Homoclinic connection

In dynamical systems, a branch of mathematics, a structure formed from the stable manifold and unstable manifold of a fixed point. 
Definition for maps.
Let formula_1 be a map defined on a manifold formula_2, with a fixed point formula_3.
Let formula_4 and formula_5 be the stable manifold and the unstable manifold
of the fixed point formula_3, respectively. Let formula_7 be a connected invariant manifold such that
Then formula_7 is called a homoclinic connection.
Heteroclinic connection.
It is a similar notion, but it refers to two fixed points, formula_3 and formula_11. The condition satisfied by
formula_7 is replaced with:
This notion is not symmetric with respect to formula_3 and formula_11.
Homoclinic and heteroclinic intersections.
When the invariant manifolds formula_4 and formula_17, possibly with formula_18, 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.
Definition for continuous flows.
For continuous flows, the definition is essentially the same.
Significance.
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.