Dynamics and synchronization analysis of coupled fractional-order nonlinear electromechanical systems

Abstract A fractional-order (FO) nonlinear model is used to describe an electromechanical system. We make capital out of the fact that for realistic modeling, the electric characteristics of a capacitor include a fractional-order time derivative. The dynamics and synchronization of coupled fractional-order nonlinear electromechanical systems are analyzed. Detailed attention is granted to the bifurcations that can occur in the dynamics of a single uncoupled electromechanical system as the fractional-order varies. For example, the fractional-order counterparts of the chaotic 4-order system are periodic at orders less than 3.985. The effect of the fractional-order on the condition of occurrence of synchronization phenomena in the network of many mutually coupled fractional-order nonlinear electromechanical systems is analyzed, especially when they are chaotic. An insight on the overall dynamics of the network is provided. It is shown that the dynamics of both the uncoupled system and the network are very sensitive to changes in the order of the fractional derivative.

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

[2]  J. D. Trolinger,et al.  Experimental verification of fractional history effects on the viscous dynamics of small spherical particles , 2005 .

[3]  Zhong Liu,et al.  A new fractional-order chaotic system and its synchronization control , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[4]  A. Jonscher.  Dielectric relaxation in solids , 1983 .

[5]  Paul Woafo,et al.  Effects of higher nonlinearity on the dynamics and synchronization of two coupled electromechanical devices , 2008 .

[6]  Junwei Wang,et al.  Inverse synchronization of coupled fractional-order systems through open-plus-closed-loop control , 2011 .

[7]  Wei-Ching Chen,et al.  Nonlinear dynamics and chaos in a fractional-order financial system , 2008 .

[8]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

[9]  Paul Woafo,et al.  Stability of synchronized network of chaotic electromechanical devices with nearest and all-to-all couplings , 2008 .

[10]  Lothar Gaul,et al.  Finite Element Formulation of Viscoelastic Constitutive Equations Using Fractional Time Derivatives , 2002 .

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

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

[13]  Michelle Pierri,et al.  Weighted -Asymptotically -Periodic Solutions of a Class of Fractional Differential Equations , 2010 .

[14]  I. Schäfer,et al.  Modelling of lossy coils using fractional derivatives , 2008 .

[15]  Ivo Petras,et al.  An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications , 2011 .

[16]  J. A. Tenreiro Machado,et al.  Analysis of the Van der Pol Oscillator Containing Derivatives of Fractional Order , 2007 .

[17]  Carlos F.M. Coimbra,et al.  A variable order constitutive relation for viscoelasticity , 2007 .

[18]  Junyi Cao,et al.  Nonlinear Dynamics of Duffing System With Fractional Order Damping , 2009 .

[19]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[20]  Om P. Agrawal,et al.  Application of Fractional Derivatives in Thermal Analysis of Disk Brakes , 2004 .

[21]  Julien Clinton Sprott,et al.  Bifurcations and Chaos in fractional-Order Simplified Lorenz System , 2010, Int. J. Bifurc. Chaos.

[22]  S. Westerlund,et al.  Capacitor theory , 1994 .

[23]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[24]  Zahra Yaghoubi,et al.  Projective Synchronization of a Class of Fractional-Order Chaotic Systems , 2011 .

[25]  Dominik Sierociuk,et al.  Some applications of fractional order calculus , 2010 .

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

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

[28]  Isabel S. Jesus,et al.  Development of fractional order capacitors based on electrolyte processes , 2009 .

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

[30]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .