Complexity

Volume 2017 (2017), Article ID 6020213, 8 pages

https://doi.org/10.1155/2017/6020213

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

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

Correspondence should be addressed to Tamás Kalmár-Nagy

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 .

#### 1. Introduction

Linear systems have commonly been thought to exhibit relatively simple behavior. Surprisingly, infinite-dimensional linear systems can have complex dynamics. In particular, Rolewicz in 1969 [1] showed that the backward shift multiplied by (i.e., ) on the space of square-summable sequences exhibits transitivity (and thus gives rise to chaotic dynamics). A nice exposition of dynamics of infinite-dimensional operators is given in [2, 3] and the recent books [4, 5]. While chaoticity of linear operators is at first puzzling and the backward shift example seems contrived, these operators are not rare. In fact, Herrero [6] and Chan [7] showed that chaotic linear operators are dense (with respect to pointwise convergence) in the set of bounded linear operators. In addition to there are many examples of chaotic linear operators including weighted shifts [8], composition operators [9], and differentiation and translations [10–12]. It has also been argued in [13, 14] that nonlinearity is not necessarily required for complex behavior; an infinite-dimensional state space can also provide the ingredients of chaotic dynamics.

Several recent papers explore chaotic behavior of linear systems (see, e.g., [15, 16]). Bernardes et al. [17], for example, obtain new characterizations of Li-Yorke chaos for linear operators on Banach and Fréchet spaces.

Here we construct a chaotic linear operator by “lifting” to the space of square-integrable functions (more precisely to the Hilbert space of -periodic odd functions). Our main tool in finding the expression for the backward shift is utilizing a smidgen of distribution theory and Cauchy’s principal value, a method for obtaining a finite result for a singular integral. The principal value (PV) integral (see, e.g., [18], p. 457) of a function about a point is given by

The PV integral is commonly used in many fields of physics. A review of developments in the mathematics and methods for Principal Value Integrals is presented in [19]. Cohen et al. [20] examine first-order PV integrals and analyze several of their important properties. The structure of the paper is the following. In Section 2 we relate the backward shift on to a shift on . We state and prove a theorem about expressing this shift on in terms of a PV integral. In Section 3 we define and analyze the corresponding chaotic operator on , including finding its eigenvectors and periodic points. We provide examples of unbounded and chaotic trajectories of . In Section 4 we draw conclusions. We also show that utilizing the representation of operator one can obtain principal values of certain integrals.

#### 2. A Chaotic Linear Operator on the Space of -Periodic Odd Functions

The backward shift on the infinite-dimensional Hilbert space of square-summable sequence is defined aswhere , such that . The Hilbert space of square-integrable functions is isomorphic with (by the Riesz-Fischer theorem) and is a natural functional representation of the sequence space . By odd extension, elements of can be viewed as odd -periodic square-integrable functions so that is also isomorphic with the space of odd -periodic square-integrable functions. Now we “lift” to by the summationClearly, the th Fourier coefficient of is expressed as We define the backward shift acting on asThereforeOur main result is the following.

Theorem 1. * can be expressed as *

The strategy of the proof is the following: let us denote by the right-hand side of (7) and by the projection from onto the linear span of . The sequence converges strongly to . In particular, for every (this is the space of test functions, see Definition 2 in the Appendix), in . Then a subsequence tends to almost everywhere. Hence if we prove that tends to for all fixed , then almost everywhere as functions in ; that is, on . Finally, is a dense set in ; thus on the whole space .

*Proof. *We start fromWe first rewrite the “kernel” of (8) asFor test functions and we getTaking limit and utilizing (A.1) and (A.2) yieldSincethe limit calculated in (11) is the same as .

Our “chaotic” operator (twice the backward shift) is now defined as

#### 3. Analysis of

The eigenfunctions of can be found from the eigenvalue relation Instead of using (13), we revert to (5) to write From this we haveand thusThe functions corresponding to eigenvalue arethat is, the functions are left invariant under the action of . In other words ’s are fixed points of operator . A family of eigenfunctions is displayed in Figure 1 (we set ).