Chaos in a modified van der Pol system and in its
fractional order systems
Zheng-Ming Ge
*, An-Ray Zhang
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Accepted 2 December 2005
Communicated by Prof. Thomaz Kapitaniak
Abstract
Chaos in a modified van der Pol system and in its fractional order systems is studied in this paper. It is found that
chaos exists both in the system and in the fractional order systems with order from 1.8 down to 0.8 much less than the
number of states of the system, two. By phase portraits, Poincare´ maps and bifurcation diagrams, the chaotic behaviors
of fractional order modified van der Pol systems are presented.
Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction
The topic of fractional calculus is enjoying growing interest not only among mathematicians, but also among
phys-icists and engineers. In recent years, many scholars have devoted themselves to study the applications of the fractional
order system to physics and engineering such as viscoelastic systems
[1]
, dielectric polarization, and electromagnetic
waves. More recently, there is a new trend to investigate the control
[2]
and dynamics
[3–10]
of the fractional order
dynamical systems
[11–13,5]
. In
[1]
it has been shown that nonlinear chaotic systems can still behave chaotically when
their models become fractional. In
[11]
, chaos control was investigated for fractional chaotic systems by the
‘‘backstep-ping’’ method of nonlinear control design. In
[12,13]
, it was found that chaos exists in a fractional order Chen system
with order less than 3. Linear feedback control of chaos in this system was also studied. In
[5]
, chaos synchronization
of fractional order chaotic systems were studied. The existence and uniqueness of solutions of initial value problems for
fractional order differential equations have been studied in the literature
[14–17]
. In this paper, chaotic behaviors of a
fractional order modified van der Pol system are studied by phase portraits
[18–23]
, Poincare´ maps
[24–27]
and
bifur-cation diagrams
[28–37]
. It is found that chaos exists in this system with order from 1.8 down to 0.8 much less than the
number of states of the system. Linear transfer function approximations of the fractional integrator block are calculated
for a set of fractional orders in [0.1, 0.9] based on frequency domain arguments
[38]
.
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.024
*Corresponding author. Tel.: +886 357 12121; fax: +886 357 20634.
E-mail address:zmg@cc.nctu.edu.tw(Z.-M. Ge).
Chaos, Solitons and Fractals 32 (2007) 1791–1822
This paper is organized as follows. In Section
2
, the fractional derivative and its approximation are introduced. In
Section
3
, a modified van der Pol system and the corresponding fractional order system are presented. In Section
4
,
numerical simulations are given. In Section
5
, conclusions are drawn.
2. A fractional derivative and its approximation
There are several definitions of fractional derivatives. The commonly used definition for a general fractional
deriv-ative is the Riemann–Liouville definition
[39]
, which is given by
d
qfðtÞ
dt
q¼
1
Cðn qÞ
d
ndt
nZ
t 0f
ðsÞ
ðt sÞ
qnþ1ds
ð1Þ
where C(Æ) is the gamma function and n is an integer such that n
1 < q < n. This definition is different from the usual
intuitive definition of derivative. Fortunately, the basic engineering tool for analyzing linear systems, the Laplace
trans-form, is still applicable and works as one would expect
L
d
qf
ðtÞ
dt
q¼ s
qLff ðtÞg
X
n1 k¼0s
kd
q1kf
ðtÞ
dt
q1k t¼0;
for all q;
ð2Þ
where n is an integer such that n
1 < q < n . Upon considering the initial conditions to be zero, this formula reduces to
the more expected form
L
d
qf
ðtÞ
dt
q¼ s
qLff ðtÞg
ð3Þ
An efficient method is to approximate fractional operators by using standard integer order operators. In
[40–44]
, an
effective algorithm is developed to approximate fractional order transfer functions. Basically the idea is to approximate
the system behavior based on frequency domain arguments. By utilizing frequency domain techniques based on Bode
diagrams, one can obtain a linear approximation of the fractional order integrator, the order of which depends on the
desired bandwidth and discrepancy between the actual and the approximate magnitude Bode diagrams. In Table 1 of
[38]
, approximations for
1sq
with q = 0.1–0.9 in steps 0.1 are given, with errors of approximately 2 dB. These
approxi-mations are used in the following simulations.
3. A modified van der Pol system and the corresponding fractional order system
Firstly, a van der Pol
[45–47]
oscillator driven by a periodic force is considered. The equation of motion can be
writ-ten as
€
x
þ ux þ a_xðx
21Þ b sin xt ¼ 0
ð4Þ
In Eq.
(4)
, the linear term stands for a conservative harmonic force which determines the intrinsic oscillation frequency.
The self-sustaining mechanism which is responsible for the perpetual oscillation rests on the nonlinear term. Energy
exchange with the external agent depends on the magnitude of displacement
jxj and on the sign of velocity _x. During
a complete cycle of oscillation, the energy is dissipated if displacement x(t) is large than one, and that energy is fed-in if
jxj < 1. The time-dependent term stands for the external driving force with amplitude b and frequency x. Eq.
(4)
can be
rewritten as two first order equations
_x
¼ y
_y
¼ ux þ að1 x
2Þy þ b sin xt
ð5Þ
The modified van der Pol system and its fractional order system studied in this paper are
d
ax
dt
a¼ y
d
by
dt
b¼ x þ að1 x
2Þy þ bz
_z
¼ w
_
w
¼ cz dz
38
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
ð6Þ
where a, b are integer numbers and fractional numbers, respectively. System
(6)
can be separated into two parts
d
ax
dt
a¼ y
d
by
dt
b¼ x þ að1 x
2Þy þ bz
8
>
>
<
>
>
:
ð7Þ
and
_z
¼ w
_
w
¼ cz dz
3ð8Þ
In Eq.
(5)
changing the integral order derivatives to the fractional order derivations and replacing sin xt by z which is
the periodic time function solution of the nonlinear oscillator
(8)
, we obtain system
(7)
. In Eq.
(8)
if d = 0, z is a
sinu-soidal function of time. Now d 5 0, z is a periodic motion of time but not a sinusinu-soidal function of time. As a result,
system
(7)
can be considered as a nonautonomous system with two states, while system
(6)
consisting of Eq.
(7)
and Eq.
(8)
can be considered as an autonomous system with four states. When a = b = 1, Eq.
(6)
is the modified van der Pol
system.
4. Numerical simulations
In this section, phase portraits and bifurcation diagrams are studied for system
(6)
for a + b 6 2. Three parameters
a, c, d are chosen as a = 5, c = 0.01, d = 0.001. A time step of 0.001 is used.
Nine cases are studied:
Case 1: Let a = b = 1.
Fig. 1
shows the bifurcation diagram of the 2 order system. It is shown that chaos exists when
b
2 [0, 1.0].
Fig. 2
is the phase portrait of chaotic motion with b = 1.0.
Figs. 3–5
are phase portraits of periodic
motions with b = 1.1, 1.5, 3, respectively.
Case 2: Let a = 0.9, b = 0.9.
Fig. 6
shows the bifurcation diagram of the 1.8 order system. It is shown that chaos exists
when b
2 [0, 9.7].
Fig. 7
is the phase portrait of chaotic motion with b = 9.7.
Figs. 8–11
are phase portraits of
periodic motions with b = 9.8, 14, 23, 40, respectively.
Case 3: Let a = 0.9, b = 0.8.
Fig. 12
shows the bifurcation diagram of the 1.7 order system. It is shown that chaos
exists when b
2 [0, 9.8].
Fig. 13
is the phase portrait of chaotic motion with b = 9.8.
Figs. 14–17
are phase
por-traits of periodic motions with b = 9.9, 12, 25, 30, respectively.
Case 4: Let a = 0.8, b = 0.9.
Fig. 18
shows the bifurcation diagram of the 1.7 order system. It is shown that chaos
exists when b
2 [0, 10.1].
Fig. 19
is the phase portrait of chaotic motion with b = 10.1.
Figs. 20–24
are phase
portraits of periodic motions with b = 10.2, 15, 23, 30, 45, respectively.
Case 5: Let a = 0.8, b = 0.8.
Fig. 25
shows the bifurcation diagram of the 1.6 order system. It is shown that chaos
exists when b
2 [0, 10.1].
Fig. 26
is the phase portrait of chaotic motion with b = 10.1.
Figs. 27–31
are phase
portraits of periodic motions with b = 10.2, 14.5, 20, 35, 40, respectively.
Case 6: Let a = 0.7, b = 0.7.
Fig. 32
shows the bifurcation diagram of the 1.4 order system. It is shown that chaos
exists when b
2 [0, 7.9].
Fig. 33
is the phase portrait of chaotic motion with b = 7.9.
Figs. 34–37
are phase
por-traits of periodic motions with b = 8.0, 15, 35, 40, respectively.
Case 7: Let a = 0.6, b = 0.6.
Fig. 38
shows the bifurcation diagram of the 1.2 order system. It is shown that chaos
exists when b
2 [0, 6.0].
Fig. 39
is the phase portrait of chaotic motion with b = 6.0.
Figs. 40–44
are phase
por-traits of periodic motions with b = 6.1, 6.5, 9.5, 20, 45, respectively.
Case 8: Let a = 0.5, b = 0.5.
Fig. 45
shows the bifurcation diagram of the 1.0 order system. It is shown that chaos
exists when b
2 [0, 4.2].
Fig. 46
is the phase portrait of chaotic motion with b = 2.
Figs. 47–49
are phase
por-traits of periodic motions with b = 6.0, 10, 15, respectively.
Case 9: Let a = 0.4, b = 0.4.
Fig. 50
shows the bifurcation diagram of the 0.8 order system. It is shown that chaos
exists when b
2 [0, 1.8].
Fig. 51
is the phase portrait of chaotic motion with b = 0.7.
Figs. 51–53
are phase
por-traits of periodic motions with b = 3.5, 6.0 respectively. When we tried to reduce the total order to 0.6, the
phase portraits become periodic motions, as shown in
Fig. 54
and
Fig. 55
, for any b value.
Fig. 2. The phase portrait for a = b = 1, b = 1.0. Fig. 1. The bifurcation diagram for a = b = 1.
Fig. 4. The phase portrait for a = b = 1, b = 1.5. Fig. 3. The phase portrait for a = b = 1, b = 1.1.
Fig. 6. The bifurcation diagram for a = 0.9, b = 0.9. Fig. 5. The phase portrait for a = b = 1, b = 3.
Fig. 8. The phase portrait for a = 0.9, b = 0.9, b = 9.8. Fig. 7. The phase portrait for a = 0.9, b = 0.9, b = 9.7.
Fig. 9. The phase portrait for a = 0.9, b = 0.9, b = 14.
Fig. 10. The phase portrait for a = 0.9, b = 0.9, b = 23. 1798 Z.-M. Ge, A.-R. Zhang / Chaos, Solitons and Fractals 32 (2007) 1791–1822
Fig. 11. The phase portrait for a = 0.9, b = 0.9, b = 40.
Fig. 12. The bifurcation diagram for a = 0.9, b = 0.8.
Fig. 13. The phase portrait for a = 0.9, b = 0.8, b = 9.8.
Fig. 14. The phase portrait for a = 0.9, b = 0.8, b = 9.9. 1800 Z.-M. Ge, A.-R. Zhang / Chaos, Solitons and Fractals 32 (2007) 1791–1822
Fig. 15. The phase portrait for a = 0.9, b = 0.8, b = 12.
Fig. 16. The phase portrait for a = 0.9, b = 0.8, b = 25.
Fig. 17. The phase portrait for a = 0.9, b = 0.8, b = 30.
Fig. 18. The bifurcation diagram for a = 0.8, b = 0.9.
Fig. 19. The phase portrait for a = 0.8, b = 0.9, b = 10.1.
Fig. 20. The phase portrait for a = 0.8, b = 0.9, b = 10.2.
Fig. 21. The phase portrait for a = 0.8, b = 0.9, b = 15.
Fig. 22. The phase portrait for a = 0.8, b = 0.9, b = 23. 1804 Z.-M. Ge, A.-R. Zhang / Chaos, Solitons and Fractals 32 (2007) 1791–1822
Fig. 23. The phase portrait for a = 0.8, b = 0.9, b = 30.
Fig. 24. The phase portrait for a = 0.8, b = 0.9, b = 45.
Fig. 25. The bifurcation diagram for a = 0.8, b = 0.8.
Fig. 26. The phase portrait for a = 0.8, b = 0.8, b = 10.1. 1806 Z.-M. Ge, A.-R. Zhang / Chaos, Solitons and Fractals 32 (2007) 1791–1822
Fig. 27. The phase portrait for a = 0.8, b = 0.8, b = 10.2.
Fig. 28. The phase portrait for a = 0.8, b = 0.8, b = 14.5.
Fig. 29. The phase portrait for a = 0.8, b = 0.8, b = 20.
Fig. 30. The phase portrait for a = 0.8, b = 0.8, b = 35. 1808 Z.-M. Ge, A.-R. Zhang / Chaos, Solitons and Fractals 32 (2007) 1791–1822
Fig. 31. The phase portrait for a = 0.8, b = 0.8, b = 40.
Fig. 32. The bifurcation diagram for a = 0.7, b = 0.7.
Fig. 33. The phase portrait for a = 0.7, b = 0.7, b = 7.9.
Fig. 34. The phase portrait for a = 0.7, b = 0.7, b = 8.0. 1810 Z.-M. Ge, A.-R. Zhang / Chaos, Solitons and Fractals 32 (2007) 1791–1822
Fig. 35. The phase portrait for a = 0.7, b = 0.7, b = 15.
Fig. 36. The phase portrait for a = 0.7, b = 0.7, b = 35.
Fig. 37. The phase portrait for a = 0.7, b = 0.7, b = 40.
Fig. 38. The bifurcation diagram for a = 0.6, b = 0.6.
Fig. 39. The phase portrait for a = 0.6, b = 0.6, b = 6.0.
Fig. 40. The phase portrait for a = 0.6, b = 0.6, b = 6.1.
Fig. 41. The phase portrait for a = 0.6, b = 0.6, b = 6.5.
Fig. 42. The phase portrait for a = 0.6, b = 0.6, b = 9.5. 1814 Z.-M. Ge, A.-R. Zhang / Chaos, Solitons and Fractals 32 (2007) 1791–1822
Fig. 43. The phase portrait for a = 0.6, b = 0.6, b = 20.
Fig. 44. The phase portrait for a = 0.6, b = 0.6, b = 45.
Fig. 45. The bifurcation diagram for a = 0.5, b = 0.5.
Fig. 46. The phase portrait for a = 0.5, b = 0.5, b = 2.
Fig. 47. The phase portrait for a = 0.5, b = 0.5, b = 6.
Fig. 48. The phase portrait for a = 0.5, b = 0.5, b = 10.
Fig. 49. The phase portrait for a = 0.5, b = 0.5, b = 15.
Fig. 50. The bifurcation diagram for a = 0.4, b = 0.4.
Fig. 51. The phase portrait for a = 0.4, b = 0.4, b = 0.7.
Fig. 52. The phase portrait for a = 0.4, b = 0.4, b = 3.5.
Fig. 53. The phase portrait for a = 0.4, b = 0.4, b = 6.0.
Fig. 54. Phase portrait (b = 0.01).
5. Conclusions
Chaos in modified van der Pol system and in its fractional order systems is studied in this paper. It is found that the
range of the chaos in the system gradually decreases as the total order number a + b decreases. Nine cases for
0.8 6 (a + b) 6 2.0 are studied. The lowest total order for chaos existence in the system is found to be 0.8.
Acknowledgement
This research was supported by the National Science Council, Republic of China under grant number NSC
94-2212-E-009-013.
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