= Stable equilibria of a conjugately forced Landau equation
For real $\mu,\beta$ and $\lambda>0$, writing $A=x+iy$ gives a <gradient flow> with potential $V=-(\mu+\beta)x^2/2-(\mu-\beta)y^2/2+\lambda(x^2+y^2)^2/4$. The origin has exponential <asymptotic stability> for $\mu<-|\beta|$, is algebraically attracting at equality, and is unstable above it. Nonzero stable equilibria are the real pair $\pm\sqrt{(\mu+\beta)/\lambda}$ for $\beta>0$, or the imaginary pair $\pm i\sqrt{(\mu-\beta)/\lambda}$ for $\beta<0$, when their squared amplitudes are positive. Differentiating the two real equations gives <eigenvalues> $-2(\mu+\beta),-2\beta$ on the real branch and $-2(\mu-\beta),2\beta$ on the imaginary branch. For $\beta=0$, $\mu>0$ gives a radially attracting circle with neutral phase, so individual <equilibrium points> have <Lyapunov stability> but are not individually asymptotically attracting.
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