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.

KCC Analysis of a One-Dimensional System During Catastrophic Shift of the Hill Function: Douglas Tensor in the Nonequilibrium Region

“…Following Antonelli's approach [Antonelli et al, 1993;Yamasaki & Yajima, 2017], we introduce the concept of time-like potential x i , defined as…”

“…where a(̸ = 0) is a constant. This approach is well-suited to the study of bifurcation in dynamical systems [Yamasaki & Yajima, 2017]. Here, the deviation curvature (4) can be simplified as follows…”

“…The theory derives geometrical invariants of an ordinary differential equation, such as Eq. ( 1), and applies them to various systems, such as a dynamic system with bifurcations in the non-equilibrium region (e.g., [Yamasaki & Yajima, 2017]). In this study, KCC theory is applied to Eq.…”

“…( 1) for stability analysis during a catastrophic shift. In this application, the Douglas tensor, one of the invariant quantities in KCC theory, is used to investigate stability in the non-equilibrium region ( [Yamasaki & Yajima, 2017])…”

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

“…Nowadays, Jacobi stability analysis has become a useful tool in the study of the complexity of typical chaotic systems. This type of geometric analysis concerning many chaotic systems, such as Lorenz system, Chen system, Rössler system, Hamiltonian system, modified Chua circuit system, Rikitake system, Navier-Stokes system, Rabinovich-Fabrikant system, an unusual Lorenz-like chaotic system, can be found in the recent literature [18,24,3,19,47,20,50,26,21,16]. Research has shown that Jacobi instability can be used to interpret the nature of chaos.…”