Generalized Hopf bifurcation ($\dot{r}=r(\tau +a{r}^2+b{r}^4),b\ne 0$)

A generalized Hopf bifurcation has a vanishing cubic radial coefficient and nonzero quintic coefficient. The two parameters unfold the eigenvalue real part and cubic coefficient. With $R=r^2$, periodic orbits satisfy $\tau +aR+b{R}^2=0$. Their saddle-node bifurcation curve is $\tau =a^2/(4b)$ with $R=-a/(2b)>0$, hence $ab<0$. For $b>0$ the smaller-radius limit cycle is stable and the larger one unstable where both exist; their collision explains why the criticality change of a Hopf bifurcation curve brings a saddle-node bifurcation of periodic orbits.

// Destination: partial-differential-equation.bigb