dc.contributor.author |
Szalai, Róbert |
en |
dc.contributor.author |
Osinga, Hinke |
en |
dc.date.accessioned |
2012-03-11T23:48:14Z |
en |
dc.date.issued |
2009 |
en |
dc.identifier.citation |
SIAM Journal on Applied Dynamical Systems 8(4):1434-1461 2009 |
en |
dc.identifier.issn |
1536-0040 |
en |
dc.identifier.uri |
http://hdl.handle.net/2292/13773 |
en |
dc.description.abstract |
The Neımark–Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At this bifurcation a periodic orbit loses stability, and, except at certain “strong” resonances, an invariant torus is born. The dynamics on the torus is organized by Arnol'd tongues in parameter space; inside the Arnol'd tongues phase-locked periodic orbits exist that disappear in saddle-node bifurcations on the tongue boundaries, and outside the tongues the dynamics on the torus is quasi-periodic. In this paper we investigate whether a piecewise-smooth system with sliding regions may exhibit an equivalent of the Neımark–Sacker bifurcation. The vector field defining such a system changes from one region in phase space to the next, and the dividing (or switching) surface contains a sliding region if the vector fields on both sides point toward the switching surface. We consider the grazing-sliding bifurcation at which a periodic orbit becomes tangent to the sliding region and provide conditions under which it can be thought of as a Neımark– Sacker bifurcation. We find that the normal form of the Poincar´e map derived at the grazing-sliding bifurcation is, in fact, noninvertible. The resonances are again organized in Arnol'd tongues, but the associated periodic orbits typically bifurcate in border-collision bifurcations that can lead to more complicated dynamics than simple quasi-periodic motion. Interestingly, the Arnol'd tongues of piecewise-smooth systems look like strings of connected sausages, and the tongues close at double border-collision points. Since in most models of physical systems nonsmoothness is a simplifying approximation, we relate our results to regularized systems. As one expects, the phase-locked solutions deform into smooth orbits that, in a neighborhood of the Ne˘ımark–Sacker bifurcation, lie on a smooth torus. The deformation of the Arnol'd tongues is more complicated; in contrast to the standard scenario, we find several coexisting pairs of periodic orbits near the points where the Arnol'd tongues close in the piecewise-smooth system. Nevertheless, the unfolding near the double border-collision points is also predicted as a typical scenario for nondegenerate smooth systems. |
en |
dc.publisher |
Society for Industrial and Applied Mathematics |
en |
dc.relation.ispartofseries |
SIAM Journal on Applied Dynamical Systems |
en |
dc.rights |
Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. Details obtained from http://www.sherpa.ac.uk/romeo/issn/1536-0040/ |
en |
dc.rights.uri |
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm |
en |
dc.title |
Arnol′d tongues arising from a grazing-sliding bifurcation |
en |
dc.type |
Journal Article |
en |
dc.identifier.doi |
10.1137/09076235X |
en |
pubs.begin-page |
1434 |
en |
pubs.volume |
8 |
en |
dc.rights.holder |
Copyright: Society for Industrial and Applied Mathematics |
en |
pubs.end-page |
1461 |
en |
dc.rights.accessrights |
http://purl.org/eprint/accessRights/RestrictedAccess |
en |
pubs.subtype |
Article |
en |
pubs.elements-id |
250687 |
en |
pubs.org-id |
Science |
en |
pubs.org-id |
Mathematics |
en |
pubs.number |
4 |
en |
pubs.record-created-at-source-date |
2011-12-05 |
en |