Dynamical Systems and Numerical Analysis

TL;DR: In this article, the authors focus on the initial-value problem of nonlinear dynamics and provide a good introduction to the main ideas and results of the theory of dynamical systems.

Abstract: T hat numerical analysis is an extremely useful tool for solving problems and exploring fundamental concepts comes as no surprise to any student of dynamical systems. However , preoccupation with learning the mathematical theory of nonlinear dynamics often precludes a deeper understanding of the variety of numerical methods available and prevents a proper appreciation of the possibilities and limitations of these methods. On the other hand, many numerical analysts spend a large part of their careers not realizing that they are actually solving problems of nonlinear dynamics and struggling with issues of dynamical-systems and chaos theory. Clearly, their work would benefit greatly from a closer familiarity with the main ideas and results of that theory. These are precisely the two scientific communities that Stuart and Hum-phries' book aims to address. Its purpose is evidently to help researchers in these communities get better acquainted with each other. It is a timely publication which, in my opinion, will appeal to many members of the two groups and succeed in achieving its purpose to a considerable extent. Of differential equations The book is exclusively concerned with the solution of the initial-value problem of ordinary differential equations, (1) with u(t) ∈ ‫ޒ‬ p , t > 0; and f : ‫ޒ‬ p → ‫ޒ‬ P where f is (at least) continuous and Lipschitz, so that a solution of Equation 1 exists and is unique. Since an analytical expression of this solution is hardly ever available, one attempts to solve Equation 1 numerically, by writing it as a dynamical system (2) with G : D → ‫ޒ‬ p. The solution of Equation 2 exists, if the U n 's remain bounded within D ⊆ ‫ޒ‬ p for all n, and is unique as long as G is single-valued. U n is, of course, the approximation of u(t n) at the nth time step, t n = n∆t, n = 0, 1, 2, …, and is expected to be increasingly accurate as ∆t → 0. Clearly, the two most crucial issues facing a researcher who attempts this approximation are convergence and stability of the numerical strategy adopted. Convergence here means the ability to determine bounds for the norm . of the error: (3) where c 1 , c 2 , and r are appropriate constants. Stability refers to one's capacity to control the effect of small perturbations on the particular method chosen. More precisely, let us …