Dynamic analysis of a new chaotic system with fractional order and its generalized projective synchronization

Based on the stability theory of the fractional order system, the dynamic behaviours of a new fractional order system are investigated theoretically. The lowest order we found to have chaos in the new three-dimensional system is 2.46, and the period routes to chaos in the new fractional order system are also found. The effectiveness of our analysis results is further verified by numerical simulations and positive largest Lyapunov exponent. Furthermore, a nonlinear feedback controller is designed to achieve the generalized projective synchronization of the fractional order chaotic system, and its validity is proved by Laplace transformation theory.

[1]  Yang Jie,et al.  The feedback control of fractional order unified chaotic system , 2010 .

[2]  I. Podlubny.  Fractional differential equations , 1998 .

[3]  Sachin Bhalekar,et al.  Chaos in fractional ordered Liu system , 2010, Comput. Math. Appl..

[4]  Chunguang Li,et al.  Chaos and hyperchaos in the fractional-order Rössler equations , 2004 .

[5]  R. Bagley,et al.  Fractional order state equations for the control of viscoelasticallydamped structures , 1991 .

[6]  邵仕泉,et al.  Projective synchronization in coupled fractional order chaotic Rossler system and its control , 2007 .

[7]  Lokenath Debnath.  Transform Methods with Applications to Engineering and Operations Research , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[8]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[9]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[10]  Changpin Li,et al.  Chaos in Chen's system with a fractional order , 2004 .

[11]  R. Koeller.  Polynomial operators, stieltjes convolution, and fractional calculus in hereditary mechanics , 1986 .

[12]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[13]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[14]  Guanrong Chen,et al.  A New Chaotic System and its Generation , 2003, Int. J. Bifurc. Chaos.

[15]  Zheng-Ming Ge,et al.  Chaos in a modified van der Pol system and in its fractional order systems , 2007 .

[16]  刘颖,et al.  A fabrication and magnetic properties study on Al doped Zn 0.99 Co 0.01 O dilution ferromagnetic semiconductors , 2009 .

[17]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[18]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[19]  S. Hara,et al.  Robust PID control using generalized KYP synthesis: direct open-loop shaping in multiple frequency ranges , 2006, IEEE Control Systems.

[20]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[21]  C. F. Lorenzo,et al.  Chaos in a fractional order Chua's system , 1995 .

[22]  Chunguang Li,et al.  Chaos in the fractional order Chen system and its control , 2004 .

[23]  Guanrong Chen,et al.  Bifurcation Analysis of Chen's equation , 2000, Int. J. Bifurc. Chaos.

[24]  Shangbo Zhou,et al.  Chaos in a fractional-order Rössler system , 2009 .

[25]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .