Hybrid Projective Synchronization of Fractional-Order Chaotic Complex Nonlinear Systems With Time Delays

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

[1]  Wei Zhu,et al.  Function projective synchronization for fractional-order chaotic systems , 2011 .

[2]  Mark J. McGuinness,et al.  The real and complex Lorenz equations and their relevance to physical systems , 1983 .

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

[4]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

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

[6]  K. Moore,et al.  Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[7]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[8]  Gamal M. Mahmoud,et al.  Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system , 2007 .

[9]  C. Y. Chee,et al.  Chaos-based M-ary digital communication technique using controlled projective synchronisation , 2006 .

[10]  S. Bhalekar,et al.  Synchronization of different fractional order chaotic systems using active control , 2010 .

[11]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[12]  Yubing Gong,et al.  Synchronization transitions on complex thermo-sensitive neuron networks with time delays. , 2010, Biophysical chemistry.

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

[14]  Emad E. Mahmoud,et al.  Complete synchronization of chaotic complex nonlinear systems with uncertain parameters , 2010 .

[15]  Manfeng Hu,et al.  Hybrid projective synchronization in a chaotic complex nonlinear system , 2008, Math. Comput. Simul..

[16]  Xingyuan Wang,et al.  CHAOS GENERATED FROM THE FRACTIONAL-ORDER COMPLEX CHEN SYSTEM AND ITS APPLICATION TO DIGITAL SECURE COMMUNICATION , 2013 .

[17]  H. Haken,et al.  Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[18]  Junwei Sun,et al.  Adaptive anti-synchronization of chaotic complex systems and chaotic real systems with unknown parameters , 2016 .

[19]  Emad E. Mahmoud,et al.  Chaotic synchronization of two complex nonlinear oscillators , 2009 .

[20]  A. Galip Ulsoy,et al.  Analysis of a System of Linear Delay Differential Equations , 2003 .

[21]  Guanrong Chen,et al.  Onset of synchronization in weighted scale-free networks. , 2009, Chaos.

[22]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

[23]  Emad E. Mahmoud,et al.  On modified time delay hyperchaotic complex Lü system , 2015 .

[24]  Junguo Lu.  Chaotic dynamics of the fractional-order Lü system and its synchronization , 2006 .

[25]  Guoguang Wen,et al.  Hybrid projective synchronization of time-delayed fractional order chaotic systems , 2014 .

[26]  Mohammad Pourmahmood Aghababa,et al.  Synchronization and stabilization of fractional second-order nonlinear complex systems , 2014, Nonlinear Dynamics.

[27]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[28]  Diyi Chen,et al.  Synchronization and Antisynchronization of a Class of Chaotic Systems With Nonidentical Orders and Uncertain Parameters , 2015 .

[29]  Emad E. Mahmoud,et al.  Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems , 2010 .

[30]  Mark J. McGuinness,et al.  The complex Lorenz equations , 1982 .

[31]  R. Koeller.  Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[32]  Young-Jai Park,et al.  Anti-synchronization of chaotic oscillators , 2003 .

[33]  Mark J. McGuinness,et al.  The real and complex Lorenz equations in rotating fluids and lasers , 1982 .

[34]  Ping Liu,et al.  Adaptive Hybrid Function Projective Synchronization of General Chaotic Complex Systems With Different Orders , 2015 .

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

[36]  Ronnie Mainieri,et al.  Projective Synchronization In Three-Dimensional Chaotic Systems , 1999 .

[37]  Tiedong Ma,et al.  Hybrid synchronization of coupled fractional-order complex networks , 2015, Neurocomputing.

[38]  Xiangdong Wang,et al.  On the chaotic synchronization of Lorenz systems with time-varying lags , 2009 .

[39]  Parlitz,et al.  Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. , 1996, Physical review letters.

[40]  Yongming Yang,et al.  Wireless Sensor Network Time Synchronization Design for Large Generator On-Line Monitoring , 2011 .

[41]  Junwei Sun,et al.  Modified projective and modified function projective synchronization of a class of real nonlinear systems and a class of complex nonlinear systems , 2014 .

[42]  Emad E. Mahmoud,et al.  Synchronization and control of hyperchaotic complex Lorenz system , 2010, Math. Comput. Simul..

[43]  Fangfang Zhang,et al.  Lag Synchronization of Complex Lorenz System with Applications to Communication , 2015, Entropy.

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

[45]  Hao Zhang,et al.  Bivariate Module-Phase Synchronization of a Fractional-Order Lorenz System in Different Dimensions , 2013 .

[46]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[47]  Chunhua Yuan,et al.  Adaptive complex modified projective synchronization of complex chaotic (hyperchaotic) systems with uncertain complex parameters , 2015 .

[48]  Dumitru Baleanu,et al.  Fractional Hyperchaotic Telecommunication Systems: A New Paradigm , 2013 .

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