# Complexity in Linear Systems: A Chaotic Linear Operator on the Space of Odd2π-Periodic Functions

Authors: Tamás Kalmár-Nagy; Márton Kiss
Journal: Complexity
 (2017)

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## 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 sequencesl2 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 of C.


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## Body

## 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 B multiplied by 2 (i.e., 2B) on the space of square-summable sequences l2 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 2B 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”2B to the space L2 of square-integrable functions (more precisely to the Hilbert space L20,π of 2π-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 f about a point c∈a,b is given by(1)PV∫abfxdx=limε→0+∫ac-εfxdx+∫c+εbfxdx.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 l2 to a shift on L20,π. We state and prove a theorem about expressing this shift on L20,π in terms of a PV integral. In Section 3 we define and analyze the corresponding chaotic operator C on L20,π, 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.


## 2. A Chaotic Linear Operator on the Space of2π-Periodic Odd Functions

The backward shiftB on the infinite-dimensional Hilbert space l2 of square-summable sequence is defined as(2)Ba=a2,a3,…,where a=(a1,a2,…), such that ∑n=0∞|an|2<∞. The Hilbert space L20,π of square-integrable functions is isomorphic with l2 (by the Riesz-Fischer theorem) and is a natural functional representation of the sequence space l2. By odd extension, elements of L20,π can be viewed as odd 2π-periodic square-integrable functions so that L20,π is also isomorphic with the space of odd 2π-periodic square-integrable functions. Now we “lift” a∈l2 to L20,π by the summation(3)ft=∑n=1∞ansin⁡nt.Clearly, the nth Fourier coefficient of ft is expressed as (4)an=2π∫0πfξsin⁡nξdξ. We define the backward shift B acting on L20,π as(5)Bft=∑n=1∞an+1sin⁡nt=∑n=1∞ansin⁡n-1t.Therefore(6)Bft=2π∑n=1∞∫0πfξsin⁡n+1ξdξsin⁡nt=2π∑n=1∞∫0πfξsin⁡nξdξsin⁡n-1t.Our main result is the following.Theorem 1.
B f ( t ) can be expressed as (7)Bft=ftcos⁡t-1πPV∫0πsin⁡tsin⁡ξcos⁡t-cos⁡ξfξdξ.The strategy of the proof is the following: let us denote byAf(t) the right-hand side of (7) and by Pn the projection from L20,π onto the linear span of sin⁡t,sin⁡2t,…,sin⁡nt. The sequence BPn converges strongly to B. In particular, for every φ∈D0,π (this is the space of test functions, see Definition 2 in the Appendix), BPnφ→Bφ in L20,π. Then a subsequence tends to Bφ almost everywhere. Hence if we prove that BPnφ(t) tends to Aφ(t) for all fixed t, then Aφ=Bφ almost everywhere as functions in L20,π; that is, A=B on D0,π. Finally, D0,π is a dense set in L20,π; thus A=B on the whole space L20,π.Proof.
We start from(8)BPnft=2π∑k=1n∫0πfξsin⁡kξdξsin⁡k-1t=2π∫0πfξ∑k=1nsin⁡kξsin⁡k-1tdξ.We first rewrite the “kernel” of (8) as(9)sin⁡kξsin⁡k-1t=sin⁡kξsin⁡ktcos⁡t-sin⁡kξcos⁡ktsin⁡t=cos⁡kξ-t-cos⁡kξ+t2cos⁡t-sin⁡kξ-t+sin⁡kξ+t2sin⁡t.For test functions φ∈D0,π and t∈0,π we get(10)BPnφt=1πcos⁡t∫0πφξ∑k=1ncos⁡kξ-t-cos⁡kξ+tdξ-1πsin⁡t∫0πφξ∑k=1nsin⁡kξ-t+sin⁡kξ+tdξ.Taking n→∞ limit and utilizing (A.1) and (A.2) yield(11)limn→∞BPnφt=cos⁡tδξ-t-δξ+t,φξ-12πsin⁡tPcot⁡ξ-t2+Pcot⁡ξ+t2,φξ.Since(12)cot⁡ξ-t2+cot⁡ξ+t2=2sin⁡ξ-t/2+ξ+t/22sin⁡ξ-t/2sinξ+t/2=2sin⁡ξcos⁡t-cos⁡ξ,the limit calculated in (11) is the same as Af(t).Our “chaotic” operator (twice the backward shift) is now defined as(13)Cft=2Bft=2ftcos⁡t-2πPV∫0πsin⁡tsin⁡ξcos⁡t-cos⁡ξfξdξ.


## 3. Analysis ofC

The eigenfunctions ofC can be found from the eigenvalue relation(14)Cf∗t=λf∗t. Instead of using (13), we revert to (5) to write(15)∑n=1∞2an+1sin⁡nt=∑n=1∞λansin⁡nt. From this we have(16)an+1=λ2an,and thus(17)f∗t=a1∑n=1∞λ2n-1sin⁡nt=a14sin⁡t4+λ2-4λcos⁡t.The functions corresponding to eigenvalue λ=1 are(18)f∗t=a14sin⁡t5-4cos⁡t;that is, the functions a1(4sin⁡(t)/(5-4cos⁡(t))) are left invariant under the action of C. In other words f∗(t)’s are fixed points of operator C. A family of eigenfunctions is displayed in Figure 1 (we set a1=1).Figure 1
Eigenfunctions ofC for λ=0,0.5,1,and1.25.To better characterize the action ofC we want to understand how a given function is “shaped” under the repeated application of C. For f∈L20,π the orbit of f is defined as OrbC,f=f,Cf,C2f,…, where Ck=C∘⋯∘C︸k is the kth composition of C with itself. The k-fold composition acts on ft as(19)Ckft=2k∑n=1∞an+ksin⁡nt. A given f∈L20,π is a T-periodic point of C if CTf=f for some T≥1 (a fixed point is a 1-periodic point; i.e., T=1). We are now in the position to construct T-periodic points of C.Introducingy=y1,…,yT, a T-periodic point of 2B (acting on l2) can be written as [3](20)a=y1,…,yT,y12T,…,yT2T,y122T,…,yT22T,…, whose nth component is given by(21)an=yn-1mod⁡T+12n-1/TT.Using the linearity of C we can easily find a period-2 point of C, that is, a function gt such that C2gt=gt:(22)gt=y1∑n=1∞sin⁡2n-1t22n-2+y2∑n=1∞sin⁡2nt22n-2=y120sin⁡t17-8cos⁡2t+y216sin⁡2t17-8cos⁡2t. In general, we find a period-T point of C as(23)ft=y1sin⁡t+sin⁡T+1t2T+⋯+sin⁡nT+1t2nT+⋯+y2sin⁡2t+sin⁡T+2t2T+⋯+sin⁡nT+2t2nT+⋯+⋯++yTsin⁡Tt+sin⁡2Tt2T+⋯+sin⁡n+1Tt2nT+⋯. By defining the “basis functions”(24)ϕl,T=∑n=1∞sin⁡n-1T+1+lt2n-1T=2T2Tsin⁡l+1t+sin⁡T-1-lt1+4T-2T+1cos⁡Tt, a period-T point of C can be expressed as the linear combination(25)ft=∑l=1Tylϕl-1,T. The first few basis functions are (shown in Figure 2)(26)ϕ0,1=4sint5-4costT=1,ϕ0,2=20sin⁡t17-8cos⁡2t,ϕ1,2=16sin⁡2t17-8cos⁡2tT=2,ϕ0,3=64sin⁡t+8sin⁡2t65-16cos⁡3t,ϕ1,3=8sin⁡t+64sin⁡2t65-16cos⁡3t,ϕ2,3=64sin⁡3t65-16cos⁡3tT=3,ϕ0,4=256sin⁡t+16sin⁡3t257-32cos⁡4t,ϕ1,4=272sin⁡2t257-32cos⁡4t,ϕ2,4=16sin⁡t+256sin⁡3t257-32cos⁡4t,ϕ3,4=256sin4t257-32cos4tT=4.Figure 2
Basis functionsϕl,T for T=1,…,4andl=0,…,T-1.Now we turn to creating a function that gives rise to a chaotic orbit under the action ofC. First, we note that for 2B (on l2) the point(27)y=y11,y22,y34,…, where yi is the ith digit of a normal irrational number (whose digits are uniformly distributed) and generates a chaotic orbit. π is believed to be normal, so we take yi to be the ith digit of π. We now lift this point to L20,π using (3):(28)Ψt=∑i=1∞yi2i-1sin⁡nt. Figure 3 shows the first 10 elements of the orbit of Ψ under the action of C, that is, OrbC,Ψ. The first element of the orbit is Ψ itself.Figure 3
The first10 elements of OrbC,Ψ.Figure4 shows the orbit OrbC,Ψ evaluated at three different t’s (π/20,π/2,19π/20) for 200 iterations.Figure 4
The orbitOrbC,Ψ for 200 iterations evaluated at t=π/20,π/2,19π/20.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 ofC on some commonly used periodic functions, namely, the ramp, the square-wave, and the triangle. The Fourier series of these functions are the following: (29)Rampt=t=∑k=1∞2-1k+1ksin⁡kt-π<t<π,Sqwt=-π-π<t<0π0<t<π=∑k=1∞2ksin2kπ2sin⁡kt=∑k=1∞22k+1⁡sin⁡2k+1t,Trianglet=t0<t<π2π2-tπ2<t<π=∑k=1∞4k2πsinkπ2sin⁡kt=∑k=1∞4-1k+12k+12πsin⁡2k+1t.Figure5 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 (C2n) of Ramp(t),Sqw(t), and Triangle(t) are similar to the graph of tan⁡(t/2), -tan⁡(t+π/2), and ±(-1)t/π-1/2tan⁡t, respectively. This is not too surprising, since the Fourier expansion of tan⁡(t/2) is 2∑k=1∞-1k+1sin⁡kt which is close in some sense to(30)2n22nC2nRampt=2∑k=1∞-1k+11-k2n+ksin⁡kt for large enough n. Unbounded orbits of differential equations (the so-called escape orbits) play an important role in Newtonian gravitation [23].Figure 5
Iterates of the ramp, square-wave, and triangle functions.


## 4. Conclusions

Contrary to common belief, linear systems can display complicated dynamics. Starting from twice the backward shift onl2 we constructed the corresponding shift operator C on L20,π (the space of odd, 2π-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 C.We note here that utilizing representation (7) of operator C one can actually calculate principal values. To wit, rearranging (7) yields(31)PV∫0πsin⁡tsin⁡ξcos⁡t-cos⁡ξfξdξ=πftcos⁡t-π2Cft. For the simplest case, when ft is the eigenfunction of operator C, that is, Cft=λft, we have (cf. (17))(32)PV∫0πsin⁡tsin⁡ξcos⁡t-cos⁡ξ4sin⁡ξ4+λ2-4λcos⁡ξdξ=2πsin⁡t2cos⁡t-λ4+λ2-4λcos⁡t. The basis functions ϕl,T 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.



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