Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in with Three Zones

Cheng, Yanyan

doi:10.1155/2013/385419

Cited by 2 publications

(1 citation statement)

References 19 publications

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“…Ponce et al studied bifurcations leading to LCs in PWL planar systems in [51]. Existence of LCs has been shown for planar PWL systems with two regions in [24] and [36], for planar PWL systems with an arbitrary but finite number of separate regions in [62], and for a PWL system in R 4 with three regions in [6]. Stability of PWL LCs in R n with m + 1 regions is analyzed in [33] using Poincaré map techniques.…”

Section: Further Applicationsmentioning

confidence: 99%

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Park

Shaw

Chiel³

et al. 2018

Eur. J. Appl. Math

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The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

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Section: Further Applicationsmentioning

confidence: 99%

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Park

Shaw

Chiel³

et al. 2018

Eur. J. Appl. Math

View full text Add to dashboard Cite

show abstract

Periodic Solutions for a Four-Dimensional Coupled Polynomial System with N-Degree Homogeneous Nonlinearities

Tian¹,

Li²

2019

Mathematics

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This paper studies the periodic solutions of a four-dimensional coupled polynomial system with N-degree homogeneous nonlinearities of which the unperturbed linear system has a center singular point in generalization resonance 1 : n at the origin. Considering arbitrary positive integers n and N with n ≤ N and N ≥ 2 , the new explicit expression of displacement function for the four-dimensional system is detected by introducing the technique on power trigonometric integrals. Then some precise and detailed results in comparison with the existing works, including the existence condition, the exact number, and the parameter control conditions of periodic solutions, are obtained, which can provide a new theoretical description and mechanism explanation for the phenomena of emergence and disappearance of periodic solutions. Results obtained in this paper improve certain existing results under some parameter conditions and can be extensively used in engineering applications. To verify the applicability and availability of the new theoretical results, as an application, the periodic solutions of a circular mesh antenna model are obtained by theoretical method and numerical simulations.

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Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in with Three Zones

Cited by 2 publications

References 19 publications

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems

Periodic Solutions for a Four-Dimensional Coupled Polynomial System with N-Degree Homogeneous Nonlinearities

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