Coordinate-independent criteria for Hopf bifurcations

TL;DR

This paper introduces a coordinate-independent method for analyzing Hopf bifurcations in parameter-dependent ODEs, simplifying the computation of critical parameters and bifurcation nature without relying on special coordinate systems.

Contribution

It presents a streamlined approach to compute bifurcation coefficients in arbitrary coordinates, reducing computational complexity compared to existing methods.

Findings

Method effectively computes bifurcation coefficients in examples

Coordinate-independent approach simplifies bifurcation analysis

Reduces computational effort compared to normal form reduction

Abstract

We discuss the occurrence of Poincar\'e-Andronov-Hopf bifurcations in parameter dependent ordinary differential equations, with no a priori assumptions on special coordinates. The first problem is to determine critical parameter values from which such bifurcations may emanate, a solution for this problem was given by W.-M. Liu. We add a few observations from a different perspective. Then we turn to the second problem, viz., to compute the relevant coefficients which determine the nature of the Hopf bifurcation. As shown by J. Scheurle and co-authors, this can be reduced to the computation of Poincar\'e-Dulac normal forms (in arbitrary coordinates) and subsequent reduction, but feasibility problems quickly arise. In the present paper we present a streamlined and less computationally involved approach to the computations. The efficiency and usefulness of the method is illustrated by examples.