Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch

Author:Morris W. Hirsch
Language: eng
Format: epub, pdf
ISBN: 9780123820112
Publisher: Elsevier Science
Published: 2012-03-11T16:00:00+00:00

10.1 Limit Sets

We begin by describing the limiting behavior of solutions of systems of differential equations. Recall that is an ω-limit point for the solution through X if there is a sequence tn → ∞ such that . That is, the solution curve through X accumulates on the point Y as time moves forward. The set of all ω-limit points of the solution through X is the ω-limit set of X and is denoted by ω (X). The α-limit points and the α-limit set α(X) are defined by replacing tn → ∞ with tn → −∞ in the above definition. By a limit set we mean a set of the form ω(X) or α(X).

Here are some examples of limit sets. If X* is an asymptotically stable equilibrium, it is the ω-limit set of every point in its basin of attraction. Any equilibrium is its own α- and ω-limit set. A periodic solution is the α-limit and ω-limit set of every point on it. Such a solution may also be the ω-limit set of many other points.

Example

Consider the planar system given in polar coordinates by

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