Abstract

INIGUEZ, Anibal  and  RUIZ LEAL, Bladismir. Attractors in piecewise increassing linear maps in the real line. Novasinergia [online]. 2021, vol.4, n.2, pp.48-61.  Epub Dec 01, 2021. ISSN 2631-2654.  https://doi.org/10.37135/ns.01.08.03.

Abstract: Piecewise linear maps appear in mathematical models to describe systems from electrical engineering, physical sciences and economics recently also appears in models of neural activity. It is considered a family with 4 parameters of piecewise increasing linear maps on the real line ℝ using the theory of piecewise continuous increasing transformations on the compact intervals to study the existence of an attractor set and describe the dynamics of the attractor, verifying the existence of periodic orbits, transitivity and the existence of ergodic invariant measures. The different values of the parameter where the family exhibits an attractor interval are specifically demonstrated. The necessary and sufficient conditions are tested for the attractor set to be globally attractor. in fact, in this case it is proved that the dynamics of said attractor behaves like the dynamics of the Poincaré rotation on the unit circle. Also, it is described under what conditions in the parameters the family exhibits a topological attractor. Finally, the existence of absolutely continuous ergodic invariant measures to the Lebesgue measure associated with the family restricted to the attractor is proven, the case in which the measure is equivalent to the Lebesgue measure is even tested.

Keywords : Topological attractor; Global Attractor; Piecewise linear maps; Transitivity; invariant ergodic measure.

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