 Periodic orbit

A periodic orbit is a nonconstant trajectory that returns to the same complete state after a positive period. Its least positive period $T$ gives $x(t+T)=x(t)$ for all times along the solution. A constant equilibrium is excluded by the nonconstant condition. For a discrete map, the analogous least period is the number of iterations required to return.

A harmonic oscillator gives a family of periodic ellipses, one for each positive mechanical energy. A limit cycle is instead an isolated periodic orbit. This distinction matters for perturbations: changing an undamped oscillator's energy selects another ellipse, while an attracting limit cycle can restore nearby trajectories toward its own orbit. Their phases need not coincide with one selected reference solution as time advances, which motivates orbital asymptotic stability rather than demanding convergence to the same timed point on the cycle.