# Omega limit set vs. stable manifold of a point?

What is relationship between omega limit of a point and its stable manifold ? On mathoverflow I would the exact same question that hasn't an answer, but only a comment as explanation saying a quick answer is that if $x$ is in the omega limit set of $\xi$, then $\xi$ is in the stable manifold of $x$ and vice versa.

Can someone please illustrate this assertion on an example (please also illustrate how you computed these sets as I don't have much practice in doing this!) and give me reference (or provide) its proof ?
(Does an analoguous statement hold for alpha limit sets and unstable manifolds ?)

What other theorems are there concerning the relationship between these objects ?

## 1 Answer

That comment is wrong.

Since this was tagged differential equations, I assume you're talking about an autonomous system of differential equations $\dot{x} = F(x)$ on $\mathbb R^n$.

Let $\phi(x_0,t)$ be the value $x(t)$ at time $t$ for initial condition $x(0) = x_0$. A point $p$ is in the omega limit set of $\xi$ if there is some sequence $t_n \to +\infty$ such that $\phi(\xi, t_n) \to p$. In particular, if $\lim_{t \to \infty} \phi(\xi, t)$ exists, the omega limit set consists of that one point, which is a fixed point (aka an equilibrium point). But there are many other possibilities for an omega limit set, e.g. a limit cycle or the empty set.

On the other hand, the stable set for a fixed point $p$ is the set of all $\xi$ such that $\lim_{t \to \infty} \phi(\xi, t) = p$. The term "stable manifold" is generally used for a hyperbolic fixed point, in which case the Stable Manifold Theorem guarantees that the stable set actually is a submanifold.

So if $p$ is a hyperbolic fixed point and $\xi$ is in the stable manifold of $p$, the omega limit set of $\xi$ is $\{p\}$. However, there may be other points $\xi$ which have $p$ in their omega limit sets but are not in the stable manifold. This can occur, for example, in the case of a homoclinic cycle with an unstable fixed point inside it.

EDIT: Here is a picture of this situation. The trajectory starting at $\xi$ (shown in brown) spirals around, approaching the homoclinic cycle (blue). Points on this trajectory get arbitrarily close to the hyperbolic fixed point $p$, but $\xi$ is not on the stable manifold of $p$. EDIT: I used Maple to make this picture.

I don't remember exactly, but the equations might have been \eqalign{\dot{x} &= -4\,{x}^{2}y-4\,{y}^{3}+x-10\,x \left( {x}^{4}+2\,{x}^{2}{y}^{2}+{y}^{ 4}-xy \right) \cr \dot{y} & = 4\,{x}^{3}+4\,x{y}^{2}-y\cr} These are chosen so that the curve $E(x,y) = 0$ is invariant, where $E(x,y) = x^4+2 x^2 y^2+y^4-x y$. The two "leaves" of this curve, one in the first quadrant and one in the third, are homoclinic cycles.

• Thanks, this was a really great answer! Can you tell me how you made this drawing and which equations it represents ? Having a concrete example for which I can (maybe even analytically ?) check all this would improve my understanding. – temo Apr 5 '15 at 16:44
• Your assumption that I was referring to autonomouse ODEs was indeed correct. Since you stated it like that, is it possible to define these concepts also for non-autonomous ODES (I assume yes, since every non-autonomouse system can be reduced to an autonomous one, the non-autonomouse case can be formally perceived as a special case of autonomouse systems, so it should be possible to translate concepts from the autonomous case to the non-autonomous one) or even PDEs ? – temo Apr 5 '15 at 16:46
• Could you please answer my two questions ? That would be great. – temo Apr 8 '15 at 13:36