Timeline for Conditions for convergence to non-isolated fixed points
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 7, 2021 at 18:05 | comment | added | Fabian Wirth | Unfortunately, the assumption that $K$ does not contain entire trajectories (other than the trajectory in $0$ I guess) rules out the case that the OP is interested in. If you have a set of nonisolated fixed points, then every such fixed point gives rise to an entire trajectory. So if $K$ is this set then it completely consists of entire trajectories. | |
| Sep 23, 2016 at 11:20 | comment | added | Nawaf Bou-Rabee | I added the idea behind the proof. Since the proof relies on a lyapunov function and the fact that K does not contain an entire trajectory of the ODE, I think it can be adapted to your setting where the set of fixed points is non-isolated. | |
| Sep 23, 2016 at 3:16 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Sep 22, 2016 at 23:06 | comment | added | Ludwig | I'm still a bit confused. I'm not talking about the set $K$ but the set of fixed points of $f$, namely $\mathrm{Fix}(f):=\{x\in X : f(x)=0\}$. In particular, in your example the fixed point at the origin is isolated. | |
| Sep 22, 2016 at 21:19 | comment | added | Nawaf Bou-Rabee | As in Lasalle's principle, the set K (for Krasovsky) is allowed to be non-isolated. Krasovsky's theorem is a general purpose result for autonomous ODEs that guarantees asymptotic convergence to fixed points and not relative equilibria, as requested. | |
| Sep 22, 2016 at 19:46 | comment | added | Ludwig | Thanks for the answer. In my question, I assume that the dynamical system possesses a set of non-isolated fixed points, and I ask whether there exist conditions which implies convergence to a point in the set of fixed points (and not to the set itself). It's not clear to me if your answer applies to this framework. | |
| Sep 21, 2016 at 0:37 | history | answered | Nawaf Bou-Rabee | CC BY-SA 3.0 |