Homoclinic orbit
= Homoclinic orbit
For an <equilibrium point> $p$, a <homoclinic orbit> is a nonconstant trajectory $x(t)$ with
$$
x(t)\longrightarrow p\qquad\text{as }t\to-\infty\text{ and as }t\to+\infty.
$$
At a <hyperbolic equilibrium point>, the trajectory leaves a neighborhood of $p$ along its <unstable manifold> and returns along its <stable manifold>. It never reaches $p$ in finite time when local solution uniqueness holds. Its closure includes the equilibrium and can form a loop-shaped <separatrix>.
A completely explicit example is
$$
\dot x=v,\qquad\dot v=x-x^3,
$$
with <Hamiltonian function> $H(x,v)=v^2/2-x^2/2+x^4/4$. Its derivative is zero along every trajectory, and the origin is a <saddle point> with <eigenvalues> $\pm1$. The curve
$$
x(t)=\sqrt2\,\operatorname{sech}t,\qquad
v(t)=-\sqrt2\,\operatorname{sech}t\tanh t
$$
lies on $H=0$. Differentiating gives $\dot x=v$ and $\ddot x=x-x^3$, while the <hyperbolic secant> tends to zero at both time infinities. Thus it is a homoclinic orbit. Changing both signs gives the second loop.
\Image[/duffing-homoclinic-loops.png]
{title=Two exact <homoclinic orbits> of the double-well oscillator return to the same saddle at infinite time}
Each loop bounds <periodic orbits> around one potential minimum. This is an <integrable Hamiltonian system>, so a homoclinic loop by itself does not establish <chaos>. A <Shilnikov bifurcation> and a simple-zero <Melnikov energy-balance method> calculation require additional dimensional, spectral or forcing hypotheses.