Can a limit cycle be stable and unstable and the same time?

Question

I’m wondering about the possibility of a system that has a limit cycle whose stability only depends on the initial conditions, i.e., the limit cycle can be stable, semi stable and unstable at the same time for a fixed a set of parameters, varying just the initial conditions. Is it possible to exist? There is any suitable theorem to show this (non) existence?

Answer 1

No, that’s not possible for more or less the same reason that a number cannot be positive and negative at the same time.

Stability of a limit cycle is not a property of a trajectory, and thus of initial conditions, but a property of the dynamical system or the limit cycle, respectively. More specifically, a limit cycle is stable if all trajectories in some neighbourhood converge to it. The definitions of unstable vary, but all of them involve that in any neighbourhood, there is a trajectory diverging from it. Thus the definitions are contradictory and thus cannot both be fulfilled at the same time.

The closest you can get is a limit cycle that is only attracting trajectories from one of its sides, i.e., where the limit cycle is a saddle, e.g.: $$ \begin{alignat}{1} \dot{r} &= (r-1)^2,\\ \dot{θ} &= 1, \end{alignat}$$ in polar coordinates. The limit cycle is at $r=1$. All trajectories starting with $r<1$ will converge to it. All trajectories starting with $r>1$ will diverge from it.