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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?

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what do you mean by locally asymptotically stable as opposed to globally stable? Also is the stationary point assumed to be inside the absorbing ball? –  Willie Wong May 18 '10 at 19:27
    
Locally asymptotically stable means that there is a neighborhood of the stationary point which such that starting in this neighborhood you approach the stationary point as $t\rightarrow \infty$. Globally asymptotically stable means that this holds for starting from any point. The stationary point must be in the absorbing ball because if it were outside it, when you start at the stationary point you would remain there, contradicting the definition of absorbing ball. –  Guy Katriel May 18 '10 at 20:36
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2 Answers 2

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.

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Thank you, this is very interesting. Let me add that my question is motivated by particular systems of ODE's for which one can show that assumptions (1)-(3) hold, and one would like to prove global stability. In these systems, the ODE's are polynomial ones. So I wonder if it is possible to get a positive answer by restricting the class of vector fields to polynomial ones. Since a "plugging" construction cannot be performed with polynomial vector fields, maybe there is a chance.. –  Guy Katriel May 19 '10 at 7:26
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Good question, I don't know the answer for polynomials. If I were looking for a polynomial counterexample, I might start in dimension n=4, and try to write down a system where the dynamics were (a) an attracting fixed point at the origin; (b) an irrational rotation flow on a 2-torus embedded on the unit sphere; and (c) outside of the 2-torus and the origin, the distance to the origin would always decrease along orbits. But I don't really know if this approach would work. –  Martin M. W. May 19 '10 at 13:50
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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.

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Thank you for the pointer! Is it clear that the results for discrete dynamical systems extend to continuous-time ones? Is there any chance of constructing an explicit ODE system (say with polynomial nonlinearities) which will exhibit this behavior? –  Guy Katriel May 18 '10 at 20:32
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I'd like to note that the answer to my question is positive in dimensions $n=1$ (trivially) and $n=2$ (using the Poincare-Bendixon theorem), so the counterexample can only work in dimension 3 or higher –  Guy Katriel May 18 '10 at 20:58
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There is a standard procedure to embed discrete maps in flows, which is called suspension. Let M be the quotient of R^n x R by the relation (x,t)->(Tx,t-1). The flow is defined on M as T_s(x,t)=(x,t+s). The manifold M is compact, so it can be embedded in some R^m. Then extend the flow outside M by pushing radially toward M. This works but it is a bit artificial. Starting from the Henon map, you should get something algebraic. I am sure that there are explicit examples in the litterature but I don't have a reference at hand. Try Keywords "adding machine" attractor flow. –  user6129 May 18 '10 at 21:30
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For the "pushing in" part - don't you need some restrictions on the topology of M so that you can contract the whole of $R^n$ to it? –  Guy Katriel May 19 '10 at 7:07
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