Study On Fractional Order Chaotic System With The Curvature Index

A large number of studies have verified that fractional order chaotic system has more abundant and complex dynamical behavior,which can describe physical phenomena more accurately,and it is more valuable in engineering applications.In this paper,the dynamic behavior of fractional order chaotic systems is studied by using a convenient and effective index of curvature index proposed by Chen&Chang(2012).The work and conclusions of this article are as follows:1.On the basis of empirical mode decomposition(EMD),we improved the limitation of Hilbert-Huang transform(HHT),for which could not get the similar frequency in a synthetic signal,and put forward the idea of finding the signal frequency in the sense of modal product.2.The curvature index is defined as the limit of the average curvature of a trajectory during evolution for a dynamical system,which lumps all the bending effects of the trajectory to a number,and estimates its average size in virtue of an inscribed space ball.In an N-dimensional system,one may actually define N-1 curvature indices.The curvature index is applied to a low and high dimensional fractional order dynamic system.It is found that the curvature index can effectively determine the motion state of the fractional order dynamic system.3.Using the improved HHT,the time-frequency characteristics of the convergent process of the curvature exponent are analyzed.The results show that the convergence process of the curvature indices is periodic when the system is in regular motion,while it is out of order and no cyclic change when the system is in a chaotic state.4.It is considered for the complete synchronization of coupling fractional order chaotic system.It is found the curvature indices can accurately manifest the threshold of the coupling factor when the coupling system achieves complete synchronization,while the Lyapunov exponents,which can determine whether a system is in the state of chaos,have no sign of complete synchronization.The curvature index has more advantages over the Lyapunov exponent in the problem of coupling synchronization.