Chaos in the fractional order Chen system and its control
Chunguang Li
a,*,Guanrong Chen b a
Institute of Electronic Systems,School of Electronic Engineering,University of Electronic Science and Technology of China,Chengdu 610054,PR China b Department of Electronic Engineering,City University of Hong Kong,83Tat Chee Avenue,Kowloon,Hong Kong,PR China
Accepted 26February 2004
Abstract
In this letter,we study the chaotic behaviors in the fractional order Chen system.We found that chaos exists in the fractional order Chen system with order less than 3.The lowest order we found to have chaos in this system is 2.1.Linear feedback control of chaos in this system is also studied.
Ó2004Elvier Ltd.All rights rerved.
1.Introduction
Fractional calculus is a 300-year-old mathematical topic.Although it has a long history,the applications of frac-tional calculus to physics and engineering are just a recent focus of interest [1,2].Many systems are known to display fractional order dynamics,such as viscoelastic systems [3–5],dielectric polarization [6],electrode–electrolyte polari-zation [7],and electromagnetic waves [8].More recently,there is a new trend to investigate the control [9–14]and dynamics [15–24,31]of fractional order dynamical systems.In [15],it is shown that the fractional order Chua’s circuit of order as low as 2.7can produce a chaotic attractor.In [16],it is shown that nonautonomous Duffing systems of order less than 2can still behave in a chaotic manner.In [17],the fractional order Wien bridge oscillator is studied,where it is shown that limit cycle can be generated for any fractional order,with a proper value of the amplifier gain.In [18],chaotic behaviors of the fractional order ‘‘jerk’’model is studied,in which chaotic attractor can be obtained with the system order as low as 2.1,and in [19]chaos control of this fractional order chaotic system is investigated.In [20],chaotic behavior of the fractional order Lorenz system is studied,but unfortunately,the results prented in this paper are not correct [30].In [21,22],bifurcation and chaotic dynamics of the fractional order cellular neural networks are
studied.In [23],chaos and hyperchaos in fractional order R €o
ssler equations are discusd,in which it is shown that chaos can exist in the fractional R €o
ssler equation with order as low as 2.4,and hyperchaos can also exist in the fractional order R €o
ssler hyperchaos equation with order as low as 3.8.In [24],chaos synchronization of fractional order chaotic systems are studied.In [31],the author prents a broad review of existing models of fractional kinetics and their connection to dynamical models,pha space topology,and other characteristics of chaos.
In this letter,we study the chaotic behaviors in the fractional order Chen system [25].A chaos control approach is also prented for this fractional order system.
2.Fractional derivative and its approximation
There are veral definitions of fractional derivatives [1].Perhaps the best known is the Riemann–Liouville defini-tion,which is given by
*Corresponding author.
E-mail address:cgli@uestc.edu (C.Li),gchen@ee.cityu.edu.hk (G.Chen).
0960-0779/$-e front matter Ó2004Elvier Ltd.All rights rerved.
doi:10.1016/j.chaos.2004.02.035
/locate/chaos
d a fðtÞd t a ¼
1
CðnÀaÞ
d n
d t n
Z t
fðsÞ
ðtÀsÞaÀnþ1
d sð1Þ
where CðÁÞis the gamma function and nÀ16a<n.This definition is significantly different from the classical definition of derivative.
Fortunately,the Laplace transform is still applicable and works as one would expect.Upon considering all the initial conditions to be zero,the Laplace transform of the Riemann–Liouville fractional derivative satisfies
L
d a fðtÞ
d t a
¼s a L f fðtÞgð2Þ
Thus,the fractional integral operator of order‘‘a’’can be reprented by the transfer function FðsÞ¼1
栖息的近义词是什么s a .
The standard definition of fractional differintegral does not allow direct implementation of fractional operators in time-domain simulations.An efficient method to circumvent this problem is to approximate fractional operators by using standard integer order operators.In[26],an effective algorithm is developed to approximate fractional order transfer functions.Basically,the idea is to approximate the system behavior in the frequency domain.By utilizing frequency domain techniques bad 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.This approximation approach was adopted in[15,18,21–23].In Table1of[15], approximations for1=s q with q¼0:1–0:9in steps0.1are given,with errors of approximately2dB.We also u the approximations in the following simulations.
3.The fractional order Chen system
Here,we consider the fractional order Chen system.The standard derivative is replaced by a fractional derivative as follows:
创业找哪些投资d a x
d t a
¼aðyÀxÞ
d a y
d t a
¼ðcÀaÞxÀxzþcy
d a z d t a ¼xyÀbz
ð3Þ
where a is the fractional order.When a¼1,system(3)is the original integer order Chen system,which is chaotic when ða;b;cÞ¼ð35;3;28Þ可爱小猪
.
Fig.1.Chaotic attractor of the fractional order Chen system with order a¼0:9and a¼35,b¼3,c¼28.
550 C.Li,G.Chen/Chaos,Solitons and Fractals22(2004)549–554
Simulations are performed for a¼0:9,0.8,0.7,0.6.The simulation results demonstrate that chaos indeed exist in the fractional order Chen system with order less than3.When a¼0:9,0.8and0.7,chaotic attractors are found and the pha portraits are shown in Figs.1,2and3,respectively.We can e that the chaotic attractors with a¼0:9,0.8and 0.7in the fraction
al order Chen system are similar and look like the Lorenz attractor.When a¼0:6,no chaotic behavior is found,which indicates that the lowest limit of the fractional order for this system to be chaotic is a¼0:6–0:7.Thus,the lowest order we found for this system to yield chaos is2.1.
4.Chaos control
最后一次讲演
Chaos control attracts more and more attentions from various disciplines in science and engineering since the pioneering work of Ott et al.[27].For state-of-the-art reviews,e,for example,[28].In this paper,we briefly discuss the issue of controlling the fractional order chaotic Chen system to its equilibria.
Recasting the fractional order Chen system(3)in a compact vector form,we have
d a X d t a ¼fðXÞð4
ÞFig.2.Chaotic attractor of the fractional order Chen system with order a¼0:8and a¼35,b¼3,c¼
28.
Fig.3.Chaotic attractor of the fractional order Chen system with order a¼0:7and a¼35,b¼3,c¼40.
C.Li,G.Chen/Chaos,Solitons and Fractals22(2004)549–554551
552 C.Li,G.Chen/Chaos,Solitons and Fractals22(2004)549–554
with X¼½x;y;z T.The above system with a simple linear state feedback controller can be written as
d a X
¼fðXÞþuð5Þ
d t a
where u is a linear state feedback controller of the form[29]
u¼
k100
0k20
00k3
2
4
蛋糕3
5ðXÀXÞð6Þ
where X is the control target,and k1,k2,k3are constant parameters.
Clearly,ð0;0;0Þis always an equilibrium of(3).In the following simulation,we stabilize system(3)to this equi-librium.Standard stability analysis easily shows that withðk1;k2;k3Þ¼ð19;À13;2Þ,the equilibriumð0;0;0Þof the controlled integer order Chen system is locally stable.Simulation results show that this controller can also stabilize the fractional order Chen systems to this equilibrium.The trajectories of the controlled fractional order Chen system with orders a¼0:9and a¼0:8are shown in Figs.4and5,respectively.In bothfigures,the control signal is added at t¼20. As we can e from the twofigures,the designed chaos controller can effectively control the fractional order Chen systems to its equilibrium X¼ð0;0;0Þ.
5.Conclusions
In this letter,we have studied the chaotic dynamics of the fractional order Chen system.We found that chaos exists in this system with order as low as2.1.A simple,but effective,linear feedback controller is also designed to stabilize the fractional order chaotic Chen system.
Future works regarding this topic include theoretical analysis of the dynamics of the fractional order Chen system, as well as in-depth studies on chaos control and synchronization of this system.
Acknowledgements
This rearch was supported by the Hong Kong Rearch Grant Council under the CERG grant CityU1115/03E, the National Natural Science Foundation of China under Grant60271019,and the Youth Science and Technology Foundation of UESTC under Grant YF020207.
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