The Geometry of Isochrons in Planar Systems

Probing the response of an oscillator to a perturbation is an important tool for understanding the underlying dynamics of the oscillator. Mathematically, we can determine these responses by finding the so-called isochrons of the underlying periodic orbit. Each isochron is the set of points that converge to the periodic orbit in phase with each other. In this thesis, we investigate planar systems, for which isochrons are curves that can have very complex geometries; we study how these complexities arise. We start by considering the isochrons of a periodic orbit in the FitzHugh-Nagumo system, an example from the 1970s of a system with two time scales for which the overall structure of the isochrons could not be determined with the computational methods of that time. We show, via numerical continuation, that the isochrons exhibit extreme sensitivity and feature sharp turns. We observe that the sharp turns and sensitivity of the isochrons are associated with the slow-fast nature of the FitzHugh-Nagumo system; more specifically, they occur near a repelling (unstable) slow manifold. In order to characterise such sharp turns, we introduce the notion of backward-time isochrons and also extend the concept of an isochron to isochrons of a focus equilibrium. The backward-time isochrons are isochrons of the reversed-time system, and by considering their interaction with the forward-time isochrons, we are able to identify a new bifurcation: a cubic tangency of curves, which gives rise to non-transverse intersections between forward-time and backward-time isochrons. We call this bifurcation a cubic isochron foliation tangency, or CIFT bifurcation. It is not a local feature but happens along trajectories in the annulus where both sets of isochrons coexist. We construct and discuss examples of three mechanisms for a CIFT bifurcation: a global time-scale separation; a perturbation that increases the velocity along trajectories in a local region of phase space; and a canard explosion (in the Van der Pol system). Finally, we present our method for computing isochrons of periodic orbits and focus equilibria, and a novel approach for computing phase response curves, which is accurate for any size of the perturbation.

Name thesis:degree_name

PhD

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Applied Mathematics

Grantor dc:publisher

ResearchSpace@Auckland

Year dc:date.issued

2015

Author dc:creator

• Langfield, Peter

Advisors dc:contributor.advisor

• Osinga, H
• Krauskopf, B

Statement 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.

Licence dc:rights.uri

• https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm

Handle dc:identifier.uri

https://hdl.handle.net/2292/26209

OAI identifier oai:identifier

oai:researchspace.auckland.ac.nz:2292/26209