Symmetric and non-symmetric periodic orbits for the digital filter map |
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Authors: | CJ Vowden BJ Vowden |
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Affiliation: | 1. U4EA Technologies Limited , Bristol , UK;2. Institute of Mathematics, Statistics and Actuarial Science, University of Kent , Canterbury , UK |
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Abstract: | 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 θ, 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(+?)2 0 N?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. |
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Keywords: | piecewise rotation time-reversal symmetry symbolic dynamics admissible sequence non-symmetric orbit |
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