Can an unstable limit cycle be contained directly within a stable one?

Question

Can both the alpha and omega point sets of a trajectory be part of two different limit cycles? I.e. can trajectories being 'repelled' from one limit cycle be pulled into an 'attracting' (stable) limit cycle? The reading I've done seems to indicate no, but I don't understand why.

Answer 1

Sure. E.g. in polar coordinates, $$ \eqalign{\dfrac{dr}{dt} &= r(r^2-1)(r^2-2)\cr \dfrac{d\theta}{dt} &= 1\cr} $$ which translates to rectangular coordinates as $$ \eqalign{\dfrac{dx}{dt} &= x (x^2+y^2-1)(x^2+y^2-2) - y\cr \dfrac{dy}{dt} &= y (x^2 + y^2 - 1)(x^2 + y^2 - 2) + x\cr}$$ The circles $x^2 + y^2 = 1$ and $x^2 + y^2 = 2$ are limit cycles, stable and unstable respectively, and the trajectories between the two are repelled from the unstable cycle and attracted to the stable one.

EDIT: Multiply $dr/dt$ by $-1$ if you want the unstable limit cycle inside the stable one.