= Quartic double-well phase portrait with linear damping
{title2=$\dot H=2\alpha x^2$}
For $\dot x=\alpha x-y+y^3$, $\dot y=-x$, the <energy> $H=x^2-y^2+y^4/2$ satisfies $\dot H=2\alpha x^2$. At $\alpha=0$, its compact <level sets> give two families of closed <periodic orbits> around $(0,\pm1)$ and an outer family around both; $H=0$ is a figure-eight <homoclinic orbit> pair through the <saddle point> at the origin. The two minima of $H$ are <center equilibria> for zero damping, <stable foci> for small negative $\alpha$, and unstable <foci> for small positive $\alpha$. For $\alpha\ne0$, strict monotonicity along nonconstant solutions excludes <periodic orbits> and <homoclinic orbits>.
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