Nonhyperbolic equilibrium

An equilibrium is nonhyperbolic when its Jacobian matrix has at least one eigenvalue on the imaginary axis. Linearization alone then need not determine the local dynamics, and a center manifold often supplies the relevant reduction.

Purely imaginary nonzero eigenvalues make a real planar linearization a centre equilibrium. They do not control the nonlinear radial drift. For example,has the same linearization as a uniform rotation. In polar coordinates, however, $\dot{r}=-r^3$ and $\dot{\theta }=1$. Every sufficiently small nonzero orbit spirals towards the origin, so the origin is an asymptotically stable weak focus, and no nonzero orbit is periodic. Reversing the signs of the cubic terms gives a repelling weak focus. Removing them gives a centre. Thus a nonlinear centre equilibrium requires a further argument, such as a suitable first integral; the imaginary pair alone is insufficient.

A weak focus is an isolated planar equilibrium point with nonzero imaginary eigenvalues and nearby trajectories that spiral toward or away from it because of nonlinear terms. Its linearization is a center equilibrium, so linear analysis alone does not settle stability. The first nonzero radial drift in its Poincaré return map decides the direction.

For example, if the radial return map is $P(r)=r+\kappa r^{2m+1}+O(r^{2m+2})$ with $\kappa \ne 0$, sufficiently small positive $r$ decreases under successive returns when $\kappa <0$ and increases when $\kappa >0$. The first case gives asymptotic stability; the second gives instability. Equivalently, if a positive local function $W=r^2+O(r^3)$ satisfies $\dot{W}=c{r}^{2m+2}+O(r^{2m+3})$ and the angular velocity stays positive, then the sign of $c$ settles the same question. A quadratic planar system can have its cubic radial drift vanish and still have a nonzero fifth-order radial drift.

This real planar dynamical system has a saddle point at $(-1,0)$ and a repelling weak focus at $(1,0)$. The latter has purely imaginary linearized eigenvalues $\pm 2i$, so its linearization has closed elliptical orbits. Nonlinear terms nevertheless produce outward drift.

Set $X=2(x-1)$, $Y=y$, $\tau =2t$ and $r^2=X^2+Y^2$. The cubic expansion isThe linear rotation acts on scalar functions by $L=Y{\partial }_X-X{\partial }_Y$. For $H_2=r^2/2$, its cubic derivative is $X{Y}^2-3X^{2}Y/4-Y^3/2$. The homogeneous correction $H_3=7X^3/12+X{Y}^2/2+Y^3/3$ satisfies $L{H}_3$ equal to the negative of that derivative. The remaining quartic derivative of $H_2+H_3$ is $3X^2{Y}^2/2-X^{3}Y$. Withdirect differentiation givesfor all sufficiently small nonzero $r$. Meanwhile $H_2+H_3+H_4=r^2/2+O(r^3)>0$. In reversed time it is a strict Lyapunov function, proving asymptotic stability of the reversed system and hence repulsion forward in time. The angular speed stays close to $-1$, so the nearby nonzero trajectories spiral rather than approaching along a fixed direction. This explicit calculation illustrates why a centre in the linearization need not be a nonlinear centre.

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 $H_4=x{P}_3+y{Q}_3$. Let $\kappa$ be the angular average of $H_4$ on the unit circle. A homogeneous quartic correction $V_4$ solvinggives $\dot{V}=\kappa r^4+O(r^6)$ 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.