Control of finite-time anti-synchronization for variable-order fractional chaotic systems with unknown parameters

Fractional-order chaotic system with variable-order and unknown parameters, as an excellent tool to describe the memory and hereditary characteristics of the complex phenomena in reality, remains important, but nowadays there exist few results about this system. This paper presents a finite-time anti-synchronization of two these systems based on the Mittag-Leffler stable theory and norm theory, in which the order varies with time and the unknown parameters of the systems are estimated. Moreover, a corollary about the monotone effect of variable order on the norm of the error system is deduced. We take different nonlinear variable orders for two identical Lü fractional chaotic systems and for two different Lü and Chen–Lee fractional chaotic systems as examples. The simulations illustrate the effectiveness and feasibility of the proposed control scheme.

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

[2]  Naveen Eluru,et al.  Modeling vehicle operating speed on urban roads in Montreal: a panel mixed ordered probit fractional split model. , 2013, Accident; analysis and prevention.

[3]  S. Zhang,et al.  EXISTENCE AND UNIQUENESS RESULT OF SOLUTIONS TO INITIAL VALUE PROBLEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS OF VARIABLE-ORDER , 2013 .

[4]  Hongguang Sun,et al.  Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. , 2014, Journal of contaminant hydrology.

[5]  Romi Mankin,et al.  Memory effects for a trapped Brownian particle in viscoelastic shear flows. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Andrew Y. T. Leung,et al.  Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method , 2014 .

[7]  Robin De Keyser,et al.  Fractional order model parameters for the respiratory input impedance in healthy and in asthmatic children , 2011, Comput. Methods Programs Biomed..

[8]  M. Rapaić,et al.  Variable-Order Fractional Operators for Adaptive Order and Parameter Estimation , 2014, IEEE Transactions on Automatic Control.

[9]  B. Ross,et al.  Integration and differentiation to a variable fractional order , 1993 .

[10]  Hadi Taghvafard,et al.  Phase and anti-phase synchronization of fractional order chaotic systems via active control , 2011 .

[11]  Alain Oustaloup,et al.  Transients of fractional-order integrator and derivatives , 2012, Signal Image Video Process..

[12]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[13]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[14]  Muhammad Rehan,et al.  Synchronization and anti-synchronization of chaotic oscillators under input saturation , 2013 .

[15]  Shutang Liu,et al.  Chaotic behaviour of nonlinear coupled reaction–diffusion system in four-dimensional space , 2014 .

[16]  W. Chen,et al.  A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems , 2011 .

[17]  Tom T Hartley,et al.  Preface. Special issue: Initialization, conceptualization, and application in the generalized fractional calculus. , 2007, Critical reviews in biomedical engineering.

[18]  T. Hartley,et al.  Initialization, conceptualization, and application in the generalized (fractional) calculus. , 2007, Critical reviews in biomedical engineering.

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

[20]  Yufeng Xu,et al.  Synchronization of variable-order fractional financial system via active control method , 2013 .

[21]  Wei Xu,et al.  Synchronization of two chaotic nonlinear gyros using active control , 2005 .

[22]  Mayank Srivastava,et al.  Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method , 2013 .

[23]  Sergey S. Postnov,et al.  Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation , 2013, Automation and Remote Control.

[24]  Elsayed Ahmed,et al.  On chaos control and synchronization of the commensurate fractional order Liu system , 2013, Commun. Nonlinear Sci. Numer. Simul..

[25]  Hadi Delavari,et al.  Chaos in fractional-order Genesio–Tesi system and its synchronization , 2012 .

[26]  Ruoxun Zhang,et al.  Adaptive synchronization of fractional-order chaotic systems via a single driving variable , 2011 .

[27]  Yangquan Chen,et al.  A Survey on Fractional-Order Iterative Learning Control , 2013, J. Optim. Theory Appl..

[28]  Dov Ingman,et al.  Control of damping oscillations by fractional differential operator with time-dependent order , 2004 .

[29]  Abdon Atangana,et al.  On the stability and convergence of the time-fractional variable order telegraph equation , 2015, J. Comput. Phys..

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

[31]  Sergey S. Postnov,et al.  Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation , 2013, Automation and Remote Control.

[32]  Jian-Liung Chen,et al.  Maximum photovoltaic power tracking for the PV array using the fractional-order incremental conductance method , 2011 .

[33]  Uchechukwu E. Vincent,et al.  Synchronization of identical and non-identical 4-D chaotic systems using active control , 2008 .

[34]  S. Momani,et al.  Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order , 2006 .

[35]  Shihua Chen,et al.  Adaptive control for anti-synchronization of Chua's chaotic system , 2005 .

[36]  F. Wilson.  The structure of the level surfaces of a Lyapunov function , 1967 .

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

[38]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[39]  K. Vishal,et al.  Chaos control of fractional order Rabinovich–Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich–Fabrikant system , 2014 .

[40]  Li Zhang,et al.  Control of Weierstrass-Mandelbrot Function Model with Morlet Wavelets , 2014, Int. J. Bifurc. Chaos.

[41]  Manuel A. Duarte-Mermoud,et al.  Lyapunov functions for fractional order systems , 2014, Commun. Nonlinear Sci. Numer. Simul..

[42]  Runfan Zhang,et al.  Control of a class of fractional-order chaotic systems via sliding mode , 2012 .

[43]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .