fig. 48. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.5, c = 1.1, x = 0.06.
Fig. 49. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.5, c = 1.1, x = 0.038.
Fig. 50. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.5, c = 1.1, x = 0.001.
Fig. 51. The bifurcation diagram of the autonomous fractional order system with order a = b = 0.4, c = 1.1.
Fig. 52. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.4, c = 1.1, x = 0.04.
Fig. 53. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.4, c = 1.1, x = 0.026.
Fig. 54. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.4, c = 1.1, x = 0.014.
Fig. 55. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.4, c = 1.1, x = 0.005.
Fig. 56. The bifurcation diagram of the autonomous fractional order system with order a = b = 0.3, c = 1.1.
Fig. 57. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.3, c = 1.1, x = 0.011.
Fig. 58. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.3, c = 1.1, x = 0.0085.
Fig. 59. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.3, c = 1.1, x = 0.006.
the phase portraits and Poincare´ maps with a = b = 0.7. According to the results of simulation in Part 1, it is found that
the chaotic motions exist in the nonautonomous system with fractional orders. The lowest total fractional order for
chaos existence in this system is 1.4 (2
· 0.7). When the total fractional order is 1.2 (2 · 0.6), no chaos exists.
In Part 2, a, b, c take the same fractional numbers, vary from 1.1 to 0.6 in steps of 0.1.
Fig. 27
shows the bifurcation
diagram and
Figs. 28–31
are the phase portraits and Poincare´ maps with a = b = c = 1.1. When a = b = c and vary
from 0.9 to 0.6 in steps of 0.1, no chaos exists.
In Part 3, c = 1.1, a, b take same fractional numbers and vary from 1.1 to 0.3 in steps of 0.1.
Fig. 32
shows the
bifur-cation diagram and
Figs. 33 and 34
are the phase portraits and Poincare´ maps with a = b = 0.9, c = 1.1.
Fig. 35
shows
the bifurcation diagram and
Figs. 36 and 37
are the phase portraits and Poincare´ maps with a = b = 0.8, c = 1.1.
Fig. 38
shows the bifurcation diagram and
Figs. 39–41
are the phase portraits and Poincare´ maps with a = b = 0.7, c = 1.1.
Fig. 42
shows the bifurcation diagram and
Figs. 43–45
are the phase portraits and Poincare´ maps with a = b = 0.6,
c
= 1.1.
Fig. 46
shows the bifurcation diagram and
Figs. 47–50
are the phase portraits and Poincare´ maps with
a
= b = 0.5, c = 1.1.
Fig. 51
shows the bifurcation diagram and
Figs. 52–55
are the phase portraits and Poincare´ maps
with a = b = 0.4, c = 1.1.
Fig. 56
shows the bifurcation diagram and
Figs. 57–60
are the phase portraits and Poincare´
maps with a = b = 0.3, c = 1.1. According to the results of simulation in Part 3, it is found that the chaotic motions
exist when c takes 1.1 and a, b vary from 0.9 to 0.3 in steps of 0.1. When a, b take the fractional number less than
0.3, no chaos is found.
c
= 0.9, a, b take the same fractional numbers 1.1, 1.2, 1.3 and 1.4, only periodic motions are found, no chaos exists.
5. Concluding remarks
In this paper, chaos of a generalized van der Pol system both with integer order and with fractional orders is studied.
Both autonomous and nonautonomous systems are studied in some detail. It is found that chaotic motions exist in the
nonautonomous generalized van der Pol system excited by a sinusoidal time function. For fractional order systems,
when c = 1, a, b vary from 1.1 to 0.7 in the steps of 0.1, chaos exists. Next, chaotic motions exist when the fraction
orders a = b = c = 1.1. When c = 1.1 with a = b varying from 1.1 to 0.3, chaotic motions also exist.
Fig. 60. The phase portrait and Poincare´ maps of the autonomous fractional order system with order a = b = 0.3, c = 1.1, x = 0.0017.
Acknowledgement
This research was supported by the National Science Council, Republic of China, under Grant No.
NSC94-2212-E-009-013.
References
[1] Glass L. Theory of heart. New York, Heidelberg, Berlin: Springer; 1990.
[2] Mahmoud Gamal M, Farghaly Ahmed AM. Chaos control of chaotic limit cycles of real and complex van der Pol oscillators. Chaos, Solitons & Fractals 2004;21:915–24.
[3] Ge Zheng-Ming, Chen Yen-Sheng. Synchronization of unidirectional coupled chaotic systems via partial stability. Chaos, Solitons & Fractals 2004;21:101–11.
[4] Ge Zheng-Ming, Leu Wei-Ying. Anti-control of chaos of two-degrees-of-freedom louderspeaker system and chaos synchroni-zation of different order systems. Chaos, Solitons & Fractals 2004;20:503–21.
[5] Ge Zheng-Ming, Leu Wei-Ying. Chaos synchronization and parameter identification for louderspeaker system. Chaos, Solitons & Fractals 2004;21:1231–47.
[6] Ge Zheng-Ming, Chen Chien-Cheng. Phase synchronization of coupled chaotic multiple time scale systems. Chaos, Solitons & Fractals 2004;20:639–47.
[7] Ge Z-M, Chang C-M. Chaos synchronization and parameter identification of single time scale brushless dc motors. Chaos, Solitons & Fractals 2004;20:883–903.
[8] Ge Zheng-Ming, Cheng Jui-Wen, Chen Yen-Sheng. Chaos anticontrol and synchronization of three time scales brushless dc motor system. Chaos, Solitons & Fractals 2004;22:1165–82.
[9] Ge Z-M, Cheng J-W. Chaos synchronization and parameter identification of three time scales brushless dc motor system. Chaos, Solitons & Fractals 2005;24:597–616.
[10] Ge Zheng-Ming, Lee Ching-I. Control, anticontrol and synchronization of chaos for an autonomous rotational machine system with time-delay. Chaos, Solitons & Fractals 2005;23:1855–64.
[11] Ge Zheng-Ming, Chen Yen-Sheng. Adaptive synchronization of unidirectional and mutual coupled chaotic systems. Chaos, Solitons & Fractals 2005;26:881–8.
[12] Chen Hsien-Keng, Ge Zheng-Ming. Bifurcation and chaos of a two-degree-of freedom dissipative gyroscope. Chaos, Solitons & Fractals 2005;24:125–36.
[13] Jiang G, Zheng W, Chen G. Global chaos synchronization with channel time-delay. Chaos, Solitons & Fractals 2004;20:267–75. [14] Chen G, Liu S. On generalized synchronization of spatial chaos. Chaos, Solitons & Fractals 2003;15:311–8.
[15] Li Z, Xu D. A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractals 2004;22:477–81. [16] Jiang Guo-Ping, Tang Wallace Kit-Sang, Chen Guanrong. A simple global synchronization criterion for coupled chaotic systems.
Chaos, Solitons & Fractals 2003;15:925–35.
[17] Ge Z-M, Yu C-Y, Chen Y-S. Chaos synchronization and anti-control of a rotational supported simple pendulum. JSME Int J, Ser C 2004;47:233–41.
[18] Ge Z-M, Hsiao C-L, Chen Y-S. Nonlinear dynamics and chaos control for a time delay Duffing system. Int J Nonlinear Sci Numer Simul 2005;6:187–99.
[19] Bagley RL, Calico RA. Fractional order state equations for the control of viscoelastically damped structures. J Guid Control Dynam 1991;14:304–11.
[20] Li C, Chen G. Chaos in the fractional order Chen system and its control. Chaos, Solitons & Fractals 2004;22:549–54. [21] Ahmad Wajdi M, Harb Ahmad M. On nonlinear control design for autonomous chaotic systems of integer and fractional orders.
Chaos, Solitons & Fractals 2003;18:693–701.
[22] Ahmad Wajdi M, Reyad El-Khazali, Yousef Al-Assaf. Stabilization of generalized fractional order chaotic systems using state feedback control. Chaos, Solitons & Fractals 2004;22:141–50.
[23] Ahmad WM. Hyperchaos in fractional order nonlinear systems. Chaos, Solitons & Fractals 2005;26:1459–65.
[24] Nimmo S, Evans AK. The effects of continuously varying the fractional differential order of chaotic nonlinear systems. Chaos, Solitons & Fractals 1999;10:1111–8.
[25] Ahmad Wajdi M, Sprott JC. Chaos in a fractional order autonomous nonlinear systems. Chaos, Solitons & Fractals 2003;16:339–51.
[26] Barbosa Ramiro S, Tenreiro Machado JA, Ferreira Isabel M, Tar Jo´zsef K. Dynamics of the fractional order van der Pol oscillator. In: Proceedings of the second IEEE International Conference on Computational Cybernetics (ICCC’04), ISBN 3-902463-01-5, 2004. p. 373–8.
[27] Li Changpin, Peng Guojun. Chaos in Chen’s system with a fractional order. Chaos, Solitons & Fractals 2004;22:443–50. [28] Charef A, Sun HH, Tsao YY, Onaral B. Fractal system as represented by singularity function. IEEE Trans Automat Contr
1992;37:1465–70.
[29] Hartley TT, Lorenzo CF, Qammer HK. Chaos in a fractional order Chua’s system. IEEE Trans CAS—I 1995;42:485–90. [30] van der Pol B. Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode). London, Edinburgh, and
Dublin Philosphical Magazine 1927;3:65–80.
[31] Bi Qinsheng. Dynamical analysis of two coupled parametrically excited van der Pol oscillators. Int J Nonlinear Mech 2004;39:33–54.
[32] dos Santos AM, Lopes SR, Viana RL. Rhythm synchronization and chaotic modulation of coupled van der Pol oscillators in a model for the heartbeat. Physica A 2004;338:335–55.
[33] Addo-Asah W, Akpati HC, Mickens RE. Investigation of a generalized van der Pol oscillator differential equation. J Sound Vibr 1995;179:733–5.
[34] Waluya SB, van Horssen WT. On the periodic solutions of a generalized nonlinear van der Pol oscillator. J Sound Vibr 2003;268:209–15.
[35] Mickens RE. Analysis of nonlinear oscillators having non-polynomial elastic terms. J Sound Vibr 2002;255:789–92.