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Encyclopedia of Complexity and Systems Science pp 1–31Cite as

Stability Theory of Ordinary Differential Equations

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Definition of the Subject

An orbit or set of orbits of a dynamical system is stable if all solutions starting nearby remain nearby for all future times. This concept is of fundamental importance in applied mathematics: the stable solutions of mathematical models of physical processes correspond to motions that are observed in nature.

Introduction

Stability theory began with a basic question about the natural world: Is the solar system stable? Will the present configuration of the planets and the sun remain forever; or, might some planets collide, radically change their orbits, or escape from the solar system?

With the advent of Isaac Newton’s second law of motion and the law of universal gravitation, the motions of the planets in the solar system were understood to correspond to the solutions of the Newtonian system of ordinary differential equations that modeled the positions and velocities of the planets and the sun according to their mutual gravitational attractions. Short-term...

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Abbreviations

Dynamical system:

A set and a law of evolution for its elements. The first-order differential equation \( \dot{u}=f\left(u,t\right) \), where f:U × J → ℝ n, is the law of evolution for the set U ⊆ ℝ n; it defines a continuous (time) dynamical system: Given (v,s) ∈ U × J, the solution t → ψ(t,s,v) such that ψ(s,s,v) = v determines the evolution of the state v: the state v at time s evolves to the state ψ(t,s,v) at time t. Similarly, a continuous function f:X → X on a metric space X defines a discrete dynamical system. The state x ∈ 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 → ψ(t,s,v) and k → f k(x) are called the orbits of the corresponding states v and x.

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 \( \dot{u}=f\left(u,t\right) \), where f is a given smooth function f:U × J → ℝ n, U is an open subset of ℝn, and J is an open subset of 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:K → ℝ n, where K is an open subset of J such that \( \frac{ du}{ dt}(t)=f\left(u(t),t\right) \) for all t ∈ K.

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, MO, USA

    Carmen Chicone

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    Robert A. Meyers

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Chicone, C. (2013). Stability Theory of Ordinary Differential Equations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_516-3

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