Perturbation Theory for Strongly Perturbed Dynamical Systems. 1.

1968 ◽  
Vol 73 ◽  
pp. 522 ◽  
Author(s):  
William H. Jefferys
Keyword(s):  
Perturbation Theory ◽  
Dynamical Systems

On the Application of Geometric Singular Perturbation Theory to Some Classical Two Point Boundary Value Problems

1998 ◽  
Vol 08 (02) ◽  
pp. 189-209 ◽  
Author(s):  
Michael Hayes ◽  
Tasso J. Kaper ◽  
Nancy Kopell ◽  
Kinya Ono
Keyword(s):  
Perturbation Theory ◽  
Dynamical Systems ◽  
Singular Perturbation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation

In this tutorial, we illustrate how geometric singular perturbation theory provides a complementary dynamical systems-based approach to the method of matched asymptotic expansions for some classical singularly-perturbed boundary value problems. The central theme is that the criterion of matching corresponds to the criterion of transverse intersection of manifolds of solutions. This theme is studied in three classes of problems, linear: ∊y″+αy′+βy=0, semilinear: ∊y″+αy′+f(y)=0, and quasilinear: ∊y″+g(y) y′+f(y)=0, on the interval [0,1], where t∈[0,1], ′=d/dt, 0<∊≪1, and general boundary conditions y(0)=A, y(1)=B hold. Chosen for their relatively simple structure, these problems provide a useful introduction to the methods of geometric singular perturbation theory that are now widely used in dynamical systems, from reaction-diffusion equations with traveling waves to perturbed N-degree-of-freedom Hamiltonian systems, and in applications to a variety of fields.

scholarly journals Perturbation theory (dynamical systems)

Scholarpedia ◽  
2008 ◽  
Vol 3 (9) ◽  
pp. 2399
Author(s):  
Henk Broer ◽  
Heinz Hanßmann
Keyword(s):  
Perturbation Theory ◽  
Dynamical Systems
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