Homoclinic bifurcation

Remarks

The orbit $\Gamma$ lies in both the stable and the unstable manifold of $x_{\mu_0}$, given by the stable manifold theorem. It is a global bifurcation in the sense of bifurcation theory, since it depends on a whole orbit and not only on the equilibrium. In the plane, let $\lambda_1 < 0 < \lambda_2$ be the eigenvalues at the saddle and $\sigma = \lambda_1 + \lambda_2$. If $\sigma \ne 0$ and the loop splits transversally in $\mu$, a unique limit cycle appears near $\Gamma$ on one side of $\mu_0$, stable if $\sigma < 0$ and unstable if $\sigma > 0$. Example: for $\dot x = y$, $\dot y = x - x^2 + \mu y$, let $H = y^2/2 - x^2/2 + x^3/3$. Along solutions $\dot H = \mu y^2$. At $\mu = 0$ the level set $H = 0$ contains a homoclinic loop to the saddle $(0, 0)$, in the half-plane $x > 0$. For $\mu \ne 0$, $H$ is strictly monotone along nonconstant orbits, so no orbit can leave and return to the saddle, and the loop is broken. Here $\sigma = \mu$ vanishes at $\mu_0 = 0$, so this example is degenerate and no limit cycle is born. Non-example: the constant family $\dot x = y$, $\dot y = x - x^2$, which does not depend on $\mu$, fails clause 2, since the loop persists for every $\mu$.