Bifurcation of Limit Cycles from Boundary Equilibria in Impacting Hybrid Systems

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A semianalytical method is derived for finding the existence and stability of single-impact periodicorbits born in a boundary equilibrium bifurcation in a generaln-dimensional impacting hybridsystem. Known results are reproduced for planar systems and general formulae derived for three-dimensional (3D) systems. A numerical implementation of the method is illustrated for several 3Dexamples and for an 8D wing-flap model that shows coexistence of attractors. It is shown how themethod can easily be embedded within numerical continuation, and some remarks are made aboutnecessary and sufficient conditions in arbitrary dimensional systems
Original languageEnglish
Article number4
Pages (from-to)3320 - 3357
Number of pages38
JournalSIAM Journal on Applied Dynamical Systems
Issue number4
Publication statusPublished - 6 Dec 2023

Bibliographical note

Funding Information:
\ast Received by the editors February 8, 2023; accepted for publication (in revised form) by V. Kirk July 28, 2023; published electronically December 6, 2023. Funding: The work of the first author was supported by the University of Bristol and Chinese Scholarship Council joint studentship 202006120007. \dagger Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK (,

Publisher Copyright:
© 2023 Society for Industrial and Applied Mathematics Publications. All rights reserved.

Structured keywords

  • Engineering Mathematics Research Group


  • impact
  • hybrid system
  • periodic orbit
  • boundary equilibrium bifurcation


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