= Quartic Lyapunov correction for a weak focus
{title2=$V=\tfrac12r^2+V_4,\quad\dot V=\kappa r^4+O(r^6)$}
Consider a smooth planar system $\dot x=y+P_3(x,y)+O(r^5)$, $\dot y=-x+Q_3(x,y)+O(r^5)$, where $P_3,Q_3$ are homogeneous cubic <polynomials> and $r^2=x^2+y^2$. The leading <derivative> of $r^2/2$ is the quartic <polynomial> $H_4=xP_3+yQ_3$. Let $\kappa$ be the angular average of $H_4$ on the unit circle. A homogeneous quartic correction $V_4$ solving
$$
(y\partial_x-x\partial_y)V_4=\kappa r^4-H_4
$$
gives $\dot V=\kappa r^4+O(r^6)$ for $V=r^2/2+V_4$. The right side has zero angular average. Since $y\partial_x-x\partial_y=-\partial_\theta$, its nonconstant angular Fourier modes can be integrated individually to obtain a periodic quartic correction; only modes of orders two and four can occur after removing the constant average.
The correction is $O(r^4)$, so $V$ remains positive definite close to zero. If $\kappa<0$, its orbital <derivative> is strictly negative there, and the <Second Lyapunov theorem> proves <asymptotic stability>. If $\kappa>0$, apply the same argument in reversed time to obtain repulsion. If $\kappa=0$, this quartic calculation alone does not decide stability. This method converts the averaged damping of a <weak focus> into a pointwise sign for a <Lyapunov function>.
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