Homoclinic connection

In dynamical systems, a branch of mathematics, a homoclinic connection is a structure formed by the stable manifold and unstable manifold of a fixed point.

It is a similar notion, but it refers to two fixed points,

is replaced with: This notion is not symmetric with respect to

, intersect but there is no homoclinic/heteroclinic connection, a different structure is formed by the two manifolds, sometimes referred to as the homoclinic/heteroclinic tangle.

The figure has a conceptual drawing illustrating their complicated structure.

The theoretical result supporting the drawing is the lambda-lemma.

Homoclinic tangles are always accompanied by a Smale horseshoe.

For continuous flows, the definition is essentially the same.

When a dynamical system is perturbed, a homoclinic connection splits.

It becomes a disconnected invariant set.

Near it, there will be a chaotic set called Smale's horseshoe.

Thus, the existence of a homoclinic connection can potentially lead to chaos.

For example, when a pendulum is placed in a box, and the box is subjected to small horizontal oscillations, the pendulum may exhibit chaotic behavior.