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Complexity
Volume 2017, Article ID 6020213, 8 pages
https://doi.org/10.1155/2017/6020213
Research Article

Complexity in Linear Systems: A Chaotic Linear Operator on the Space of Odd -Periodic Functions

1Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Budapest, Hungary
2Institute of Mathematics, Faculty of Natural Sciences, Budapest University of Technology and Economics, Budapest, Hungary

Correspondence should be addressed to Tamás Kalmár-Nagy; moc.yganramlak@epm

Received 28 July 2016; Revised 6 December 2016; Accepted 28 December 2016; Published 22 February 2017

Academic Editor: Sylvain Sené

Copyright © 2017 Tamás Kalmár-Nagy and Márton Kiss. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Not just nonlinear systems but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequences displays chaotic dynamics. Here we construct the corresponding operator on the space of -periodic odd functions and provide its representation involving a Principal Value Integral. We explicitly calculate the eigenfunction of this operator, as well as its periodic points. We also provide examples of chaotic and unbounded trajectories of .