Fundamental aspects of curvature indices for characterizing dynamical systems

Abstract

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-known 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. The curvature index (proposed by Chen and Chang in Chaos 22(2):371–383 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. Once the system undergoes a structural change, there are corresponding changes in the first and/or higher curvatures. The study is aimed to examine fundamental aspects of the curvature indices with further applications to some outstanding examples of dynamical systems in the literature, in parallel to the analysis by the Lyapunov exponents.