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 ).
To better characterize the action of we want to understand how a given function is “shaped” under the repeated application of . For the orbit of is defined as , where is the th composition of with itself. The -fold composition acts on as A given is a -periodic point of if for some (a fixed point is a 1-periodic point; i.e., ). We are now in the position to construct -periodic points of .
Introducing , a -periodic point of (acting on ) can be written as [3] whose th component is given byUsing the linearity of we can easily find a period-2 point of , that is, a function such that : In general, we find a period- point of as By defining the “basis functions” a period- point of 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 . First, we note that for (on ) 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 using (3): Figure 3 shows the first elements of the orbit of under the action of , that is, . The first element of the orbit is itself.
Figure 4 shows the orbit evaluated at three different ’s () for 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 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 . Second, the graphs of the even iterates () of , and are similar to the graph of , , and , respectively. This is not too surprising, since the Fourier expansion of is which is close in some sense to for large enough . Unbounded orbits of differential equations (the so-called escape orbits) play an important role in Newtonian gravitation [23].
4. Conclusions
Contrary to common belief, linear systems can display complicated dynamics. Starting from twice the backward shift on we constructed the corresponding shift operator on (the space of odd, -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 nontrivial fixed point) and provided examples of chaotic and unbounded trajectories of .
We note here that utilizing representation (7) of operator one can actually calculate principal values. To wit, rearranging (7) yields For the simplest case, when is the eigenfunction of operator , that is, , we have (cf. (17)) 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.
Appendix
In this section we give a short introduction on distributional derivatives. The following definitions together with Proposition 5 can be found in standard textbooks on partial differential equations; see, for example, [24, 25].
Notation 1 (multi-index). Let be an open subset of and let be a smooth function. Then, for a vector , denotes the partial derivative of order .
Definition 2 (test functions). Let denote the space of test functions on the open set , that is, , endowed with the following convergence: if there is a compact subset of containing the support of for all and for every multi-index uniformly.
Definition 3 (distributions). Let denote the space of continuous linear functionals on . The convergence on induces a weak or pointwise convergence on ; namely, if for all .
Note that every function locally integrable on acts as a distribution via . Then is called a regular distribution.
Definition 4 (derivatives of distributions). Let . Then .
Note the similarity with the integration by parts formula for regular distributions.
Proposition 5. Differentiation is a continuous operation with respect to the pointwise convergence of distributions. As a consequence, derivatives of infinite series of distributions can be calculated by term-by-term differentiation.
To aid the proof of Theorem 1 we state and prove the following.
Proposition 6. Consider the following:where the distribution acts on a function aswhere the principal values are given in the sense of (1) with , , and .
Proof. We start with the identities [26]By periodicity, both equalities extend to . Term-by-term differentiation of the left hand sides of (A.4) and (A.5) (using Proposition 5) results immediately in the left hand sides of (A.1) and (A.2). Concerning the right-hand side of (A.4), let us denote the -periodic extension of by . ThenNote that this is actually a finite sum as the test function is compactly supported. One partial integration in all terms leads to which is the right-hand side of (A.1) applied to . Similarly, let us define as the -periodic extension of (the right-hand side of (A.5)). The ordinary derivative of is , but, in contrast with the previous case, this is not locally integrable near the integer multiples of . Cutting the singularities first and then integrating by parts,This is the same as (A.3), that is, the distribution in (A.2) applied to .
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work relates to the scientific program of the project “Development of Quality-Oriented and Harmonized R+D+I Strategy and the Functional Model at BME,” supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002). The work was also supported by the project “Talent Care and Cultivation in the Scientific Workshops of BME” project (Project ID: TÁMOP-4.2.2/B-10/1-2010-0009). The authors would like to thank Prof. Tibor Illés for his encouragement of this collaboration.