Consider the continuous mappings f from a compact real interval to itself. We show that when / has a positive topological entropy (or equivalently, when f has a cycle of order 2 j n = 0, 1, 2, ...) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level e > 0 . Recent results of the second author then imply that any continuous map has exactly one of the following properties:

For dynamical systems generated by iterates of continuous maps of the interval with trajectories subjected to small random perturbations, we prove that with probability 1, any trajectory is transitive in a system of compact intervals forming a periodic orbit, but generically these intervals are arbitrarily small when the perturbations are sufficiently small.

The principal measure of chaos of a continuous map of an interval is generated by a pair of points at least one of which belongs to an omega limit set of the second kind if such set exists, or is zero in the case that such set does not exist.

In this paper, we study random dynamical systems generated by two Allee maps. Two models are considered -with and without small random perturbations. It is shown that the behavior of the systems is very similar to the behavior of the deterministic system if we use strictly increasing Allee maps. However, in the case of unimodal Allee maps, the behavior can dramatically change irrespective of the initial conditions.

Let f be a distributionally chaotic map of the interval such that the endpoints of the minimal periodic portions of any basic set are periodic. Then the principal measure of chaos, µp(f ), is not greater than twice the spectral measure of chaos µs(f ). This proves an assertion of Schweizer et al. in a special case.