Chaos Synchronization of the Modified Autonomous Van der Pol-Duffing Circuits via Active Control

In this work, we study the dynamics and synchronization of a chaotic system describes the Modified Autonomous Van der Pol-Duffing (MAVPD) circuit. The detailed bifurcation diagrams are given to show the rich dynamics of the proposed system. Lyapunov exponents are calculated to verify the existence of chaos in this system. Chaos synchronization of MAVPD system is obtained using active control method. According to the qualitative theory of fractional differential equations, the existence and uniqueness of solutions for a class of commensurate fractional-order MAVPD systems are investigated. Furthermore, based on the stability theory of fractional-order systems, the conditions of local stability of linear fractional-order system are discussed. Moreover, the existence of chaotic behaviors in the fractional-order MAVPD system is shown. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than three. Phase synchronization of the fractional-order MAVPD system is also achieved using an active control technique. Numerical simulations show the effectiveness of the proposed synchronization schemes.

[1]  L. Chua.  The Genesis of Chua's circuit , 1992 .

[2]  Louis M. Pecora,et al.  Synchronizing chaotic circuits , 1991 .

[3]  Ivo Petras,et al.  A note on the fractional-order Chua’s system , 2008 .

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

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

[6]  Ahmed Sadek Hegazi,et al.  Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system , 2011, Appl. Math. Lett..

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

[8]  Xinghuo Yu,et al.  Experimental confirmation of a new chaotic attractor , 2004 .

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

[10]  Alan V. Oppenheim,et al.  Circuit implementation of synchronized chaos with applications to communications. , 1993, Physical review letters.

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

[12]  Atsushi Uchida,et al.  Synchronization of chaos in two microchip lasers by using incoherent feedback method , 2003 .

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

[14]  Wang Yangyuan,et al.  A CMOS-compatible silicon substrate optimization technique and its application in radio frequency crosstalk isolation , 2008 .

[15]  A. E. Matouk,et al.  Chaos Synchronization between Two Different Fractional Systems of Lorenz Family , 2009 .

[16]  Liu Chong-Xin,et al.  Circuit implementation of a new hyperchaos in fractional-order system , 2008 .

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

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

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

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

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

[22]  M. Lakshmanan,et al.  Chaos in Nonlinear Oscillators: Controlling and Synchronization , 1996 .

[23]  Er-Wei Bai,et al.  Synchronization of two Lorenz systems using active control , 1997 .

[24]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[25]  Lan Chen,et al.  Study on chaos synchronization in the Belousov–Zhabotinsky chemical system , 2003 .

[26]  Liu Chong-Xin,et al.  Study on the fractional-order Liu chaotic system with circuit experiment and its control , 2008 .

[27]  King,et al.  Bistable chaos. I. Unfolding the cusp. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[28]  O. Winter,et al.  Gravitational Capture of Asteroids by Gas Drag , 2009 .

[29]  Bernd Blasius,et al.  Complex dynamics and phase synchronization in spatially extended ecological systems , 1999, Nature.

[30]  Zhang Shulian,et al.  Intensity modulation characters of orthogonally polarized HeNe lasers with different optical feedback level , 2008 .

[31]  N. Laskin.  Fractional market dynamics , 2000 .

[32]  Leon O. Chua,et al.  EXPERIMENTAL CHAOS SYNCHRONIZATION IN CHUA'S CIRCUIT , 1992 .

[33]  Liu Chong-Xin,et al.  Circuit realization of the fractional-order unified chaotic system , 2008 .

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

[35]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[36]  A. E. Matouk,et al.  Chaos, feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit , 2011 .

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

[38]  H. N. Agiza,et al.  Synchronization and adaptive synchronization of nuclear spin generator system , 2001 .

[39]  H. N. Agiza,et al.  Adaptive synchronization of Chua's circuits with fully unknown parameters , 2006 .