Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

2001 ◽  
pp. 29-63 ◽  
Author(s):  
Freddy Dumortier ◽  
Robert Roussarie
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Normal Hyperbolicity ◽  
Geometric Singular Perturbation

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

2020 ◽  
pp. 2150003
Author(s):  
Ali Atabaigi
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Limit Cycles ◽  
Heteroclinic Orbits ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Generalist Predator ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Geometric Singular Perturbation

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

scholarly journals Propagation of hexagonal patterns near onset

2003 ◽  
Vol 14 (1) ◽  
pp. 85-110 ◽  
Author(s):  
ARJEN DOELMAN ◽  
BJÖRN SANDSTEDE ◽  
ARND SCHEEL ◽  
GUIDO SCHNEIDER
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Bifurcation Theory ◽  
Spatial Dynamics ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Pattern Forming

For a pattern-forming system with two unbounded spatial directions that is near the onset to instability, we prove the existence of modulated fronts that connect (i) stable hexagons with the unstable trivial pattern, (ii) stable hexagons with unstable roll solutions, (iii) stable hexagons with unstable hexagons, and (iv) stable roll solutions with unstable hexagons. Our approach is based on spatial dynamics, bifurcation theory, and geometric singular perturbation theory.

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