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arXiv · 2006.13373

Rigorous verification of Hopf bifurcations via desingularization and continuation

Jan Bouwe van den Berg·Jean-Philippe Lessard·Elena Queirolo
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Abstract

In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.

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Jan Bouwe van den Berg, Jean-Philippe Lessard, Elena Queirolo. 2020-06-23. Rigorous verification of Hopf bifurcations via desingularization and continuation. https://arxiv.org/abs/2006.13373

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