Chaos, feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit

Abstract In this work, stability analysis of the fractional-order modified Autonomous Van der Pol–Duffing (MAVPD) circuit is studied using the fractional Routh–Hurwitz criteria. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than 3. Furthermore, the fractional Routh–Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh–Hurwitz conditions and using specific choice of linear controllers, it is shown that the fractional-order MAVPD system is controlled to its equilibrium points; however, its integer-order counterpart is not controlled. Moreover, chaos synchronization of MAVPD system is found only in the fractional-order case when using a specific choice of nonlinear control functions. This shows the effect of fractional order on chaos control and synchronization. Synchronization is also achieved using the unidirectional linear error feedback coupling approach. Numerical results show the effectiveness of the theoretical analysis.

[1]  K. Diethelm.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[2]  Qionghua Wang,et al.  A fractional-order hyperchaotic system and its synchronization , 2009 .

[3]  A. Matouk.  Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system , 2009 .

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

[5]  Shangbo Zhou,et al.  Chaos synchronization of the fractional-order Chen's system , 2009 .

[6]  Xing-yuan Wang,et al.  Dynamic analysis of the fractional-order Liu system and its synchronization. , 2007, Chaos.

[7]  Guojun Peng,et al.  Synchronization of fractional order chaotic systems , 2007 .

[8]  B. Onaral,et al.  Linear approximation of transfer function with a pole of fractional power , 1984 .

[9]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[10]  Ahmad Harb,et al.  On nonlinear control design for autonomous chaotic systems of integer and fractional orders , 2003 .

[11]  Elsayed Ahmed,et al.  On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems , 2006 .

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

[13]  P. Butzer,et al.  AN INTRODUCTION TO FRACTIONAL CALCULUS , 2000 .

[14]  Ahmed M. A. El-Sayed,et al.  On the fractional-order logistic equation , 2007, Appl. Math. Lett..

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

[16]  Xing-yuan Wang,et al.  Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control , 2009 .

[17]  Juebang Yu,et al.  Synchronization of two coupled fractional-order chaotic oscillators , 2005 .

[18]  E. Ahmed,et al.  On fractional order differential equations model for nonlocal epidemics , 2007, Physica A: Statistical Mechanics and its Applications.

[19]  D. Kusnezov,et al.  Quantum Levy Processes and Fractional Kinetics , 1999, chao-dyn/9901002.

[20]  A. E. Matouk,et al.  Dynamical analysis, feedback control and synchronization of Liu dynamical system , 2008 .

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

[22]  P. Arena,et al.  Nonlinear Noninteger Order Circuits and Systems — An Introduction , 2000 .

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

[24]  Hao Zhu,et al.  Chaos and synchronization of the fractional-order Chua’s system , 2009 .

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

[26]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

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

[28]  Reyad El-Khazali,et al.  Fractional-order dynamical models of love , 2007 .

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

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

[31]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[32]  Qing-ju Fan,et al.  Horseshoe in a modified Van der Pol-Duffing circuit , 2009, Appl. Math. Comput..

[33]  W. Deng,et al.  Chaos synchronization of the fractional Lü system , 2005 .

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

[35]  Guanrong Chen,et al.  A simple global synchronization criterion for coupled chaotic systems , 2003 .

[36]  H. Agiza,et al.  Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor , 2008 .

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

[38]  M. Ichise,et al.  An analog simulation of non-integer order transfer functions for analysis of electrode processes , 1971 .

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

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

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

[42]  Jun-Guo Lu,et al.  Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal , 2006 .

[43]  Yongguang Yu,et al.  The synchronization of fractional-order Rössler hyperchaotic systems☆ , 2008 .

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

[45]  Shiquan Shao,et al.  Controlling general projective synchronization of fractional order Rossler systems , 2009 .

[46]  Yaolin Jiang,et al.  Generalized projective synchronization of fractional order chaotic systems , 2008 .

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