Global stability for dynamical systems in \$R^n\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:12:06Z http://mathoverflow.net/feeds/question/25162 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25162/global-stability-for-dynamical-systems-in-rn Global stability for dynamical systems in \$R^n\$ Guy Katriel 2010-05-18T18:36:12Z 2010-05-18T23:44:10Z <p>Suppose we have a smooth dynamical system on \$R^n\$ (defined by a system of ODEs). Assume that:</p> <p>(1) The system has an absorbing ball, that is every trajectory eventually enters this ball and stays in it. </p> <p>(2) The system has a unique stationary point, and this stationary point is locally asymptotically stable.</p> <p>(2) The system has no period orbits.</p> <p>Can we conclude that the stationary point is in fact <em>globally</em> stable?</p> http://mathoverflow.net/questions/25162/global-stability-for-dynamical-systems-in-rn/25167#25167 Answer by coudy for Global stability for dynamical systems in \$R^n\$ coudy 2010-05-18T19:26:01Z 2010-05-18T19:26:01Z <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/25162/global-stability-for-dynamical-systems-in-rn/25189#25189 Answer by Martin M. W. for Global stability for dynamical systems in \$R^n\$ Martin M. W. 2010-05-18T23:44:10Z 2010-05-18T23:44:10Z <p>As the questioner notes in a comment, the answer is Yes for n&lt;3. </p> <p>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.</p> <p>For details on various plug constructions, <a href="http://www.geom.uiuc.edu/docs/forum/seifert/se2.html" rel="nofollow">this note from the Geometry Center</a> is very readable and also has references to the original papers of Wilson and Kuperberg. </p>