Computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence

Huguet, Gemma; Llave, Rafael de la

doi:10.1137/120901210

Cited by 61 publications

(75 citation statements)

References 30 publications

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“…Indeed, the case of limit cycles has been considered in [17,20] and related algorithms have been presented in [21] for the symplectic case. We observe that, if we write…”

Section: Computational Aspectsmentioning

confidence: 99%

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

2013

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Abstract. We study the behavior of conformally symplectic systems near rotational Lagrangian tori. We recall that conformally symplectic systems appear for example in mechanical models including a friction proportional to the velocity.We show that in a neighborhood of these quasi-periodic solutions (either transitive tori of maximal dimension or periodic solutions), one can always find a smooth symplectic change of variables in which the time evolution becomes just a rotation in some direction and a linear contraction in others. In particular quasi-periodic solutions of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, C r or C ∞ . We emphasize that the results presented here are non-perturbative and apply to systems that are far from integrable; moreover, we do not require any assumption on the frequency and in particular we do not assume any non-resonance condition.We also show that the system of coordinates can be computed rather explicitly and we provide iterative algorithms, which allow to generalize the notion of "isochrones". We conclude by showing that the above results apply to quasi-periodic conformally symplectic flows.

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“…Indeed, the case of limit cycles has been considered in [17,20] and related algorithms have been presented in [21] for the symplectic case. We observe that, if we write…”

Section: Computational Aspectsmentioning

confidence: 99%

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

2013

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“…In [34] the global isochrons were computed for a two-dimensional reduced Hodgkin-Huxley model via a two point boundary value continuation method. (Other recent methods can also be found in [20,24].) As stated by the authors, a higher-dimensional version of this method, although theoretically straightforward, would present challenges numerically and has not yet been implemented.…”

Section: Introductionmentioning

confidence: 99%

Global Isochrons and Phase Sensitivity of Bursting Neurons

Mauroy¹,

Rhoads²,

Moehlis³

et al. 2014

SIAM J. Appl. Dyn. Syst.

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Abstract. Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models. One popular way of capturing phase sensitivity is through the computation of isochrons-subsets of the state space that each converge to the same trajectory on the limit cycle. However, the computation of isochrons is notoriously difficult, especially for bursting neuron models. In [W. E. Sherwood and J. Guckenheimer, SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 659-703], the phase sensitivity of the bursting Hindmarsh-Rose model is studied through the use of singular perturbation theory: cross sections of the isochrons of the full system are approximated by those of fast subsystems. In this paper, we complement the previous study, providing a detailed phase sensitivity analysis of the full (three-dimensional) system, including computations of the full (twodimensional) isochrons. To our knowledge, this is the first such computation for a bursting neuron model. This was made possible thanks to the numerical method recently proposed in [A. Mauroy and I. Mezić, Chaos, 22 (2012), 033112]-relying on the spectral properties of the so-called Koopman operator-which is complemented with the use of adaptive quadtree and octree grids. The main result of the paper is to highlight the existence of a region of high phase sensitivity called the almost phaseless set and to completely characterize its geometry. In particular, our study reveals the existence of a subset of the almost phaseless set that is not predicted by singular perturbation theory (i.e., by the isochrons of fast subsystems). We also discuss how the almost phaseless set is related to empirically observed phenomena such as addition/deletion of spikes and to extrema of the phase response of the system. Finally, through the same numerical method, we show that an elliptic bursting model is characterized by a very high phase sensitivity and other remarkable properties.

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“…Intervals of positive (negative) effective forcing are the intervals where respiration is accelerating (decelerating) the heart rate. Interesting and relevant parallels could be drawn between coupling functions and amplitude response curves, or phaseamplitude response curves (Castejón, Guillamon, and Huguet, 2013;Huguet and de la Llave, 2013). The latter are similar to phase response curves, with the difference that there is also a response to the amplitude on increasing or decreasing the strength of the oscillations.…”

Section: Relation To Phase Response Curve In Experimentsmentioning

confidence: 95%

Coupling functions: Universal insights into dynamical interaction mechanisms

et al. 2017

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The dynamical systems found in nature are rarely isolated. Instead they interact and influence each other. The coupling functions that connect them contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how an interaction occurs. A coherent and comprehensive review is presented encompassing the rapid progress made recently in the analysis, understanding, and applications of coupling functions. The basic concepts and characteristics of coupling functions are presented through demonstrative examples of different domains, revealing the mechanisms and emphasizing their multivariate nature. The theory of coupling functions is discussed through gradually increasing complexity from strong and weak interactions to globally coupled systems and networks. A variety of methods that have been developed for the detection and reconstruction of coupling functions from measured data is described. These methods are based on different statistical techniques for dynamical inference. Stemming from physics, such methods are being applied in diverse areas of science and technology, including chemistry, biology, physiology, neuroscience, social sciences, mechanics, and secure communications. This breadth of application illustrates the universality of coupling functions for studying the interaction mechanisms of coupled dynamical systems.

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Computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence

Cited by 61 publications

References 30 publications

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

Global Isochrons and Phase Sensitivity of Bursting Neurons

Coupling functions: Universal insights into dynamical interaction mechanisms

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