 Inverted quartic oscillator phase portrait

The autonomous planar system $\dot{x}=y-2y^3$, $\dot{y}=-x$ preserves $K$. Its origin is a center equilibrium and $(0,\pm 1/\sqrt{2})$ 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.