Bifurcation and Chaos in the Fractional-Order Chen System via a Time-Domain Approach

Abstract

This tutorial investigates bifurcation and chaos in the fractional-order Chen system from the time-domain point of view. The objective is achieved using the Adomian decomposition method, which allows the solution of the fractional differential equations to be written in closed form. By taking advantage of the capabilities given by the decomposition method, the paper illustrates two remarkable findings: (i) chaos exists in the fractional Chen system with order as low as 0.24, which represents the smallest value ever reported in literature for any chaotic system studied so far; (ii) it is feasible to show the occurrence of pitchfork bifurcations and period-doubling routes to chaos in the fractional Chen system, by virtue of a systematic time-domain analysis of its dynamics.

Polynomial First Integrals for the Chen and Lü Systems

“…These recent years the dynamics of the Chen system has been analyzed from many different points of view. See for instance [Bashkirtseva et al, 2010, Cafagna & Grassi, 2008,Hou et al, 2008, Mahmoud et al, 2007, Yao & Liu, 2010.…”

Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System

“…By applying the R-L fractional integral operator to both sides of Equation (2), the following equation is obtained [26,27] …”

Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci

“…There exist several typical methods to do numerical computations of fractional-order differential equations: (i) approximate fractional operators by using standard integer operators, realized via utilizing frequency domain techniques based on Bode diagrams [5]; (ii) linearize the nonlinear fractional differential equation on each step of integration and iterative application of the analytical expression of the solution of linearized fractional differential equation [6]; (iii) derive the analytic expression of the solution of the nonlinear fractional differential equation and numerically iterate the formula by using generalized Admas-Bashforth-Moulton scheme [21]; (iv) other analytical methods [22][23][24]. Among the above methods, the third one is at least super-linearly convergent and has good numerical stability, especially it preserves the inherent attribute, "memory effects," of fractional derivatives.…”