limit cycle

Let $$\dot{x}=f(x)$$ be a planar autonomous ordinary differential equation and $\Gamma$ be a periodic solution of the system. If the $\alpha$ -limit set or the $\omega$ -limit set of a solution with initial value not on $\Gamma$ and the respective limit set is $\Gamma$ then $\Gamma$ is a limit cycle. In simpler terms a limit cycle is an isolated periodic solution of the system.
A limit cycle, $\Gamma$ , is a stable limit cycle (or $\omega$ -limit cycle) if $\Gamma$ is the $\omega$ -limit set of all solutions in some neighborhood of $\Gamma$ .
A limit cycle, $\Gamma$ , is a unstable limit cycle (or $\alpha$ -limit cycle) if $\Gamma$ is the $\alpha$ -limit set of all solutions in some neighborhood of $\Gamma$ .[PL]

References

PL

Perko, Lawrence: Differential Equations and Dynamical Systems (Third Edition). Springer, New York, 2001.