= Center equilibrium
= Center of a planar dynamical system
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= Center equilibria
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= Centre equilibrium
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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 $\dot x=v$, $\dot v=-\omega_0^2x$, with conserved curves $v^2+\omega_0^2x^2=\text{constant}$. 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.
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