The synchronization of fractional-order Rössler hyperchaotic systems☆

The synchronization of fractional-order hyperchaotic systems is studied, using the Rossler system as an example. Based on the Laplace transformation theory, sufficient conditions for global synchronization of the systems are given analytically. Also, the variational iteration method is implemented to give the approximate solution for the fractional-order error system of the two identical hyperchaotic systems, which is in good agreement with the approximate solution using the classical Laplace transformation method. Numerical methods and simulations on the master–slave systems are presented to verify the results obtained.

[1]  Shaher Momani,et al.  Non-perturbative analytical solutions of the space- and time-fractional Burgers equations , 2006 .

[2]  Shaher Momani,et al.  Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method , 2006, Appl. Math. Comput..

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

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

[5]  Ji-Huan He.  Variational iteration method—Some recent results and new interpretations , 2007 .

[6]  Shaher Momani,et al.  Numerical approach to differential equations of fractional order , 2007 .

[7]  Juebang Yu,et al.  Synchronization of fractional-order chaotic systems , 2005, Proceedings. 2005 International Conference on Communications, Circuits and Systems, 2005..

[8]  S. Momani,et al.  Numerical comparison of methods for solving linear differential equations of fractional order , 2007 .

[9]  Zheng-Ming Ge,et al.  Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems , 2007 .

[10]  Davood Domiri Ganji,et al.  Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations , 2007 .

[11]  Ji-Huan He,et al.  Construction of solitary solution and compacton-like solution by variational iteration method , 2006 .

[12]  Kai Diethelm,et al.  Multi-order fractional differential equations and their numerical solution , 2004, Appl. Math. Comput..

[13]  Daolin Xu,et al.  Chaos synchronization of the Chua system with a fractional order , 2006 .

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

[15]  Changpin Li,et al.  The synchronization of three fractional differential systems , 2007 .

[16]  Ji-Huan He,et al.  Variational iteration method for delay differential equations , 1997 .

[17]  Z. Ge,et al.  Chaos in a generalized van der Pol system and in its fractional order system , 2007 .

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

[19]  Zheng-Ming Ge,et al.  Chaos in a fractional order modified Duffing system , 2007 .

[20]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

[21]  Xiaohua Xiong,et al.  Extending synchronization scheme to chaotic fractional-order Chen systems , 2006 .

[22]  Z. Ge,et al.  Chaos and Chaos Control for a Two-Degree-of-Freedom Heavy Symmetric Gyroscope , 2007 .

[23]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[24]  Changpin Li,et al.  On chaos synchronization of fractional differential equations , 2007 .

[25]  Ji-Huan He.  Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

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

[27]  Xing-yuan Wang,et al.  Projective synchronization of fractional order chaotic system based on linear separation , 2008 .

[28]  Lingpeng Yang,et al.  Document reranking by term distribution and maximal marginal relevance for chinese information retrieval , 2007, Information Processing & Management.

[29]  Changpin Li,et al.  Synchronization in fractional-order differential systems , 2005 .

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

[31]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

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

[33]  H. Sekine,et al.  General Use of the Lagrange Multiplier in Nonlinear Mathematical Physics1 , 1980 .

[34]  Ji-Huan He.  A new approach to nonlinear partial differential equations , 1997 .

[35]  Z. Ge,et al.  Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor , 2007 .

[36]  Guanrong Chen,et al.  HYPERCHAOS IN THE FRACTIONAL-ORDER NONAUTONOMOUS CHEN'S SYSTEM AND ITS SYNCHRONIZATION , 2005 .

[37]  Shaher Momani,et al.  Approximate solutions for boundary value problems of time-fractional wave equation , 2006, Appl. Math. Comput..

[38]  Davood Domiri Ganji,et al.  Some nonlinear heat transfer equations solved by three approximate methods , 2007 .

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

[40]  Ji-Huan He,et al.  Variational iteration method for autonomous ordinary differential systems , 2000, Appl. Math. Comput..

[41]  Shaher Momani,et al.  Homotopy perturbation method for nonlinear partial differential equations of fractional order , 2007 .

[42]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[43]  Ji-Huan He,et al.  Variational iteration method: New development and applications , 2007, Comput. Math. Appl..