Recurrence and LaSalle invariance principle

In this paper, we consider three different notions of recurrence, chain recurrence, strong chain recurrence and generalized recurrence, where the generalized recurrence is introduced by Auslander (1964) and the strong chain recurrence is derived from the idea of a strong chain defined by Easton (1978). We prove that for a quasi-gradient flow in a compact metric space, these three recurrences are equivalent. Finally, we establish an invariance principle, i.e., the ωω-limit set of a precompact orbit is contained in the generalized recurrent component of ’zero derivative’ set of a Lyapunov function.