A Continuation Approach to Computing Phase Resetting Curves

Langfield, Peter; Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1007/978-3-030-51264-4_1

Cited by 3 publications

(1 citation statement)

References 32 publications

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“…The construction in Section 2.6.1 can be applied to abstract autonomous periodic boundary-value problems and the observable T (the unknown period) to obtain so-called phase response curves [42,51,59,71].…”

Section: Phase Response Curves As Linear Responsementioning

confidence: 99%

Methods of Continuation and their Implementation in the COCO Software Platform with Application to Delay Differential Equations

Ahsan,

Dankowicz,

et al. 2021

Preprint

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This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the COCO software package and its explicit support for community-driven development. To this end, the paper first formalizes the COCO construction paradigm for augmented continuation problems compatible with simultaneous analysis of implicitly defined manifolds of solutions to nonlinear equations and the corresponding adjoint variables associated with optimization of scalar objective functions along such manifolds. The paper uses applications to data assimilation from finite time histories and phase response analysis of periodic orbits to identify a universal paradigm of construction that permits abstraction and generalization. It then details the theoretical framework for a COCO-compatible toolbox able to support the analysis of a large family of delay-coupled multi-segment boundary-value problems, including periodic orbits, quasiperiodic orbits, connecting orbits, initial-value problems, and optimal control problems, as illustrated in a suite of numerical examples. The paper aims to present a pedagogical treatment that is accessible to the novice and inspiring to the expert by appealing to the many senses of the applied nonlinear dynamicist. Sprinkled among a systematic discussion of problem construction, graph representations of delaycoupled problems, and vectorized formulas for problem discretization, the paper includes an original derivation using Lagrangian sensitivity analysis of phase-response functionals for periodic-orbit problems in abstract Banach spaces, as well as a demonstration of the regularizing benefits of multi-dimensional manifold continuation for near-singular problems analyzed using real-time experimental data.

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Section: Phase Response Curves As Linear Responsementioning

confidence: 99%

Methods of Continuation and their Implementation in the COCO Software Platform with Application to Delay Differential Equations

Ahsan,

Dankowicz,

et al. 2021

Preprint

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Global Manifolds of Saddle Periodic Orbits Parametrised by Isochrons

Hannam,

Krauskopf,

Osinga

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Isochron foliations and global bifurcations: a case study

Hannam

Krauskopf

Osinga

2022

Transactions of Mathematics and Its Applications

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The basins of attraction of periodic orbits or focus equilibria of a given vector field are foliated by forward-time isochrons, defined as all initial conditions that synchronize under the flow with a given phase. Similarly, backward-time isochrons of repelling periodic orbits or focus equilibria foliate their respective basins of repulsion. We present a case study of a planar system that features a sequence of bifurcations, including a saddle-node bifurcation of periodic orbits, a homoclinic bifurcation and Hopf bifurcations, that change the nature and existence of periodic orbits. We explain how the basins and isochron foliations change throughout the sequence of bifurcations. In particular, we identify structurally stable tangencies between the foliations by forward-time and backward-time isochrons, which are curves in the plane, in regions of phase space where they exist simultaneously. Such tangencies are generically quadratic and associated with sharp turns of isochrons and phase sensitivity of the system. In contrast to the earlier reported cubic isochron foliation tangency (CIFT) mechanism, which generates a pair of tangency orbits, we find isochron foliation tangencies that occur along single specific orbits in the respective basin of attraction or repulsion. Moreover, the foliation tangencies we report arise from actual bifurcations of the system, while a CIFT is not associated with a topological change of the underlying phase portrait. The properties and interactions of isochron foliations are determined and illustrated by computing a representative number of forward-time and backward-time isochrons as arclength-parametrized curves with a boundary value problem set-up. Our algorithm for computing isochrons has been further refined and implemented in the Matlab package CoCo; it is made available as Matlab code in the supplementary material of this paper, together with a guide that walks the user through the computation of two specific isochron foliations.

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A Continuation Approach to Computing Phase Resetting Curves

Cited by 3 publications

References 32 publications

Methods of Continuation and their Implementation in the COCO Software Platform with Application to Delay Differential Equations

Methods of Continuation and their Implementation in the COCO Software Platform with Application to Delay Differential Equations

Global Manifolds of Saddle Periodic Orbits Parametrised by Isochrons

Isochron foliations and global bifurcations: a case study

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