Optimal synchronization of two different in-commensurate fractional-order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional-order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional-order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. VC 2016 Wiley Periodicals, Inc. Complexity 000: 00–00, 2016

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

[2]  Tao Liu,et al.  A novel three-dimensional autonomous chaos system , 2009 .

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

[4]  Ju H. Park.  A note on synchronization between two different chaotic systems , 2009 .

[5]  Ju H. Park,et al.  Exponential synchronization for fractional-order chaotic systems with mixed uncertainties , 2015, Complex..

[6]  Swagatam Das,et al.  Chaos Synchronization in Commensurate Fractional Order Lü System via Optimal PI λ D μ Controller with Artificial Bee Colony Algorithm , 2013, SEMCCO.

[7]  Awad El-Gohary,et al.  Optimal synchronization of Rössler system with complete uncertain parameters , 2006 .

[8]  Thomas J. Osler,et al.  Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series , 1970 .

[9]  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 .

[10]  K.N. Salama,et al.  Control and switching synchronization of fractional order chaotic systems using active control technique , 2013, Journal of advanced research.

[11]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[12]  Jinde Cao,et al.  Adaptive synchronization of fractional-order memristor-based neural networks with time delay , 2015, Nonlinear Dynamics.

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

[14]  Yong Zhou,et al.  Fractional Schrödinger equations with potential and optimal controls , 2012 .

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

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

[17]  Guy Jumarie,et al.  On the representation of fractional Brownian motion as an integral with respect to (dt)alpha , 2005, Appl. Math. Lett..

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

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

[20]  Zahra Rahmani Cherati,et al.  Synchronization of different-order chaotic systems: Adaptive active vs. optimal control , 2012 .

[21]  Yu Wang,et al.  Synchronization of Fractional-Order Hyperchaotic Systems via Fractional-Order Controllers , 2014 .

[22]  Xiang-Jun Wu,et al.  Chaos in the fractional-order Lorenz system , 2009, Int. J. Comput. Math..

[23]  Mohammad Ali Badamchizadeh,et al.  An approach to achieve modified projective synchronization between different types of fractional-order chaotic systems with time-varying delays , 2015 .

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

[25]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

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

[27]  Hongtao Lu,et al.  Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication , 2012 .

[28]  H ParkJu,et al.  Synchronization of fractional-order different memristor-based chaotic systems using active control , 2014 .

[29]  B. Øksendal,et al.  FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE , 2003 .

[30]  M. Rosenstein,et al.  A practical method for calculating largest Lyapunov exponents from small data sets , 1993 .

[31]  Sachin Bhalekar,et al.  Fractional ordered Liu system with time-delay , 2010 .

[32]  A. Ruzmaikina.  Stochastic calculus with fractional Brownian motion , 1999 .

[33]  Jinde Cao,et al.  Synchronization of fractional-order delayed neural networks with hybrid coupling , 2016, Complex..

[34]  H. N. Agiza,et al.  Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization , 2015, Commun. Nonlinear Sci. Numer. Simul..

[35]  A. Hegazi,et al.  THE EFFECT OF FRACTIONAL ORDER ON SYNCHRONIZATION OF TWO FRACTIONAL ORDER CHAOTIC AND HYPERCHAOTIC SYSTEMS , 2011 .

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

[37]  Yu Wang,et al.  Stability of a Class of Fractional-Order Nonlinear Systems , 2014 .

[38]  Ahmed Ezzat Matouk,et al.  Chaos synchronization of a fractional-order modified Van der Pol-Duffing system via new linear control, backstepping control and Takagi-Sugeno fuzzy approaches , 2016, Complex..

[39]  Mohammad Pourmahmood Aghababa,et al.  Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme , 2014, Complex..

[40]  MOHAMMAD POURMAHMOOD AGHABABA,et al.  Fractional modeling and control of a complex nonlinear energy supply-demand system , 2015, Complex..

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

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

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

[44]  D. Naidu,et al.  Optimal Control Systems , 2018 .