Ordinary differential equation perturbation theory

The analysis of ordinary differential equation (ODE) systems with small parameters e (with 0 < generally referred to as perturbation analysis or perturbation theory. Perturbation theory has been the subject of many fundamental research contributions (Fenichel 1979, Ladde and Siljak 1983), finding applications in many areas, including linear and nonlinear control systems, fluid mechanics, and reaction engineering (see, e.g., Kokotovic et al. 1986, Kevorkian and Cole 1996, Verhulst 2005). The main concepts of perturbation theory are presented below, following closely the developments in (Kokotovic et al. 1986). 

Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equat., 31, 53. 

A traditional course on nonlinear ordinary differential equations, but with more emphasis on applications and less on perturbation theory than usual. Such a course would focus on Chapters 1-8. 

The three chapters in this volume deal with various aspects of singular perturbations and their numerical solution. The first chapter is concerned with the analysis of some singular perturbation problems that arise in chemical kinetics. In it the matching method is applied to find asymptotic solutions of some dynamical systems of ordinary differential equations whose solutions have multiscale time dependence. The second chapter contains a comprehensive overview of the theory and application of asymptotic approximations for many different kinds of problems in chemical physics, with boundary and interior layers governed by either ordinary or partial differential equations. In the final chapter the numerical difficulties arising in the solution of the problems described in the previous chapters are discussed. In addition, rigorous criteria are proposed for... 

Some peculiar features of the perturbation theory given here seem worth mentioning. The only assumption to be made in our theory is that the individual oscillators are weakly perturbed, and nothing is assumed about the specific nature of those oscillators. They may be quite general ones, except that they obey a system of n ordinary differential equations the system need not lie near some bifurcation point nor in any other extreme situation. Since limit cycle motion... 

If one tries to develop a perturbation theory proceeding directly from the reaction-diffusion equations this meets with serious difficulties. They arise because translation and rotation perturbation modes for the spiral wave are not spatially localized. We bypass such difficulties by using the quasi-steady approximation formulated in the previous section. In this approximation the trajectory of the tip motion can be calculated by solving a system of ordinary differential equations which depend only on the local properties of the medium in the vicinity of the tip. The perturbations which originate outside a small neighbourhood of the end point propagate quickly to the periphery and do not influence the motion of the tip. The evolution of the entire curve can then be calculated in the WR approximation using the known trajectory of the tip motion as a dynamic boundary condition. 

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