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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?

Wrzlprmft
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Herr Schrödinger
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  • Not sure if this is what you mean, but of course a system can have many different limit cycles at the same time. The usual examples in the plane are of the form $\dot r = f(r)$, $\dot\theta = 1$ in polar coordinates. – Hans Lundmark Feb 01 '18 at 18:40
  • What I wanna know is whether the same limit cycle can be stable, semi stable and unstable in the same time fixed a set of parameters, varying just the initial conditions. – Herr Schrödinger Feb 01 '18 at 18:58

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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.

Wrzlprmft
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    By the way, partially what OP asks is possible. If you pick a limit cycle $\gamma$ and consider restriction of system on $W^s_{\gamma}$, then the limit cycle $\gamma$ would be stable; for restriction on $W^u_{\gamma}$ it's purely unstable. – Evgeny Feb 01 '18 at 21:43