Timeline for Global stability for dynamical systems in $R^n$
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 19, 2010 at 7:07 | comment | added | Guy Katriel | 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? | |
| May 19, 2010 at 6:57 | vote | accept | Guy Katriel | ||
| May 19, 2010 at 6:58 | |||||
| May 18, 2010 at 21:30 | comment | added | coudy | 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. | |
| May 18, 2010 at 20:58 | comment | added | Guy Katriel | 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 | |
| May 18, 2010 at 20:32 | comment | added | Guy Katriel | 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? | |
| May 18, 2010 at 19:26 | history | answered | coudy | CC BY-SA 2.5 |