American Institute of Mathematical Sciences

Coordinate-independent criteria for Hopf bifurcations

 Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany

* Corresponding author: Niclas Kruff

Dedicated to Jürgen Scheurle on the occasion of his retirement from non-mathematical duties

Received  August 2017 Revised  January 2018 Published  April 2019

Fund Project: The frst author acknowledges support by the DFG Research Training Group GRK 1632 "Experimental and constructive algebra". Both authors thank an anonymous reviewer for helpful comments

We discuss the occurrence of Poincaré-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 Poincaré-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.

Citation: Niclas Kruff, Sebastian Walcher. Coordinate-independent criteria for Hopf bifurcations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020075
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