Geometric Singular Perturbation Theory (GSPT) and Conley Index Theory are two powerful techniques to analyze dynamical systems. Conley already realized that using his index is easier for singular perturbation problems. In this paper, we will revisit Conley’s results and prove that the GSPT technique of Fenichel Normal Form can be used to simplify the application of Conley index techniques even ...

Perturbation bounds for the relative error in the eigenvalues of diagonalizable and singular matrices are derived by using perturbation theory for simple invariant subspaces of a matrix and the group inverse of a matrix. These upper bounds are supplements to the related perturbation bounds for the eigenvalues of diagonalizable and nonsingular matrices. © 2006 Elsevier Inc. All rights reserved. ...

The renormalization group (RG) method for differential equations is one of the perturbation technique proposed by Chen, Goldenfeld, and Oono [1,2]. The RG method unifies traditional singular perturbation methods, such as the multi-scaling method, the boundary layer theory , the averaging method, the normal form theory, the center manifold theory, and the geometric singular perturbation. Chiba [...

In this paper, we investigate a singular perturbation problem including a fourth order O.D.E. with general linear boundary conditions. Firstly, we obtain the necessary conditions of solution of O.D.E. by making use of fundamental solution, then by compatibility of these conditions with boundary conditions, we determine that, for given perturbation problem, whether boundary layer is formed or not.

We study the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstr...