 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.