Uncountably Many Cases of Filippov’s Sewed Focus

Paul Glendinning, S J Hogan*, Martin E Homer, Mike R Jeffrey, Robert Szalai

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

The sewed focus is one of the singularities of planar piecewise smooth dynamical systems. Defined by Filippov in his book (Differential Equations with Discontinuous Righthand Sides, Kluwer, 1988), it consists of two invisible tangencies either side of the switching manifold. In the case of analytic focus-like behaviour, Filippov showed that the approach to the singularity is in infinite time. For the case of non-analytic focus-like behaviour, we show that the approach to the singularity can be in finite time. For the non-analytic sewed centre-focus, we show that there are uncountably many different topological types of local dynamics, including cases with infinitely many stable periodic orbits, and show how to create systems with periodic orbits intersecting any bounded symmetric closed set.
Original languageEnglish
Article number52 (2023)
Pages (from-to)1-14
Number of pages14
JournalJournal of Nonlinear Science
Volume33
Issue number4
DOIs
Publication statusPublished - 5 May 2023

Bibliographical note

Funding Information:
We are grateful to David Chillingworth (University of Southampton) for his insights on the analyticity of return maps, and to James Montaldi and Jonathan Fraser (University of Manchester) for helpful discussions, especially concerning Lemma 5. We are extremely grateful to an anonymous reviewer who supplied us with example (16), to complete the proof of Theorem 3. SJH would like to thank the Hungarian Academy of Sciences for the award of a Distinguished Guest Scientist Fellowship, during which part of this paper was written.

Funding Information:
We are grateful to David Chillingworth (University of Southampton) for his insights on the analyticity of return maps, and to James Montaldi and Jonathan Fraser (University of Manchester) for helpful discussions, especially concerning Lemma . We are extremely grateful to an anonymous reviewer who supplied us with example (), to complete the proof of Theorem . SJH would like to thank the Hungarian Academy of Sciences for the award of a Distinguished Guest Scientist Fellowship, during which part of this paper was written.

Publisher Copyright:
© 2023, The Author(s).

Structured keywords

  • Engineering Mathematics Research Group

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