KCC Analysis of the Normal Form of Typical Bifurcations in One-Dimensional Dynamical Systems: Geometrical Invariants of Saddle-Node, Transcritical, and Pitchfork Bifurcations

Abstract

The Jacobi stability of the normal form of typical bifurcations in one-dimensional dynamical systems is analyzed by introducing the concept of the production process (time-like potential) to KCC theory. This KCC theory approach shows that the geometric invariants of the system characterize the nonequilibrium dynamics of the bifurcations. For example, the deviation curvature that is one of the geometric invariants shows that the well-known two hysteresis jumps in subcritical pitchfork bifurcations differ qualitatively from each other. In the nonequilibrium region, the deviation curvature in the saddle-node and the transcritical bifurcations are a function of the bifurcation parameter alone; thus, the Jacobi stability does not depend on time. However, the deviation curvature in a pitchfork bifurcation is a function of the time-like potential, so the Jacobi stability does depend on time. This time dependence can be described by the Douglas tensor, which is a useful geometric invariant to consider how the higher-order term in the bifurcation system affects the stability structure.