Elementary Catastrophe's Chaos in One-Dimensional Discrete Systems Based on Nonlinear Connections and Deviation Curvature Statistics

This study shows, by means of numerical analysis, that the characteristics of discrete dynamical systems, in which chaos and catastrophe coexist, are closely related to the geometric statistics in Finsler geometry. The two geometric statistics introduced are nonlinear connections information, denoted as NI, and the mean deviation curvature, denoted as P. The quantity N-I can be used to determine the occurrence of chaos in terms of nonequilibrium stability. The resulting chaos is characterized by P in terms of the trajectory's robustness, which is related to the localization or globalization of chaos. The characteristics of catastrophe-induced chaos are clearly visualized through the contour topography of N-I, in which an abrupt change is represented by cliff topography (i.e. a line of critical points); initial dependence is reflected in the reversibility of topographic patterns. On overlaying the contour topography with the singularity pattern, it is evident that chaos does not arise around the singular point. Furthermore, the extensive development of cusp and butterfly chaos demands information on the nonlinear connections within the singularity pattern. The asymmetry in swallowtail chaos is less distinguishable in an equilibrated state, but becomes more evident when the system is in a state of nonequilibrium. In many analyses, chaos and catastrophe are examined separately. However, these results demonstrate that when both are present, the two have a complex relationship constrained by the singularity.