Lyapunov stability

An equilibrium is Lyapunov stable if every neighbourhood contains a smaller neighbourhood whose forward trajectories remain in the original neighbourhood for all time.

Orbital stability asks that a disturbance remain close to an invariant orbit or symmetry family, rather than to one fixed representative. In a complex Landau amplitude equation with $\beta =0$, $\mu >0$, $\lambda >0$, the circle $|A|=\sqrt{\mu /\lambda }$ attracts radial disturbances while every phase on it remains stationary. The family is attracting as a set, but its individual points do not have individual asymptotic stability because their phase is neutral.

A periodic orbit has orbital asymptotic stability if nearby trajectories remain close and their distance to the orbit tends to zero. A persistent phase shift along the orbit is allowed.

For $\dot{r}=r(a-g(r))$, $\dot{\theta }=1$, a positive root $a=g(r_0)$ gives a circular periodic orbit of period $2\pi$. The radial exponent is $-r_{0}g\prime (r_0)$, so $g\prime (r_0)>0$ gives orbital asymptotic stability and $g\prime (r_0)<0$ gives instability. The phase direction is neutral; a periodic orbit is not an asymptotically stable individual point of the flow.