= Quintic saturation of a symmetry-broken pitchfork
{title2=$\dot a=a(\eta-a+Ca^2-a^4)$}
For $C>0$, nonzero <equilibrium points> satisfy $\eta=g(a)=a-Ca^2+a^4$ and are stable exactly when $ag'(a)>0$. The origin is stable for $\eta<0$. The quadratic term removes the odd symmetry and changes the local branch crossing into a <transcritical bifurcation>, with $a=\eta+O(\eta^2)$. The negative quintic term bounds trajectories. There is one negative-amplitude fold, supplying a stable finite-amplitude branch and hysteresis with the stable origin. Positive-amplitude folds occur precisely when $C>3/2$: $g'(a)=1-2Ca+4a^3$ has two positive roots then, a double positive root at $C=3/2$, and none below it. All branches and stability assignments therefore depend on $C$; a single symmetric pitchfork sketch cannot describe this family.
Back to article page
