Symmetric and non-symmetric periodic orbits for the digital filter map

Vowden, C.J. and Vowden, Barry J. (2008) Symmetric and non-symmetric periodic orbits for the digital filter map. Dynamical Systems - an International Journal, 23 (4). pp. 437-466. ISSN 1468-9367. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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We exhibit instances of non-symmetric periodic orbits for the digital filter map, resolving a question posed in the literature as to whether such orbits can exist. This piecewise irrational rotation, depending on a parameter a = 2cos , is an isometry of [-1, 1) [-1, 1) and reflections in the two diagonals are time-reversing symmetries for the map. Symmetric orbits are plentiful and have been much investigated. Each periodic orbit is paired with a symbolic string, from the alphabet {-, 0, +}, arising under iteration of the map because of the presence of a line of discontinuity. We prove the existence of an infinite family of non-symmetric orbits where the period N starts at 29 and increases in steps of 5; they correspond to the strings (+00)5(+-)20N-19. We describe several computer algorithms to find non-symmetric periodic orbits and their symbolic strings and list non-symmetric strings both for a = 0.5, and for N 100 across the parameter range. Our evidence suggests that non-symmetric orbits, though not plentiful, are characteristic of the dynamics of the map for all parameter values.

Item Type: Article
Uncontrolled keywords: piecewise rotation; time-reversal symmetry; symbolic dynamics; admissible sequence; non-symmetric orbit
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science
Depositing User: Louise Dorman
Date Deposited: 31 Mar 2009 11:15
Last Modified: 11 Jun 2014 10:47
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