Reversal-time dynamics of two-dimensional non-autonomous systems

This study, we discusses the topological type of nonautonomous systems with periodic solutions when the time variable t increases negatively. When linear approximation holds near a fixed point of the Poincaré map, we confirm that the bifurcation points where the fixed point becomes nonhyperbolic are invariant regardless of the time direction however, the stability of the fixed point is changed. Consequently, we show that two-dimensional bifurcation diagrams obtained by the brute-force method give different results for positive and reversal-time variable systems; however, the bifurcation curves are identical. The inverted time variable system is useful for visualizing the completely unstable fixed point, because the repeller can be observed as an attractor. Furthermore, in certain models, chaotic attractors with a wide parameter range exist in reversal time variable systems.