Lyapunov exponents spectrum estimation of fractional order nonlinear systems using Cloned Dynamics

Abstract

This work presents the determination of Lyapunov spectrum of fractional order dynamical systems. The method considers the Cloned Dynamic technique, which is based on time evaluation of exact copies of dynamic equations with small initial conditions disturbances. As its calculations are simple, this method can be applied to various dynamic systems types, in particular fractional order, nonlinear and time varying systems, and non linear systems where Jacobian are impossible to evaluate. The predictor-corrector Adams-Bashforth-Moulton algorithm is used for fractional order system numerical evaluation. All Lyapunov exponents are estimated based on the convergence or divergence of the small disturbed clones with respect to the reference (fiducial) trajectory and application of Gram-Schmidt Reorthonormalization after each convenient periodic time interval. To evaluate the application of this method, three third order dynamical systems are explored: Jerk, Financial, and a Four Wing systems. Integer order, commensurate fractional order and incommensurate fractional order are explored. To confirm chaotic or non-chaotic behavior, 0-1 Test is used and Jerk system is studied with a classical method for Lyapunov exponents determination also. The advantages and contribution of the proposed method the possibility to study incommensurate orders fractional cases and explore non linear systems without need of Jacobian determination, as is proved with applications shown. The Lyapunov Exponents and the phase portraits were simulated together to validate this method and the obtained results are in good agreement.