Analysis And Study Of The Curvature Indices For Dynamical Systems

With the development of science and technology,dynamical systems have been a hot research topic of Applied Mathematics for several decades,has been hitherto unknown development and penetrated into different fields.And it has a great impact on many branches of natural science and engineering.Chaos is the typical nonlinear phenomena can not predict.As an important branch of nonlinear science,chaos dynamics have attracted wide attention in the last century.After a few years of development,the basic form of chaos theories are synchronization,bifurcation,chaos control and anti control,system identification and its application.It is important to characterize the properties of dynamical systems by a quantity that signifies their structural changes,in particular those associated with occurrence of chaos or other transitional behaviors.There are some well-k:nown indices,such as Lyapunov exponent,fractal dimension,and Kolmogorov entropy,while in this article we use a new quantifier,named the curvature index,to study the dynamical systems.(1)The curvature index(proposed by Chen and Chang 2012)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(such as an attractor)in virtue of an inscribed space ball.One may define N-1 curvature indices for an N-dimensional dynamical system.The definition and calculation method of the curvature indices are given in this paper.(2)It is considered for basic curve and in three,four,or higher dimensional dynamical systems,once the system undergoes a change of the topological structure,there are corresponding changes in at least one or more curvature indices(in higher dimensions>2).This study is aimed to examine fundamental aspects of the curvature indices with further applications to some outstanding examples of dynamical systems in the literature.(3)It is considered for the coupled chaotic synchronization problem.The curvature indices may indicate effectively the different kinds of coupling system can achieve the synchronization,and shows the threshold point of some synchronization regimes.And the coupled Rossler systems are simulated to demonstrate the effectiveness of the curvature indices.(4)In parallel to compare with the performance of the Lyapunov exponents,the curvature indices were found very effective to characterize the change of the structural structure of the dynamical system.