Complexity in Linear Systems : A Chaotic Linear Operator on the Space of Odd 2 π-Periodic Functions

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 l2 displays chaotic dynamics. Here we construct the corresponding operator C on the space of 2π-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 ofC.


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 2 (i.e., 2) on the space of square-summable sequences  2 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 2 there are many examples of chaotic linear operators including weighted shifts [8], composition operators [9], and differentiation and translations [10][11][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" 2 to the space  2 of square-integrable functions (more precisely to the Hilbert space  2 (0, ) of 2-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  2 to a shift on  2 (0, ).We state and prove a theorem 2 Complexity about expressing this shift on  2 (0, ) in terms of a PV integral.In Section 3 we define and analyze the corresponding chaotic operator C on  2 (0, ), including finding its eigenvectors and periodic points.We provide examples of unbounded and chaotic trajectories of C. In Section 4 we draw conclusions.We also show that utilizing the representation of operator C one can obtain principal values of certain integrals.

A Chaotic Linear Operator on the Space of 2𝜋-Periodic Odd Functions
The backward shift  on the infinite-dimensional Hilbert space  2 of square-summable sequence is defined as where  = ( 1 ,  2 , . ..), such that ∑ ∞ =0 |  | 2 < ∞.The Hilbert space  2 (0, ) of square-integrable functions is isomorphic with  2 (by the Riesz-Fischer theorem) and is a natural functional representation of the sequence space  2 .By odd extension, elements of  2 (0, ) can be viewed as odd 2-periodic square-integrable functions so that  2 (0, ) is also isomorphic with the space of odd 2-periodic squareintegrable functions.Now we "lift"  ∈  2 to  2 (0, ) by the summation Clearly, the th Fourier coefficient of () is expressed as We define the backward shift B acting on  2 (0, ) as Therefore Our main result is the following.
Theorem 1. B() can be expressed as The strategy of the proof is the following: let us denote by A() the right-hand side of (7) and by   the projection from  2 (0, ) onto the linear span of {sin , sin 2, . . ., sin }.The sequence B  converges strongly to .In particular, for every  ∈ D(0, ) (this is the space of test functions, see Definition 2 in the Appendix), B   → B in  2 (0, ).Then a subsequence tends to B almost everywhere.Hence if we prove that B  () tends to A() for all fixed , then A = B almost everywhere as functions in  2 (0, ); that is, A = B on D(0, ).Finally, D(0, ) is a dense set in  2 (0, ); thus A = B on the whole space  2 (0, ).
Proof.We start from We first rewrite the "kernel" of ( 8 Since the limit calculated in (11) is the same as A().
Our "chaotic" operator (twice the backward shift) is now defined as

Analysis of C
The eigenfunctions of C can be found from the eigenvalue relation Instead of using ( 13), we revert to (5) to write From this we have and thus The functions corresponding to eigenvalue  = 1 are that is, the functions  1 (4 sin()/(5 − 4 cos())) are left invariant under the action of C. In other words  * ()'s are fixed points of operator C. A family of eigenfunctions is displayed in Figure 1 (we set  1 = 1).
To better characterize the action of C we want to understand how a given function is "shaped" under the repeated application of C. For  ∈  2 (0, ) the orbit of  is defined as Orb(C, ) = {, C, C 2 , . ..},where is the th composition of C with itself.The -fold composition acts on () as A given  ∈  2 (0, ) is a -periodic point of C if C   =  for some  ≥ 1 (a fixed point is a 1-periodic point; i.e.,  = 1).
We are now in the position to construct -periodic points of C.
Introducing  = { 1 , . . .,   }, a -periodic point of 2 (acting on  2 ) can be written as [3]  = ( 1 , . . .,   ,  1 2  , . . ., whose th component is given by Using the linearity of C we can easily find a period-2 point of C, that is, a function () such that C 2 () = (): In general, we find a period- point of C as By defining the "basis functions" a period- point of C can be expressed as the linear combination  The first few basis functions are (shown in Figure 2) Now we turn to creating a function that gives rise to a chaotic orbit under the action of C. First, we note that for 2 (on  2 ) the point where   is the th digit of a normal irrational number (whose digits are uniformly distributed) and generates a chaotic orbit.
is believed to be normal, so we take   to be the th digit of .We now lift this point to  2 (0, ) using (3): Figure 3 shows the first 10 elements of the orbit of Ψ under the action of C, that is, Orb(C, Ψ).The first element of the orbit is Ψ itself.Figure 4 shows the orbit Orb(C, Ψ) evaluated at three different 's (/20, /2, 19/20) for 200 iterations.
Engineering applications of chaotic orbits include design of fuel efficient space missions [21] and efficient mixing protocols for microfluids [22].Now we examine the effect of C on some commonly used periodic functions, namely, the ramp, the square-wave, and the triangle.The Fourier series of these functions are the following: Figure 5 shows the first 4 elements of the orbits of these functions.First, we note that the norm of the iterates grows (moreover, each Fourier coefficient tends to infinity); that is, these functions have unbounded orbits under the action of C. Second, the graphs of the even iterates (C 2 ) of Ramp(), Sqw(), and Triangle() are similar to the graph of tan(/2), − tan( + /2), and ±(−1) [/−1/2] tan , respectively.This is not too surprising, since the Fourier expansion of tan(/2) is 2 ∑ ∞ =1 (−1) +1 sin  which is close in some sense to ) sin  (30) for large enough .Unbounded orbits of differential equations (the so-called escape orbits) play an important role in Newtonian gravitation [23].

Conclusions
Contrary to common belief, linear systems can display complicated dynamics.Starting from twice the backward shift on  2 we constructed the corresponding shift operator  on  2 (0, ) (the space of odd, 2-periodic functions) and provided its representation using a modicum of distribution theory and Cauchy's Principal Value Integral.We explicitly calculated the periodic points of the operator (including its   (32) The basis functions (, ) can similarly be used to obtain PV integrals.The Principal Value Integral is a tool commonly used in physics, but not in engineering-related fields.We hope that this connection between chaotic operators and Principal Value Integrals will stimulate further research.