Bifurcation equations for periodic orbits of implicit discrete dynamical systems

TL;DR

This paper derives bifurcation equations and conditions for periodic points in one-dimensional implicit discrete dynamical systems, with applications to numerical ODE methods like backward Euler and trapezoid methods.

Contribution

It provides the first comprehensive derivation of bifurcation equations and conditions for implicit systems, including practical examples involving numerical methods.

Findings

Bifurcation equations for various bifurcations are established.

Conditions for non-degeneracy and transversality are formulated.

Examples demonstrate bifurcations in numerical ODE methods.

Abstract

Bifurcation equations, non-degeneracy and transversality conditions are obtained for the fold, transcritical, pitchfork and flip bifurcations for periodic points of one dimensional implicitly defined discrete dynamical systems. The backward Euler method and the trapezoid method for numeric solutions of ordinary differential equations fall in the category of implicit dynamical systems. Examples of bifurcations are given for some implicit dynamical systems including bifurcations for the backward Euler method when the step size is changed.