Geometric singular perturbation theory in biological practice

被引:185
作者
Hek, Geertje [1 ]
机构
[1] Univ Amsterdam, Amsterdam, Netherlands
关键词
FOOD-CHAIN CHAOS; TRACKING INVARIANT-MANIFOLDS; ASYMPTOTIC STABILITY; TRAVELING-WAVES; SPIKE SOLUTIONS; SOLITARY WAVES; LIMIT-CYCLES; MULTI-BUMP; MODEL; SYSTEMS;
D O I
10.1007/s00285-009-0266-7
中图分类号
Q [生物科学];
学科分类号
07 ; 0710 ; 09 ;
摘要
Geometric singular perturbation theory is a useful tool in the analysis of problems with a clear separation in time scales. It uses invariant manifolds in phase space in order to understand the global structure of the phase space or to construct orbits with desired properties. This paper explains and explores geometric singular perturbation theory and its use in (biological) practice. The three main theorems due to Fenichel are the fundamental tools in the analysis, so the strategy is to state these theorems and explain their significance and applications. The theory is illustrated by many examples.
引用
收藏
页码:347 / 386
页数:40
相关论文
共 92 条
[11]   Food chain chaos due to transcritical point [J].
Deng, B ;
Hines, G .
CHAOS, 2003, 13 (02) :578-585
[12]   Food chain chaos due to Shilnikov's orbit [J].
Deng, B ;
Hines, G .
CHAOS, 2002, 12 (03) :533-538
[13]   Food chain chaos due to junction-fold point [J].
Deng, B .
CHAOS, 2001, 11 (03) :514-525
[14]   Travelling waves in a singularly perturbed sine-Gordon equation [J].
Derks, G ;
Doelman, A ;
van Gils, SA ;
Visser, T .
PHYSICA D-NONLINEAR PHENOMENA, 2003, 180 (1-2) :40-70
[15]   RESONANT PATTERNS THROUGH COUPLING WITH A ZERO MODE [J].
DEWEL, G ;
METENS, S ;
HILALI, M ;
BORCKMANS, P ;
PRICE, CB .
PHYSICAL REVIEW LETTERS, 1995, 74 (23) :4647-4650
[16]   THE CANARD UNCHAINED OR HOW FAST SLOW DYNAMICAL-SYSTEMS BIFURCATE [J].
DIENER, M .
MATHEMATICAL INTELLIGENCER, 1984, 6 (03) :38-49
[17]   Stabilization by slow diffusion in a real Ginzburg-Landau system [J].
Doelman, A ;
Hek, G ;
Valkhoff, N .
JOURNAL OF NONLINEAR SCIENCE, 2004, 14 (03) :237-278
[18]   Homoclinic saddle-node bifurcations in singularly perturbed systems [J].
Doelman A. ;
Hek G. .
Journal of Dynamics and Differential Equations, 2000, 12 (1) :169-216
[19]   Breaking the hidden symmetry in the Ginzburg-Landau equation [J].
Doelman, A .
PHYSICA D-NONLINEAR PHENOMENA, 1996, 97 (04) :398-428
[20]   Semistrong pulse interactions in a class of coupled reaction-diffusion equations [J].
Doelman, A ;
Kaper, TJ .
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, 2003, 2 (01) :53-96