Timeline for Convergence of ODE solutions almost everywhere to a stable equilibrium point
Current License: CC BY-SA 4.0
9 events
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| Dec 14, 2023 at 18:32 | comment | added | tony | The conclusion holds if the system is gradient, and the energy function is real analytic, and all undesirable equilibria are exponentially unstable. | |
| Jan 22, 2022 at 10:10 | history | edited | Rodrigo de Azevedo | CC BY-SA 4.0 |
added 21 characters in body
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| Jan 22, 2022 at 5:49 | comment | added | Pietro Majer | It is easy to make a smooth field in $\mathbb R^n$ whose flow has a (degenerate) rest point x, such that the unstable manifold of x is e.g. a line and the the basin of attraction of x has non-empty interior. So maybe the question needs clarification on the definitions. | |
| Jan 22, 2022 at 3:16 | comment | added | Fabian Wirth | To support the comment of @PietroMajer: We know examples of fixed points that are attractive but not stable (in the sense of Lyapunov). Here attractive means that a neighborhood of the fixed point is attracted to it. So the claim of the theorem cannot be true. A famous example is due to Vinograd (1957), explained in Hahn, Stability of Motion, p.191. | |
| Jan 21, 2022 at 23:11 | comment | added | Pietro Majer | As it is written it is false, unless you have in mind a special definition of “unstable equilibrium” (you should include it). Note that A has no role. | |
| Jan 21, 2022 at 21:29 | comment | added | RLip2 | Through a related question, generated by posting this question, I found the answer. | |
| Jan 21, 2022 at 17:48 | comment | added | RLip2 | For example a saddle point. In particular, an equilibrium point for which the derivative of g has at least eigenvalue with real part > 0 | |
| S Jan 21, 2022 at 6:11 | review | First questions | |||
| Jan 21, 2022 at 6:25 | |||||
| S Jan 21, 2022 at 6:11 | history | asked | RLip2 | CC BY-SA 4.0 |