 Solution

For the real-coefficient cubic-quintic amplitude equation, the trivial pattern $A=0$ exists for every $\mu$. A nonzero spatially uniform pattern has arbitrary constant phase and intensity $q=|A{|}^2>0$ satisfying $\mu +\alpha q-q^2=0$. ThusNonzero patterns exist for $\mu \ge -{\alpha }^2/4$. The upper branch $q_{+}$ exists throughout that range; a distinct positive lower branch $q_{-}$ exists only for $-{\alpha }^2/4<\mu <0$ and reaches zero at $\mu =0$.

For a homogeneous perturbation, write $A=R{e}^{i\varphi }$. Then $\dot{R}=R(\mu +\alpha R^2-R^4)$ and $\dot{\varphi }=0$. The radial eigenvalue at a nonzero pattern iswhile the phase eigenvalue is zero. Therefore the upper branch is stable modulo its arbitrary phase for $\mu >-{\alpha }^2/4$; the lower branch is unstable. The phase-neutral direction prevents asymptotic attraction to one particular complex amplitude, but permits orbital stability of the circle of equivalent patterns.

Conduction is asymptotically stable for $\mu <0$ and unstable for $\mu >0$. At $\mu =0$, the positive cubic term destabilizes arbitrarily small nonzero amplitudes, even though its linear eigenvalue vanishes. At $\mu =-{\alpha }^2/4$, the merged nonzero branch is semistable from above only: $\dot{R}=-R(R^2-\alpha /2{)}^2$, so amplitudes just below it eventually decrease toward zero. Thus the bistable interval is $-{\alpha }^2/4<\mu <0$, with the endpoints treated separately rather than declared stable merely because an eigenvalue is zero.