Geometric singular perturbation theory for ordinary differential equations
About: This article is published in Journal of Differential Equations.The article was published on 1979-01-01 and is currently open access. It has received 2163 citations till now. The article focuses on the topics: Poincaré–Lindstedt method & Singular solution.
Citations
More filters
01 Jan 1995
1,221 citations
TL;DR: In this article, the effect of finite thermal conductivity and recirculation in droplets can be taken into account using the so-called parabolic model, which is a reasonable compromise between accuracy and CPU efficiency.
724 citations
Cites methods from "Geometric singular perturbation the..."
...A useful analytical tool for the analysis of stiff systems of ODEs, used for modelling of spray heating, evaporation and ignition, could be based on the geometrical asymptotic approach to singularly perturbed systems (integral manifold method) as developed by Gol’dshtein and Sobolev [384,385] for combustion applications (see also [386])....
[...]
[...]
01 Jan 2001
TL;DR: Mostafa Adimy as mentioned in this paper Directeur de Recherches à l’INRIA Dir. de thèse Ionel S. CIUPERCA Mâıtre de Conférence à l'Université Lyon 1 Examinateur Michael C. MACKEY Directeur of Recherche et al.
Abstract: Mostafa ADIMY Directeur de Recherches à l’INRIA Dir. de thèse Ionel S. CIUPERCA Mâıtre de Conférence à l’Université Lyon 1 Examinateur Michael C. MACKEY Directeur de Recherche à l’Université Mc GIll Dir. de thèse Sylvie MÉLÉARD Professeur à l’Ecole Polytechnique Examinatrice Sophie MERCIER Professeur à l’Université de Pau et des Pays de l’Adour Rapportrice Laurent PUJO-MENJOUET Mâıtre de Conférence à l’Université Lyon 1 Dir. de thèse Marta TYRAN-KAMIŃSKA Professeur à l’University of Silesia Examinatrice Bernard YCART Professeur à l’Université de Grenoble Rapporteur
713 citations
TL;DR: In this paper, the interaction of an excitable system with a slow oscillation was investigated, and it was shown that such a coupled system can display bursting, i.e. a stable solution in which some variable undergoes rapid oscillations followed by a period of quiescence, with both oscillation and quyingcence continually repeated.
Abstract: We investigate the interaction of an excitable system with a slow oscillation. Under robust and general assumptions compatible with the more stringent assumptions usually made about excitable systems, we show that such a coupled system can display bursting, i.e. a stable solution in which some variable undergoes rapid oscillations followed by a period of quiescence, with both oscillation and quiescence continually repeated. Under a further weak condition, the bursting is “parabolic”, i.e. the local frequency of the fast oscillation increases and then decreases within a burst. The technique in this paper involves nonlinear changes of coordinates which transform the equations into ones which are closely related to Hill’s equation.
592 citations
TL;DR: Recent developments in modeling social-ecological systems are presented, some of these challenges are illustrated with examples related to coral reefs and grasslands, and the implications for economic and policy analysis are identified.
Abstract: Systems linking people and nature, known as social-ecological systems, are increasingly understood as complex adaptive systems. Essential features of these complex adaptive systems – such as nonlinear feedbacks, strategic interactions, individual and spatial heterogeneity, and varying time scales – pose substantial challenges for modeling. However, ignoring these characteristics can distort our picture of how these systems work, causing policies to be less effective or even counterproductive. In this paper we present recent developments in modeling social-ecological systems, illustrate some of these challenges with examples related to coral reefs and grasslands, and identify the implications for economic and policy analysis.
555 citations
Cites background from "Geometric singular perturbation the..."
...Singular perturbation analysis (e.g., Wasow, 1965; Fenichel, 1979; Berglund and Gentz, 2003) can help analyze dynamical systems evolving in a fast–slow time framework....
[...]
References
More filters
Book•
01 Jan 1941
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Abstract: Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear partial differential equations: Sobolev spaces Second-order elliptic equations Linear evolution equations Part III: Theory for nonlinear partial differential equations: The calculus of variations Nonvariational techniques Hamilton-Jacobi equations Systems of conservation laws Appendices Bibliography Index.
25,734 citations
2,725 citations
1,187 citations
"Geometric singular perturbation the..." refers background or methods in this paper
...The uniqueness results, Theorem 9.l.iv, follow from the uniqueness of the invariant manifolds and invariant families constructed in Fenichel (1971, 1974, 1977)....
[...]
...The main analytical tool used in this paper is the invariant manifold theory developed in Fenichel (1971, 1974, 1977)....
[...]