= Generalized Hopf bifurcation
{title2=$\dot r=r(\tau+a r^2+b r^4),\quad b\ne0$}
= Bautin bifurcation
{c}
{synonym}
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 $\tau+aR+bR^2=0$. 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
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