Relaxation oscillations in \({\mathbb R}^3\)
type: Item
instance of: scholarly article
title: Relaxation oscillations in \({\mathbb R}^3\)
zbMATH Open document ID: 1058.34055
author: Peter Szmolyan
author: Martin Wechselberger
published in: Journal of Differential Equations
publication date: 2004-10-01T00:00:00Z
review text: This paper is concerned with relaxation oscillations which arise from the singularly perturbed system \(\dot{x}=g_{1}(x,y,z,\varepsilon )\), \(\dot{y}=g_{2}(x,y,z,\varepsilon )\), \(\varepsilon \dot{z}=f(x,y,z,\varepsilon )\) with \(0< \varepsilon \ll 1\). The system has two slow and one fast variable. It is proved, under certain restrictions, that there exists a periodic relaxation orbit for small \(\varepsilon\). Proofs use geometric singular perturbation theory and blow-up methods. A singular form of the forced van der Pol equation is discussed in some detail.
Mathematics Subject Classification ID: 34C26
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 2104506
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: singular perturbations
zbMATH Keywords: blow-up methods
zbMATH Keywords: van der Pol equation
MaRDI profile type: Publication
full work available at URL: j.jde.2003.09.010
OpenAlex ID: W2093823001
cites work: Dynamical systems II. Ergodic theory with applications to dynamical systems and statistical mechanics. Transl. from the Russian
cites work: Singular perturbation of relaxed periodic orbits
cites work: On non-linear differential equations of the second order. II. The equation \(\ddot y+kf(y)\dot y+g(y,k)=p(t)=p_1(t)+kp_2(t)\); \(k>0,f(y)\geq 1\)
cites work: Q4280766
cites work: Canard cycles and center manifolds
cites work: Geometric singular perturbation theory for ordinary differential equations
cites work: Asymptotic expansions using blow-up
cites work: Asymptotic methods for relaxation oscillations and applications
cites work: Numerical computation of canards
cites work: Global bifurcations of periodic orbits in the forced Van der Pol equation
cites work: Q4847945
cites work: Q4840149
cites work: Extending geometric singular perturbation theory to nonhyperbolic points -- fold and canard points in two dimensions
cites work: Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator
cites work: Q4550183
cites work: Differential equations with small parameters and relaxation oscillations. Translation from the Russian by F. M. C. Goodspeed
cites work: Q4866649
cites work: Q3244598
cites work: On the Structure of Local Homeomorphisms of Euclidean n-Space, II
cites work: Canards in \(\mathbb{R}^3\)
DOI: 10.1016/J.JDE.2003.09.010
description: scientific article; zbMATH DE number 2104506