= Inverted quartic oscillator phase portrait
{title2=$K=x^2+y^2-y^4$}
The <autonomous planar system> $\dot x=y-2y^3$, $\dot y=-x$ preserves $K$. Its origin is a <center equilibrium> and $(0,\pm1/\sqrt2)$ are <saddle equilibria>. The level $K=1/4$ is $x=\pm(y^2-1/2)$: its two inner arcs are <heteroclinic orbits> enclosing periodic motion, while its outer branches connect saddles to escaping trajectories. Since $|\dot y|\sim y^2$ on escaping levels, infinity is reached in finite time. Adding $\alpha x/2$ to $\dot x$ gives $\dot K=\alpha x^2$, so positive $\alpha$ produces outward crossing of central levels and breaks the equal-energy heteroclinic connections.
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