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.

Definition for maps

Let

f

:

M

→

M

{\displaystyle f:M\to M}

be a map defined on a manifold

M

{\displaystyle M}

, with a fixed point

p

{\displaystyle p}

.

Let

W

s

(

f

,

p

)

{\displaystyle W^{s}(f,p)}

and

W

u

(

f

,

p

)

{\displaystyle W^{u}(f,p)}

be the stable manifold and the unstable manifold

of the fixed point

p

{\displaystyle p}

, respectively. Let

V

{\displaystyle V}

be a connected invariant manifold such that

V

⊆

W

s

(

f

,

p

)

∩

W

u

(

f

,

p

)

{\displaystyle V\subseteq W^{s}(f,p)\cap W^{u}(f,p)}

Then

V

{\displaystyle V}

is called a homoclinic connection.

Heteroclinic connection

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

p

{\displaystyle p}

and

q

{\displaystyle q}

. The condition satisfied by

V

{\displaystyle V}

is replaced with:

V

⊆

W

s

(

f

,

p

)

∩

W

u

(

f

,

q

)

{\displaystyle V\subseteq W^{s}(f,p)\cap W^{u}(f,q)}

This notion is not symmetric with respect to

p

{\displaystyle p}

and

q

{\displaystyle q}

.

Homoclinic and heteroclinic intersections

When the invariant manifolds

W

s

(

f

,

p

)

{\displaystyle W^{s}(f,p)}

and

W

u

(

f

,

q

)

{\displaystyle W^{u}(f,q)}

, possibly with

p

=

q

{\displaystyle p=q}

, 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.