TITLE: Existence and uniqueness of homoclinic orbit
QUESTION [7 upvotes]: Consider the following 2D autonomous system of ODEs:
$$
\left\{
\begin{array}{ll}
\dot{x} = x^2 + 2y - x \\
\dot{y} = 3xy/2 - 3x^2 - y + 2x
\end{array}
\right.
$$

How can we prove the existence and uniqueness of a homoclinic orbit (i.e. a solution $X$ of the system
for which $\lim_{t \to \infty} X(t) = \lim_{t \to -\infty} X(t) = x_0,$ where $x_0$ is an equilibrium point of the system) for the system?
It is not hard to determine the equilibrium points: $(0,0), (-3, -6)$ and $(2/3, 1/9)$. The latter two are asymptotically stable equilibrium points, so there cannot be a homoclinic orbit for those points.
However, $(0,0)$ is a saddle equilibrium, so perhaps there is a homoclinic orbit here.
We can also see that the function $H(x,y) = - y^2 + x^2(1-x)$ is constant on the solutions of the system.
The set $H(x,y) = 0$ seems to be comprised of three solutions of the system, one of which is indeed a homoclinic orbit, so this proves existence.
However, how do we prove that this is unique? I don't really know how to approach this part.

REPLY [2 votes]: Actually, proving uniqueness of the homoclinic orbit is much easier than expected, but it requires some heavy machinery.
The Hartman-Grobman theorem says that, near a hyperbolic equilibrium point, any system of ODEs is topologically conjugate to its linearization.
In the linear case, for a saddle equilibrium point, there are $2$ trajectories going out of the equilibrium and two trajectories coming in. So, this is the behaviour for a general system as well.
Using you function $H$, you already determined your trajectories, so the above discussion tells you that there is only one homoclinic orbit.