Periodic orbit

A periodic orbit is a nonconstant trajectory that returns to its initial state after some least positive period.

A limit cycle is an isolated periodic orbit of a dynamical system. It is attracting if nearby orbits approach it in forward time. A continuous family of closed orbits around a center equilibrium does not consist of limit cycles, because its members are not isolated.

The planar system $\dot{x}=y+(1-x^2-y^2)x$, $\dot{y}=-x+(1-x^2-y^2)y$ has polar coordinates $\dot{r}=r(1-r^2)$, $\dot{\theta }=-1$. Every nonzero orbit approaches the clockwise unit-circle limit cycle, of period $2\pi$. For $r(t_0)=r_0>0$,Interior orbits approach the origin backward in time, but exterior orbits have backward finite-time blow-up of an ordinary differential equation at $t=t_0+\frac{1}{2}\log (1-r_0^{-2})$. Thus forward convergence to a limit cycle does not imply existence for all negative times.

If one coordinate of every nonstationary trajectory is strictly monotone, the system has no periodic orbit: a periodic coordinate must return to its initial value. Any trajectory on which that coordinate is constant must be checked separately.