Limit set

Summary

The $\omega$-limit set of a point is the set of states that its solution approaches again and again as time goes to $+\infty$. The $\alpha$-limit set is the same notion for time going to $-\infty$. Limit sets describe the long-time behaviour of a dynamical system: they can be equilibria, periodic orbits or more complicated sets. They are closed invariant sets, and for a bounded forward orbit the $\omega$-limit set is nonempty, compact and connected.