Path between equilibrium points in a phase space
The phase portrait of the pendulum equation x ″ + sin x = 0 . The highlighted curve shows the heteroclinic orbit from (x , x ′) = (–π, 0) to (x , x ′) = (π, 0) . This orbit corresponds with the (rigid) pendulum starting upright, making one revolution through its lowest position, and ending upright again.
In mathematics , in the phase portrait of a dynamical system , a heteroclinic orbit (sometimes called a heteroclinic connection ) is a path in phase space which joins two different equilibrium points . If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit .
Consider the continuous dynamical system described by the ordinary differential equation
x
˙
=
f
(
x
)
.
{\displaystyle {\dot {x}}=f(x).}
Suppose there are equilibria at
x
=
x
0
,
x
1
.
{\displaystyle x=x_{0},x_{1}.}
Then a solution
ϕ
(
t
)
{\displaystyle \phi (t)}
is a heteroclinic orbit from
x
0
{\displaystyle x_{0}}
to
x
1
{\displaystyle x_{1}}
if both limits are satisfied:
ϕ
(
t
)
→
x
0
as
t
→
−
∞
,
ϕ
(
t
)
→
x
1
as
t
→
+
∞
.
{\displaystyle {\begin{array}{rcl}\phi (t)\rightarrow x_{0}&{\text{as}}&t\rightarrow -\infty ,\\[4pt]\phi (t)\rightarrow x_{1}&{\text{as}}&t\rightarrow +\infty .\end{array}}}
This implies that the orbit is contained in the stable manifold of
x
1
{\displaystyle x_{1}}
and the unstable manifold of
x
0
{\displaystyle x_{0}}
.