Talk:Stability theory

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Need intro[edit]

This article could really use an expanded conceptual introduction: for my part, I'd like to know a little more about the general idea, before I decide to go reading five other articles to decipher the math.Btwied 15:06, 21 August 2007 (UTC)

I agree, I really cannot get a grip on what the mathematical concept of stability actually is. ω and α are introduced without explanation and, comparing the equations containing them, there is no apparent difference. A real world example would really help, stability of mechanical oscillations for instance. SpinningSpark 09:12, 5 October 2008 (UTC)

Abstract mathematical definition[edit]

I am moving below the entire section with "definitions". I agree with the commentators above that this is for the most part a symbolic gibberish that doesn't explain anything (and uses some undefined notation, lifted straight from an unpublished Bourbaki treatise on dynamical systems, perhaps?) Arcfrk (talk) 04:15, 24 April 2009 (UTC)

Definition[edit]

Let (R, X, Φ) be a real dynamical system with R the real numbers, X a locally compact Hausdorff space and Φ the evolution function. For a Φ-invariant, non-empty and closed subset M of X we call

A_{\omega}(M) := \{x \in X : \lim_\omega \gamma_x \ne \varnothing \, \mathrm{ and } \, \lim_\omega \gamma_x \subset M\} \cup M

the ω-basin of attraction and

A_{\alpha}(M) := \{x \in X : \lim_\alpha \gamma_x \ne \varnothing \, \mathrm{ and } \, \lim_\alpha \gamma_x \subset M\} \cup M

the α-basin of attraction and

A(M):= A_{\omega}(M) \cup A_{\alpha}(M)

the basin of attraction.

We call M ω-(α-)attractive or ω-(α-)attractor if Aω(M) (Aα(M)) is a neighborhood of M and attractive or attractor if A(M) is a neighborhood of M.

If additionally M is compact we call M ω-stable if for any neighborhood U of M there exists a neighbourhood VU such that

\Phi(t,v) \in V \quad v \in V, t \ge 0

and we call M α-stable if for any neighborhood U of M there exists a neighbourhood VU such that

\Phi(t,v) \in V \quad v \in V, t \le 0.

M is called asymptotically ω-stable if M is ω-stable and ω-attractive and asymptotically α-stable if M is α-stable and α-attractive.

Notes[edit]

Alternatively ω-stable is called stable, not ω-stable is called unstable, ω-attractive is called attractive and α-attractive is called repellent.

If the set M is compact, as for example in the case of fixed points or periodic orbits, the definition of the basin of attraction simplifies to

A_{\omega}(M) := \{x \in X : \phi(t, x)_{t \to \infty} \to M\}

and

A_{\alpha}(M) := \{x \in X : \phi(t, x)_{t \to -\infty} \to M\}

with

\phi(t, x)_{t \to \infty} \to M

meaning for every neighbourhood U of M there exists a tU such that

\phi(t,x) \in U \quad t \ge t_U.

Stability/Attractivity[edit]

The first bit in the overview in the Dynamical Systems section is so misleading, that it is easy to read it to become a false statement:

"will a nearby orbit indefinitely stay close to a given orbit? will it converge to the given orbit? (this is a stronger property) In the former case, the orbit is called stable and in the latter case, asymptotically stable, or attracting."

There are several issues here: what is a nearby orbit? It should be made clear that nearby means, at a given time instant, the orbits are close. Then I can only read the following as claiming that attractivity implies stability, or what else could "this is a stronger property" mean? This is false. There are examples of stable fixed points that are not attractive and of attractive fixed points that are not stable. So what should be done is to define stability, attractivity and then asymptotic stability is the property that both stability and attractivity hold.

I would be willing to make the appropriate changes in a week's time, but I do not wish to step on the toes of the owners of this page. — Preceding unsigned comment added by 195.212.29.94 (talk) 10:28, 10 January 2014 (UTC)

Stability of a theorem?[edit]

A sentence in the introduction says: "More generally, a theorem is stable if small changes in the hypothesis lead to small variations in the conclusion." I've never heard of the "stability" of a theorem. How do you make a "small change" to a hypothesis? A typo? Regardless, it doesn't seem to be within the stated scope of this article, which is the stability of differential equations and dynamic systems. --ChetvornoTALK 01:48, 11 May 2014 (UTC)