Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch & Stephen Smale & Robert L. Devaney

Author:Morris W. Hirsch & Stephen Smale & Robert L. Devaney [Hirsch, Morris W.]
Language: eng
Format: mobi
ISBN: 9780123820112
Publisher: Elsevier Science
Published: 2012-03-11T16:00:00+00:00

As an example, all nonzero solutions of the undamped harmonic oscillator equation are periodic solutions. Like equilibrium points that are asymptotically stable, periodic solutions may also attract other solutions. That is, solutions may limit on periodic solutions just as they can approach equilibria.

In the plane, the limiting behavior of solutions is essentially restricted to equilibria and closed orbits, although there are a few exceptional cases. We will investigate this phenomenon in this chapter in the guise of the important Poincaré–Bendixson Theorem. We will see later that, in dimensions greater than two, the limiting behavior of solutions can be quite a bit more complicated.

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