Anticontrol of chaos of the fractional order modified
van der Pol systems
Zheng-Ming Ge
*, An-Ray Zhang
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC
Abstract
Anticontrol of chaos of fractional order modified van der Pol systems is studied. Addition of a constant term and
addi-tion of kjxjsin x term where x is a state of the system are used to anticontrol the system effectively. By applying numerical
results, phase portrait, Poincare´ maps and bifurcation diagrams a variety of the phenomena of the chaotic motion can be
presented. Finally, it can be find that chaos under these procedures exists in the fractional order systems of a modified van
der Pol system.
Ó 2006 Elsevier Inc. All rights reserved.
Keywords: Chaos; Fractional order system; van der Pol equation; Modified van der Pol system; Anticontrol
1. Introduction
Anticontrol
[1–7]
and synchronization
[8–16]
of chaos have received great attention for many research
activities in recent years. Anticontrol is an interesting, new and challenging phenomenon
[17–19]
. As a reverse
process of suppressing or eliminating chaotic behaviors in order to reduce the complexity of an individual
sys-tem or a coupled syssys-tem, anticontrol of chaos aims at creating or enhancing the syssys-tem complexity for some
special applications. More precisely, anticontrolling chaos is to generate some chaotic behaviors from a given
system, which is non-chaotic or even is stable originally. By fully exploiting the intrinsic non-linearity, this
‘‘control’’ technique provides another dimension for feedback systems design. Its potential applications can
be easily found in many fields, including typically physics, biology, engineering, and medical as well as social
sciences.
In addition, the topic of fractional calculus is enjoying growing interest not only among mathematicians,
but also among physicists 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
[20]
,
dielectric polarization, and electromagnetic waves. More recently, there is a new trend to investigate the
con-trol
[21]
and dynamics
[22–30]
of the fractional order dynamical systems
[31–34]
. In
[20]
it has been shown that
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.086
* Corresponding author.
non-linear chaotic systems can still behave chaotically when their models become fractional. In
[32,33]
, it was
found that chaos exists in a fractional order Chen system with order less than 3.
In this paper, anticontrol of chaos of modified van der Pol systems
[35–38]
in fractional order form are
studied. This paper is organized as follows. In Section
2
, a 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
derivative is the Riemann–Liouville definition
[39]
, which is given by
d
qf
ðtÞ
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 transform, 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
Fig. 1. (a) The bifurcation diagram for a = b = 0.9. (b) The phase portrait for a = b = 0.9, k = 0. (c) The phase portrait for a = b = 0.9, k = 1.05. (d) The phase portrait for a = b = 0.9, k = 1.1.
L
d
q
f
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
[45]
, approximations for
1sq
with q = 0.1–0.9 in steps 0.1 are given,
with errors of approximately 2 dB. These approximations are used in the following simulations.
3. A modified van der Pol system and the corresponding fractional order system
Firstly, a van der Pol oscillator driven by a periodic force is considered. The equation of motion can be
written 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
non-linear 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:
Fig. 2. (a) The bifurcation diagram for a = b = 0.8. (b) The phase portrait for a = b = 0.8, k = 0. (c) The phase portrait for a = b = 0.8, k = 0.87. (d) The phase portrait for a = b = 0.8, k = 1.05.
Fig. 4. (a) The bifurcation diagram for a = b = 0.6. (b) The phase portrait for a = b = 0.6, k = 0. (c) The phase portrait for a = b = 0.6, k = 1.291. (d) The phase portrait for a = b = 0.6, k = 1.298.
Fig. 3. (a) The bifurcation diagram for a = b = 0.7. (b) The phase portrait for a = b = 0.7, k = 0. (c) The phase portrait for a = b = 0.7, k = 1.5. (d) The phase portrait for a = b = 0.7, k = 1.6.
_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
dax dta
¼ y;
dby dtb¼ x þ að1 x
2Þy þ bz;
_z
¼ w;
_
w
¼ cz dz
3;
8
>
>
>
>
<
>
>
>
>
:
ð6Þ
where a, b are integer numbers and fractional numbers, respectively.
System
(6)
can be separated into two parts:
dax dta
¼ y;
dby dtb¼ x þ að1 x
2Þy þ bz
(
ð7Þ
and
_z
¼ w;
_
w
¼ cz dz
3:
ð8Þ
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 non-linear oscillator
(8)
, we obtain system
(7)
. In Eq.
(8)
if
d = 0, z is a sinusoidal function of time. Now d 5 0, z is a periodic motion of time but not a sinusoidal
func-tion of time. As a result, system
(7)
can be considered as a non-autonomous system with two states, while
sys-tem
(6)
consisting of Eqs.
(7) and (8)
can be considered as an autonomous system with four states. When
a
= b = 1, Eq.
(6)
is the modified van der Pol system.
Fig. 5. (a) The bifurcation diagram for a = b = 0.5. (b) The phase portrait for a = b = 0.5, k = 0. (c) The phase portrait for a = b = 0.5, k = 1.35. (d) The phase portrait for a = b = 0.5, k = 1.38.
4. Numerical simulations
Anticontrol of chaos is making a non-chaotic dynamical system chaotic. This means that the regular
behav-ior will be destroyed and replaced by chaotic behavbehav-ior.
We will show that the anticontrols by means of addition of a constant term k or addition of a non-linear
term kjxjsin x are effective. The results are shown by numerical simulations, such as phase portrait, Poincare´
maps and bifurcation diagrams.
Case 1: We add a constant term k in the second equation of system
(6)
and it become:
dax dta
¼ y;
dby dtb¼ x þ að1 x
2Þy þ bz þ k;
_z
¼ w;
_
w
¼ cz dz
3:
8
>
>
>
>
<
>
>
>
>
:
ð9Þ
In system
(9)
, the parameter b is adjusted to achieve periodic motion for different a and b when k = 0. a, c, d are
fixed and they are chosen as a = 5, c = 0.01, d = 0.001.
1. Let a = b = 0.9, b = 2.5, and k
2 [0.9, 1.2].
Fig. 1
(a) shows the bifurcation diagram of the 1.8 order system.
Fig. 1
(b)–(d) are the phase portraits with k = 0, 1.05, 1.1.
2. Let a = b = 0.8, b = 1.5, and k
2 [0.6, 1.2].
Fig. 2
(a) shows the bifurcation diagram of the 1.6 order system.
Fig. 2
(b)–(d) are the phase portraits with k = 0, 0.87, 1.05.
3. Let a = b = 0.7, b = 1.3, and k
2 [0.1, 1.6].
Fig. 3
(a) shows the bifurcation diagram of the 1.4 order system.
Fig. 3
(b)–(d) are the phase portraits with k = 0, 1.5, 1.6.
4. Let a = b = 0.6, b = 1, and k
2 [1.282, 1.312].
Fig. 4
(a) shows the bifurcation diagram of the 1.2 order
sys-tem.
Fig. 4
(b)–(d) are the phase portraits with k = 0, 1.291, 1.298.
5. Let a = b = 0.5, b = 1, and k
2 [1.32, 1.42].
Fig. 5
(a) shows the bifurcation diagram of the 1.0 order system.
Fig. 5
(b)–(d) are the phase portraits with k = 0, 1.35, 1.38.
6. Let a = b = 0.4, b = 1.5, and k
2 [1.88, 1.94].
Fig. 6
(a) shows the bifurcation diagram of the 0.8 order
sys-tem.
Fig. 6
(b)–(d) are the phase portraits with k = 0, 1.915, 1.93.
7. Let a = b = 0.3, b = 1.5, and k
2 [1.89, 1.98].
Fig. 7
(a) shows the bifurcation diagram of the 0.6 order
sys-tem.
Fig. 7
(b)–(d) are the phase portraits with k = 0, 1.9, 1.94.
Case 2: We add a non-linear term kjxjsin x in the second equation of system
(6)
and it become:
dax
dta
¼ y;
dby
dtb
¼ x þ að1 x
2Þy þ bz þ kjxj sin x;
_z
¼ w;
_
w
¼ cz dz
3:
8
>
>
>
>
<
>
>
>
>
:
ð10Þ
In system
(10)
, the parameter b is also adjusted to achieve periodic motion for different a and b when k = 0.
a, c, d are fixed as Case 1.
1. Let a = b = 0.9, b = 2.5, and k
2 [0.2, 0.4].
Fig. 8
(a) shows the bifurcation diagram of the 1.8 order system.
Fig. 8
(b)–(d) are the phase portraits with k = 0, 0.35, 0.4.
2. Let a = b = 0.8, b = 2.5, and k
2 [1.0, 1.3].
Fig. 9
(a) shows the bifurcation diagram of the 1.6 order system.
Fig. 9
(b)–(d) are the phase portraits with k = 0, 1.03, 1.2.
3. Let a = b = 0.7, b = 0.5, and k
2 [1.19, 1.24].
Fig. 10
(a) shows the bifurcation diagram of the 1.4 order
sys-tem.
Fig. 10
(b)–(d) are the phase portraits with k = 0, 1.205, 1.215.
4. Let a = b = 0.6, b = 1, and k
2 [0.8, 1.3].
Fig. 11
(a) shows the bifurcation diagram of the 1.2 order system.
Fig. 7. (a) The bifurcation diagram for a = b = 0.3. (b) The phase portrait for a = b = 0.3, k = 0. (c) The phase portrait for a = b = 0.3, k = 1.9. (d) The phase portrait for a = b = 0.3, k = 1.94.
Fig. 6. (a) The bifurcation diagram for a = b = 0.4. (b) The phase portrait for a = b = 0.4, k = 0. (c) The phase portrait for a = b = 0.4, k = 1.915. (d) The phase portrait for a = b = 0.4, k = 1.93.
Fig. 8. (a) The bifurcation diagram for a = b = 0.9. (b) The phase portrait for a = b = 0.9, k = 0. (c) The phase portrait for a = b = 0.9, k = 0.35. (d) The phase portrait for a = b = 0.9, k = 0.4.
Fig. 9. (a) The bifurcation diagram for a = b = 0.8. (b) The phase portrait for a = b = 0.8, k = 0. (c) The phase portrait for a = b = 0.8, k = 1.03. (d) The phase portrait for a = b = 0.8, k = 1.2.
Fig. 11. (a) The bifurcation diagram for a = b = 0.6. (b) The phase portrait for a = b = 0.6, k = 0. (c) The phase portrait for a = b = 0.6, k = 0.95. (d) The phase portrait for a = b = 0.6, k = 1.05.
Fig. 10. (a) The bifurcation diagram for a = b = 0.7. (b) The phase portrait for a = b = 0.7, k = 0. (c) The phase portrait for a = b = 0.7, k = 1.205. (d) The phase portrait for a = b = 0.7, k = 1.215.
Fig. 12. (a) The bifurcation diagram for a = b = 0.5. (b) The phase portrait for a = b = 0.5, k = 0. (c) The phase portrait for a = b = 0.5, k = 0.4. (d) The phase portrait for a = b = 0.5, k = 0.6.
Fig. 13. (a) The bifurcation diagram for a = b = 0.4. (b) The phase portrait for a = b = 0.4, k = 0. (c) The phase portrait for a = b = 0.4, k = 1.25. (d) The phase portrait for a = b = 0.4, k = 1.3.
5. Let a = b = 0.5, b = 0.5, and k
2 [0.1, 0.7].
Fig. 12
(a) shows the bifurcation diagram of the 1.0 order system.
Fig. 12
(b)–(d) are the phase portraits with k = 0, 0.4, 0.6.
6. Let a = b = 0.4, b = 1, and k
2 [1.16, 1.37].
Fig. 13
(a) shows the bifurcation diagram of the 0.8 order
sys-tem.
Fig. 13
(b)–(d) are the phase portraits with k = 0, 1.25, 1.3.
5. Conclusions
Anticontrol of chaos in the fractional order systems of a modified van der Pol system are studied in the
paper. An efficient way to transform a non-chaotic dynamical system into a chaotic one is easily made by
addi-tion of a constant term or by addiaddi-tion of kjxjsin x term where x is a state variable of the system. It is found that
chaos exists in the fractional order systems with order from 1.8 down to 0.6 for the addition of constant term,
and from 1.8 down to 0.8 for the addition of kjxjsin x term.
Acknowledgments
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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