The VIVID function for numerically continuing periodic orbits arising from grazing bifurcations of hybrid dynamical systems Article Swipe
Indranil Ghosh
,
David J. W. Simpson
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2510.16218
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2510.16218
Periodic orbits of systems of ordinary differential equations can be found and continued numerically by following fixed points of Poincaré maps. However, this often fails near grazing bifurcations where a periodic orbit collides tangentially with a boundary of phase space. Failure occurs when the map contains a square-root singularity and the root-finding algorithm searches beyond the domain of viable values. We show that by instead following the zeros of a function that maps Velocity Into Variation In Displacement (VIVID) this issue is circumvented and there is no such failure. We illustrate this with a prototypical one-degree-of-freedom impact oscillator model by applying Newton's method to the VIVID function to follow periodic orbits collapsing into grazing bifurcations. We also follow curves of saddle-node and period-doubling bifurcations of periodic orbits that issue from a codimension-two resonant grazing bifurcation. The VIVID function provides a simple alternative to the more sophisticated collocation method and enables periodic orbits and their bifurcations to be resolved easily and accurately near grazing bifurcations.
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- preprint
- Landing Page
- http://arxiv.org/abs/2510.16218
- https://arxiv.org/pdf/2510.16218
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- https://openalex.org/W4415950443
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The VIVID function for numerically continuing periodic orbits arising from grazing bifurcations of hybrid dynamical systemsWork title
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preprintOpenAlex work type
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2025Year of publication
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2025-10-17Full publication date if available
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Indranil Ghosh, David J. W. SimpsonList of authors in order
- Landing page
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https://arxiv.org/abs/2510.16218Publisher landing page
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https://arxiv.org/pdf/2510.16218Direct link to full text PDF
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YesWhether a free full text is available
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