Jacobi Stability of Linearized Geometric Dynamics

2017

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

2017

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

2020

This paper considers the stability of a one-dimensional system during a catastrophic shift described by the Hill function. Because the shifting process goes through a nonequilibrium region, we applied the theory of Kosambi, Cartan, and Chern (KCC) to analyze the stability of this region based on the differential geometrical invariants of the system. Our results show that the Douglas tensor, one of the invariants in the KCC theory, affects the robustness of the trajectory during a catastrophic shift. In this analysis, the forward and backward shifts can have different Jacobi stability structures in the nonequilibrium region. Moreover, the bifurcation curve of the catastrophic shift can be interpreted geometrically, as the solution curve where the nonlinear connection and the deviation curvature become zero. KCC analysis also shows that even if the catastrophic pattern itself is similar, the stability structure in the nonequilibrium region is different in some cases, from the viewpoint of the Douglas tensor.

Kosambi–Cartan–Chern Stability in the Intermediate Nonequilibrium Region of the Brusselator Model

2022

This study applies the Kosambi–Cartan–Chern (KCC) theory to the Brusselator model to derive differential geometric quantities related to bifurcation phenomena. Based on these geometric quantities, the KCC stability of the Brusselator model is analyzed in linear and nonlinear cases to determine the extent to which nonequilibrium affects bifurcation and stability. The geometric quantities of the Brusselator model have a constant value in the linear case, and are functions of spatial variables with parameter dependence in the nonlinear case. Therefore, the KCC stability of the nonlinear case shows various distribution patterns, depending on the distance from the equilibrium point (EQP), as follows: in the regions near or far enough from the EQP, the distribution of KCC stability is uniform and regular; and in the intermediate nonequilibrium region, the distribution varies and shows complex patterns with parameter dependence. These results indicate that stability in the intermediate nonequilibrium region plays an important role in the dynamic complex patterns in the Brusselator model.