Kosambi–Cartan–Chern Analysis of the Nonequilibrium Singular Point in One-Dimensional Elementary Catastrophe

Abstract

This paper analyzes the properties of the nonequilibrium singular point in one-dimensional elementary catastrophe. For this analysis, the Kosambi–Cartan–Chern (KCC) theory is applied to characterize the dynamical system based on differential geometrical quantities. When both the nonlinear connection and deviation curvature are zero, that is, when the geometric stability of the KCC theory is neutral, two bifurcation curves are obtained: one is the known curve with an equilibrium singular point, and the other is a new curve with a nonequilibrium singular point. The two singular points are distinguished based on the vanishing condition of the Berwald connection. Applied to the ecosystem described by the Hill function, the absolute value of the cuspidal curvature of the nonequilibrium singular point is larger than that of the equilibrium singular point. The ecological interpretation of this result is that the range of bistability of the ecosystem in the nonequilibrium state is greater than that in the equilibrium state. The type of singular points in equilibrium and nonequilibrium bifurcation curves are not necessarily the same. For instance, there is a combination in which even if the former has one cusp, the latter may show various types, depending on the parametric space. These results demonstrate that there are cases where simply shifting the system from the equilibrium to nonequilibrium state expands the range of bistability and changes the type of singularity. Although singularity analysis is often performed near the equilibrium point, nonequilibrium analysis, i.e. analysis based on the KCC theory, provides a useful perspective for analyzing singularity theory according to the bifurcation phenomenon.