Understanding stable and unstable manifolds

Question

I want to understand better the subject of stable and unstable manifold, I want to know what is the motivation to discuss this, and if there is way to imagine this or think about this? Because I see the definition and some theorems and I understand some proofs but not what comes behind this.

Answer 1

Consider the fixed point of a dynamical system—which is just a point where the system's state doesn't change over time.

The stable manifold of that fixed point is the collection of all states that will asymptotically approach that fixed point as time goes to infinity.

The unstable manifold of that fixed point is the collection of all states that would asymptotically approach that fixed point as time goes to infinity in the inverted system—so if you "ran the system backwards", the unstable manifold would constitute all the states that would eventually approach the fixed point asymptotically.

Knowing what the stable and unstable manifolds of a fixed point are can be useful for simple purposes—for example, knowing which initial conditions of a system of ODE's lead to which asymptotic behavior. It is also very useful for spotting "exotic" behavior in dynamical systems—for example, homoclinic orbits occur when the stable and unstable manifolds of a fixed point are connected. If the stable and unstable manifolds of a fixed point intersect non-tangentially, a structure called a homoclinic tangle forms, which is indicative of transient chaotic dynamics within the dynamical system.