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arXiv · 1708.06545

Coordinate-independent criteria for Hopf bifurcations

Niclas Kruff·Sebastian Walcher
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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.

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Niclas Kruff, Sebastian Walcher. 2017-08-22. Coordinate-independent criteria for Hopf bifurcations. https://arxiv.org/abs/1708.06545

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