Harmonic Analysis in Discrete Dynamical Systems

Gerardo Pastor, Miguel Romera, Amalia Beatriz Orue, Agustin Martin, Marius F. Danca, Fausto Montoya

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Instituto de Seguridad de la Información, CSIC, Madrid, Spain.
Romanian Institute of Science and Technology, Cluj-Napoca, Romania.

DOI: 10.4236/ijmnta.2012.11003                

Abstract

In this paper we review several contributions made in the field of discrete dynamical systems, inspired by harmonic analysis. Within discrete dynamical systems, we focus exclusively on quadratic maps, both one-dimensional (1D) and two-dimensional (2D), since these maps are the most widely used by experimental scientists. We first review the applications in 1D quadratic maps, in particular the harmonics and antiharmonics introduced by Metropolis, Stein and Stein (MSS). The MSS harmonics of a periodic orbit calculate the symbolic sequences of the period doubling cascade of the orbit. Based on MSS harmonics, Pastor, Romera and Montoya (PRM) introduced the PRM harmonics, which allow to calculate the structure of a 1D quadratic map. Likewise, we review the applications in 2D quadratic maps. In this case both MSS harmonics and PRM harmonics deal with external arguments instead of with symbolic sequences. Finally, we review pseudoharmonics and pseudoantiharmonics, which enable new interesting applications.

Keywords

Harmonic Analysis; Discrete Dynamical Systems; MSS Harmonics; PRM Harmonics; Pseudoharmonics

Conflicts of Interest

The authors declare no conflicts of interest.

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