User guide on Hopf bifurcation and time periodic orbits with pde2path

TL;DR

This paper provides a comprehensive user guide for implementing Hopf bifurcation analysis and periodic orbit continuation using the pde2path library, including details on systems with symmetries and related algorithms.

Contribution

It offers detailed implementation instructions and extends the pde2path capabilities to handle Hopf bifurcations, symmetric systems, and related bifurcation scenarios.

Findings

Guidelines for using pde2path for Hopf bifurcation analysis

Methods for continuing traveling and rotating waves in symmetric systems

Techniques for bifurcation from periodic orbits with Floquet multipliers

Abstract

We explain the setup for using the pde2path libraries for Hopf bifurcation and continuation of branches of periodic orbits and give implementation details of the associated demo directories. See [Uecker, Comm. in Comp. Phys., 2019] for a description of the basic algorithms and the mathematical background of the examples. Additionally we explain the treatment of Hopf bifurcations in systems with continuous symmetries, including the continuation of traveling waves and rotating waves in O(2) equivariant systems as relative equilibria, the continuation of Hopf bifurcation points via extended systems, and some simple setups for the bifurcation from periodic orbits associated to critical Floquet multipliers going through +-1.