In this section we present some more mathematical vocabulary that helps to refine our notions of invariant sets, limit sets (attractors and repellers), asymptotic behavior, and recurrence. For the most part we state the fundamental definitions in terms of maps. The corresponding definitions for flows are completely analogous and can be found in Wiggins [1].
Recurrence is a key theme in the study of dynamical systems.
The simplest notion of recurrence is periodicity. Recall that
a point of a map is periodic of period n if there
exists an integer n such that 
and 
, 0 ;SPMlt; i ;SPMlt; n.
This notion of recurrence is too restricted since it fails to account for
quasiperiodic motions or strange attractors. We will therefore explore more
general notions of recurrence. At the end we will argue that the ``chain
recurrent set'' is the best definition of recurrence that captures most of the
interesting dynamics.
, 
, and Nonwandering