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Stability Theory of Ordinary Differential Equations

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Mathematical Formulation of the Stability Concept and Basic Results

Stability in Conservative Systems and the KAM Theorem

Averaging and the Stability of Perturbed Periodic Orbits

Structural Stability

Attractors

Generalizations and Future Directions

Bibliography

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Abbreviations

Ordinary differential equation :

An equation for an unknown vector of functions of a single variable that involves derivatives of the unknown functions. The order of a differential equation is the highest order of the derivatives that appear. The most important class of differential equations are first-order systems of ordinary differential equations that can be written in the form \({\dot u=f(u,t)}\), where f is a given smooth function \({f \colon U\times J\to \mathbb{R}^n}\), U is an open subset of \({\mathbb{R}^n}\), and J is an open subset of \({\mathbb{R}}\). The unknown functions are the components of u, and the vector of their first-order derivatives with respect to the independent variable t is denoted by \({\dot u}\). A solution of this differential equation is a function \({u\colon K \to \mathbb{R}^n}\), where K is an open subset of J such that \({\frac{\text{d} u}{\text{d} t}(t)=f(u(t),t)}\) for all \({t\in K}\).

Dynamical system :

A set and a law of evolution for its elements. The first-order differential equation \({\dot u=f(u,t)}\), where \({f \colon U\times J\to \mathbb{R}^n}\), is the law of evolution for the set \({U\times J}\); it defines a continuous (time) dynamical system: Given \({(v,s)\in U\times J}\), the solution \({t\mapsto \psi(t,s,v)}\) such that \({\phi(s,s,v)=v}\) determines the evolution of the state v: the state v at time s evolves to the state \({\psi(t,s,v)}\) at time t. Similarly, a continuous function \({f \colon X\to X}\) on a metric space X defines a discrete dynamical system. The state \({x\in X}\) evolves to \({f^k(x)}\) (which denotes the value of f composed with itself k times and evaluated at x) after k time-steps. The images of \({t\mapsto \phi(t,s,v)}\) and \({k\mapsto f^k(x)}\) are called the orbits of the corresponding states v and x.

Stability theory :

The mathematical analysis of the behavior of the distances between an orbit (or set of orbits) of a dynamical system and all other nearby orbits.

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  1. Department of Mathematics, University of Missouri‐Columbia, Columbia, USA

    Carmen Chicone

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Chicone, C. (2012). Stability Theory of Ordinary Differential Equations. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_106

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