= Reflection-symmetric cubic double-zero normal form
{title2=$\dot q=p,\quad\dot p=-\lambda q+\kappa p+\sigma q^3-2\sigma q^2p$}
The sign $\sigma=\pm1$ distinguishes an inverted quartic potential from a double-well potential. The origin has determinant $\lambda$ and trace $\kappa$. Nonzero equilibria obey $q^2=\lambda/\sigma$, with determinant $-2\lambda$ and trace $\kappa-2\lambda$. Its energy $p^2/2+\lambda q^2/2-\sigma q^4/4$ has derivative $(\kappa-2\sigma q^2)p^2$. The origin's Hopf bifurcation is supercritical for $\sigma=1$ and subcritical for $\sigma=-1$. In the latter case the nonzero equilibria have supercritical Hopf bifurcations at $\kappa=2\lambda<0$.
// Target: dynamical-systems.bigb
Back to article page
