a center

A center is an equilibrium ringed entirely by closed loops: trajectories circle around it forever, never spiralling in, never flying out, just going round and round at a fixed distance. It is the picture of perpetual oscillation — a frictionless pendulum that swings back and forth identically for all time, or an ideal lossless circuit that rings without ever dying down. The equilibrium itself sits, untouched, at the eye of a whirlpool of closed orbits.

In a linear system a center occurs when the eigenvalues are PURELY IMAGINARY: a pair plus or minus i beta with zero real part. The zero real part is the crucial detail — there is rotation (from the imaginary part beta) but neither growth nor decay (because nothing multiplies by e^(alpha t) when alpha is zero). So a state goes around at angular rate beta keeping its distance exactly constant, tracing a closed ellipse. Every starting point gives its own nested closed orbit, and the system is neither attracted to nor repelled from the center — it is stable but not asymptotically stable (it stays near, but never settles to, the equilibrium).

Centers are the signature of conservative, energy-preserving systems: where there is a conserved quantity, its level curves form exactly these nested closed orbits. But here lies the most important caveat in the whole subject: a center is structurally fragile. The purely-imaginary condition is razor-thin, so for a nonlinear system whose linearization shows a center, the true behaviour is undecided — the smallest extra term can turn that center into a slow spiral. Linearization simply cannot certify a center; the honest verdict requires more than the eigenvalues.