Attractors in Pattern Iterations of Flat Top Tent Maps

Abstract

Flat-topped one-dimensional maps have been used in the control of chaos in one-dimensional dynamical systems. In these applications, this mechanism is known as simple limiter control. In this paper, we will consider the introduction of simple limiters u in the tent map, according to a time-dependent scheme defined by a binary sequence s, the iteration pattern. We will define local and Milnor attractors in this non-autonomous context and study the dependence of their existence and coexistence on the value of the limiter u and on the pattern s. Using symbolic dynamics, we will be able to characterize the families of pairs (u,s) for which these attractors exist and coexist, as well as fully describe them. We will observe that this non-autonomous context provides a richness of behaviors that are not possible in the autonomous case.

Keywords

• non-autonomous dynamical systems
• interval maps
• attractors
• symbolic dynamics