Global stability for dynamical systems in $R^n$ - MathOverflow

Global stability for dynamical systems in $R^n$

2010-05-18T18:36:12Z

Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs). Assume that:

(1) The system has an absorbing ball, that is every trajectory eventually enters this ball and stays in it.

(2) The system has a unique stationary point, and this stationary point is locally asymptotically stable.

(2) The system has no period orbits.

Can we conclude that the stationary point is in fact globally stable?

Answer by coudy for Global stability for dynamical systems in $R^n$

2010-05-18T19:26:01Z

No. You could have in the ball a compact attractor K containing no periodic orbits. In fact there are attractors on which the dynamic is minimal (all trajectories are dense in K) and conjuguated to (the suspension of) an adding machine.

Examples of such attractors even appear in the unidimensional setting, for unimodal maps. I think that Bruin, Keller, Liverani (1997, erg. th. dyn. sys.) give such an example. Adding a attracting fixed point to these examples is not difficult.

Answer by Martin M. W. for Global stability for dynamical systems in $R^n$

2010-05-18T23:44:10Z

As the questioner notes in a comment, the answer is Yes for n<3.

One way to create counterexamples for larger n is to use the work on the Seifert Conjecture. Start with a vector field pointing inward to the origin, and replace a little piece of it with an "aperiodic plug." This "plug" looks from the outside like a constant flow, has no periodic orbits in the interior, but there is at least one orbit that goes in and never comes out.

For details on various plug constructions, this note from the Geometry Center is very readable and also has references to the original papers of Wilson and Kuperberg.