Center equilibrium

A center equilibrium of a planar dynamical system is surrounded by closed nonstationary trajectories. Nearby initial states remain nearby, giving Lyapunov stability, but they do not converge to the equilibrium. The motion is consequently not asymptotically stable.

An undamped harmonic oscillator gives $\dot{x}=v$, $\dot{v}=-{\omega }_0^{2}x$, with conserved curves $v^2+{\omega }_0^2{x}^2=\text{constant}$. Each nonzero small energy selects a periodic ellipse around the origin. This physical energy storage explains why there is continued oscillation rather than inward spiralling. Purely imaginary linearized eigenvalues alone do not prove a nonlinear center: nonlinear terms can slowly increase or decrease the amplitude. The surrounding closed-orbit property concerns the full dynamics, not just the first-order linear approximation.

Let $F$ be a continuously differentiable planar vector field with an isolated equilibrium point $p$, and let the twice continuously differentiable first integral $H$ satisfy $F\cdot \nabla H=0$ and $\nabla H(p)=0$. If the Hessian matrix $D^{2}H(p)$ is positive or negative definite, then $p$ is a centre equilibrium.

For a positive-definite matrix, sufficiently short rays from $p$ have strictly increasing $H$ with radial derivative bounded below by a positive constant times distance. Every sufficiently small positive level difference therefore meets each ray exactly once, and the implicit function theorem makes these intersections a simple closed curve. These regular level sets shrink to $p$; use $-H$ in the negative-definite case. The vector field is tangent and nonzero on each such compact curve. Its speed is bounded away from zero, so a traversal takes finite time and each curve is a periodic orbit. An arbitrary strict local extremum without these regularity and nondegeneracy conditions does not by itself supply this argument.