Mode concept to construct diffeomorphisms representing smooth/non-smooth dynamical systems

Abstract

This research introduces an innovative approach to automatically analyzing of non-smooth dynamical systems involving continuous and discrete components. The core of this approach is the “mode concept,” a framework that systematically separates the original systems into primitive parts and combines them with the proper rule. The study successfully constructs diffeomorphisms representing entire systems by applying the mode concept, equivalent to Poincaré maps. It implies that we can numerically obtain the Jacobian matrix and Hessian tensor of the map and use them to analyze the system behavior. The algorithm requires only the modes and their transition rules. In numerical experiments, the algorithm is implemented in Python and applied to the typical non-smooth dynamical systems: the Izhikevich neuron model, the forced Alpazur oscillator, and even the smooth systems: the Duffing equation and the Hénon map. The rich analysis results, such as the bifurcation diagrams, return map, and eigenvalue transition, are obtained using the algorithm.