Timeline for Difference between unstable fixed point and chaotic point
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| May 8, 2018 at 7:12 | vote | accept | AndreaPaco | ||
| May 8, 2018 at 7:12 | vote | accept | AndreaPaco | ||
| May 8, 2018 at 7:12 | |||||
| May 8, 2018 at 7:10 | vote | accept | AndreaPaco | ||
| May 8, 2018 at 7:12 | |||||
| May 7, 2018 at 20:07 | comment | added | Wrzlprmft | Let us continue this discussion in chat. | |
| May 7, 2018 at 20:05 | comment | added | AndreaPaco | What i have in mind is the dynamics of trajectories that start in the vicinity of unstable fixed points of an Hamiltonian non-integrable system. Are these trajectories chaotic or not? | |
| May 7, 2018 at 20:01 | comment | added | Wrzlprmft | @AndreaPaco: I don’t understand your problem. Chaos is a property of a dynamics. An unstable fixed point on its own is not a dynamics. Talking about whether an unstable fixed point is chaotic makes as much sense as talking about whether a single point is integrable. (Also see my edit.) | |
| May 7, 2018 at 20:00 | history | edited | Wrzlprmft | CC BY-SA 4.0 |
added 12 characters in body
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| May 7, 2018 at 19:54 | history | edited | Wrzlprmft | CC BY-SA 4.0 |
added 238 characters in body
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| May 7, 2018 at 19:47 | comment | added | AndreaPaco | I know that Hamiltonian systems feature a zero-divergence flow. What I am interested in, is what happens near unstable fixed points, i.e. points associated to a Jacobian having at least one eigenvalue with positive real part. Are there additional conditions for the emergence of chaos? Thanks for your help. | |
| May 7, 2018 at 19:41 | comment | added | Wrzlprmft | @AndreaPaco: Sorry, replace unstable fixed point with repellor in my previous comment. Anyway, Hamiltonian systems cannot have attractors (or repellors), as this would mean contracting (or expanding) phase-space volumes, which in turn would violate Liouville’s theorem. Also see the link in the previous comment. Hamiltonian systems can have saddle points. | |
| May 7, 2018 at 19:27 | comment | added | AndreaPaco | Are you saying that Hamiltonian systems have just neutral-equilibrium fixed point? This sound strange to me. Can you please provide an explanation? | |
| May 7, 2018 at 16:13 | comment | added | Wrzlprmft | How does your answer change if we introduce the hypothesis that the dynamical system is Hamiltonian? – Hamiltonian systems neither have stable nor unstable fixed points. | |
| May 7, 2018 at 15:21 | comment | added | AndreaPaco | Thanks a lot for your very detailed answer. Just few clarifications, please correct me if I'm wrong. Suppose you have a fixed point $\vec{x}_0$ which is associated to a Jacobian having at least one eigenvalue endowed with positive real part. Therefore, this fixed point is unstable. BUT chaos does NOT necessarily emerge. In fact the trajectory, escaping from the unstable fixed point may fall, for example, in the basin of an attractive fixed point. How does your answer change if we introduce the hypothesis that the dynamical system is Hamiltonian? | |
| May 7, 2018 at 13:57 | history | answered | Wrzlprmft | CC BY-SA 4.0 |