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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

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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

a

yðtÞ ¼

d

n

dt

n

D

an

yðtÞ ¼

1

Cðn  aÞ

d

n

dt

n

Z

t 0

yðsÞ

ðt  sÞ

anþ1

ds;

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

a

f

ðtÞ

dt

a





¼ s

a

Lff ðtÞg 

X

n1 k¼0

s

k

d

a1k

f

ðtÞ

dt

a1k





t¼0

for 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

a

f

ðtÞ

dt

a





¼ s

a

Lff 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

a

with 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

1

dt

¼ x

2

;

dx

2

dt

¼ x

1

 eðx

2 1

 1Þx

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

1

dt

¼ x

2

;

dx

2

dt

¼ x

1

 eð1  x

2 1

Þðc  ax

2 1

Þx

2

þ 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

2p

x

. The corresponding

nonautonomous fractional order system is

d

a

x

1

dt

a

¼ x

2

;

d

b

x

2

dt

b

¼ x

1

 eð1  x

2 1

Þðc  ax

2 1

Þx

2

þ 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

(3)

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.

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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.

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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.

(6)

d

a

x

1

dt

a

¼ x

2

;

d

b

x

2

dt

b

¼ x

1

 eð1  x

2 1

Þðc  ax

2 1

Þx

2

þ b sin xx

3

;

d

c

x

3

dt

c

¼ 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

1

and 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

2p

x

.

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Acknowledgement

This research was supported by the National Science Council, Republic of China, under Grant No.

NSC94-2212-E-009-013.

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數據

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. 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.
+7

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