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PHYSICAL REVIEW E 95, 062214 (2017)

From dynamical systems with time-varying delay to circle maps and Koopman operators

David Müller,* Andreas Otto,† and Günter Radons‡

Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany(Received 20 January 2017; revised manuscript received 18 May 2017; published 16 June 2017)

In this paper, we investigate the influence of the retarded access by a time-varying delay on the dynamics ofdelay systems. We show that there are two universality classes of delays, which lead to fundamental differences indynamical quantities such as the Lyapunov spectrum. Therefore, we introduce an operator theoretic framework,where the solution operator of the delay system is decomposed into the Koopman operator describing the delayaccess and an operator similar to the solution operator known from systems with constant delay. The Koopmanoperator corresponds to an iterated map, called access map, which is defined by the iteration of the delayedargument of the delay equation. The dynamics of this one-dimensional iterated map determines the universalityclasses of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, weconnect the theory of time-delay systems with the theory of circle maps and the framework of the Koopmanoperator. In this paper, we extend our previous work [A. Otto, D. Müller, and G. Radons, Phys. Rev. Lett. 118,044104 (2017)] by elaborating the mathematical details and presenting further results also on the Lyapunovvectors.

DOI: 10.1103/PhysRevE.95.062214

I. INTRODUCTION

Time-delay systems emerge, for example, in control the-ory [1], engineering [2], life science [3,4], chaos control [5],and synchronization of networks [6]. In all of these fields,effects of time-varying delays are investigated. Introducingtime-varying delays can improve the stability of machiningprocesses [7] and of systems in general via time-delayedfeedback control [8]. Furthermore, in biological models,time-varying delays arise naturally by taking into accountfluctuations of the environment [3,9], and also the effect onsynchronization is studied [10]. The functional structure ofthe delay itself influences mathematical properties of generaldelay systems such as causality, minimality [11], and spectralreachability [12].

For some systems with time-varying delay it is knownthat they can be transformed to systems with constant delaybecause the delay is defined by an intrinsic constant delay anda known time scale transformation [13]. This type of systemarises, for example, in biological models with threshold delays,where the intrinsic constant delay represents the evolutionarysteps, which have to be passed to reach the state of adulthoodand the corresponding time scale represents the grade ofevolution [3,14]. Further examples are systems with variabletransport delays, where the constant delay is given by thelength of the transport line and the corresponding time scalerepresents the covered distance [15]. The question arises as towhether one can find a transformation for a general systemwith time-varying delay, such that the initial system withtime-varying delay is equivalent to a resulting system withconstant delay, or not. In other words, is there a well-behavedtransformation in the sense that dynamical quantities are welldefined for the new system with constant delay? And, are theyinvariant under the transformation and does the new system

*[email protected]†[email protected]‡[email protected]

correspond to a reasonable physical model, characterized bytransport or fixed evolutionary steps? In general, this is nottrue and the existence of such a well-behaved transformationdepends on the functional properties of the time-varying delay.In fact, the resulting delay classification implies fundamentaldifferences in the dynamics of systems that are equivalent tosystems with constant delay and the remaining systems, asdemonstrated in [16].

So, let us consider a general d-dimensional dynamicalsystem defined by the system of delay differential equations(DDE) with time-varying delay τ (t):

z(t) = f (z(t),z(R(t)),t), where R(t) := t − τ (t) (1)

is called retarded argument. Hence, the results presented belowhold for all DDEs with one time-varying delay τ (t). In otherwords, the results are independent of the specific form of f andapply to autonomous and nonautonomous systems. Althoughthe present theory holds only for systems with one delay, it pro-vides a basis for a more general theory including systems withmultiple delays and, even more general, distributed delays.

In this paper, we focus on the influence of the functionalstructure of the retarded argument on the state dynamics asgiven by Eq. (1) and extend the theory established in [16].We will show that there is a natural connection between thedynamics of systems with time-varying delay, the dynamics ofone-dimensional iterated maps, and the spectral properties ofthe Koopman operator associated with these maps. This con-nection itself is interesting because it relates well understoodand distinct theories and may lead to a better understandingof delay systems with time-varying delay. The analysis ofdynamical systems in the framework of the Koopman operatoris a wide topic of past and recent research [17,18] and isapplied, for example, in the computation of isostables andisochrons [19,20] or in global stability analysis [21].

The paper is organized as follows. In Sec. II we derive thetwo delay classes as sketched in [16] and give a short overviewon the differences in the dynamics. Sections III and IV dealwith the separation of the dynamics related to the variable

2470-0045/2017/95(6)/062214(16) 062214-1 ©2017 American Physical Society

https://doi.org/10.1103/PhysRevLett.118.044104




https://doi.org/10.1103/PhysRevE.95.062214

DAVID MÜLLER, ANDREAS OTTO, AND GÜNTER RADONS PHYSICAL REVIEW E 95, 062214 (2017)

delay from the dynamics of the delay system. The influenceof the delay classes on the Lyapunov spectrum is analyzedin Sec. V and on the Lyapunov vectors in Sec. VI, therebyexplaining also the differences in the numerical properties ofspectral methods.

II. DELAY CLASSIFICATION

We showed in [16] that two different types of delays canbe identified, which result in fundamental differences in thedynamics of the corresponding delay system. In the following,we briefly introduce this classification by the analysis of theexistence of a time scale transformation that transforms thesystem of Eq. (1) to a system with constant delay. Furthermore,we elaborate the differences in the dynamics and numerics ofsystems belonging to one or the other delay type.

Every system of differential equations is equivalent toa large set of systems of similar type that are connectedamong each other by time scale transformations. Such atransformation is easily done by the substitution of the originaltime scale t by an invertible and at least once differentiablefunction t = �(t ′) of the new time scale t ′. In other words, oneintroduces the new variable y(t ′) by

y = z ◦ �, (2)

where ◦ denotes function composition and z(t) is the originalvariable. By the substitution of the new variable y(t ′) into theoriginal delay Eq. (1) and by some rearrangements we obtainan equivalent delay system [22]

y(t ′) = �(t ′) · f ( y(t ′), y(Rc(t ′)),�(t ′)), (3)

where the new retarded argument Rc(t ′) is derived from theoriginal retarded argument by

Rc = �−1 ◦ R ◦ �. (4)

If we find at least one function �(t ′) such that

Rc(t ′) = t ′ − c, with c > 0 (5)

we are able to transform the system with time-varying delay,defined by Eq. (1), into a system with constant delay, Eq. (3).Equation (4) takes the structure of a conjugacy equation,which is known from the theory of one-dimensional iteratedmaps [23]. From this point of view, such an equation deter-mines the topological conjugacy between the iterated maps

θk = R(θk−1) = Rk(θ0), (6)

θ ′k = Rc(θ ′

k−1) = Rkc (θ ′

0). (7)

Since these maps describe the dynamics of the access to thedelayed state of our delay system, we call them access maps.

In the following, we only consider invertible retardedarguments R(t), i.e., τ (t) < 1, because noninvertible retardedarguments cause problems with minimality and causality, andare even considered unphysical [11]. Now, let us assumefurther that the delay τ (t) is periodic with period 1, i.e., wemeasure time in units of the delay period. Since the relatedmaps can be reduced to maps acting on the interval [0,1] we areable to apply the theory of circle maps [23] and it follows thatfor some retarded arguments R(t) the corresponding accessmap is topological conjugate to a pure rotation. The topological

A

FIG. 1. Parameter space of an arbitrary system with time-varyingdelay τ (t) = τ0 + A

2πsin(2π t). The white regions correspond to a

conservative delay, where the system is equivalent to a system withconstant delay. On the other hand, the black regions correspond to adissipative delay, where the system is not equivalent to a system withconstant delay.

conjugacy between one-dimensional iterated maps acting onthe same interval preserves the rotation number, which isdefined by [23]

ρ = limk→∞

Rk(t) − t

k= −c. (8)

If the retarded argument R(t) defines a map that is topologicalconjugate to a pure rotation, the constant delay c in Eq. (5) isdetermined by Eq. (8) and it follows directly from Eq. (4) thatR(t) fulfills the criterion

limj→∞

Rqj (t) + pj = t ∀ t, (9)

where pj

qjis the j th convergent of the continued fraction

expansion [24] of c = −ρ. Circle maps fulfilling this criterionexhibit marginally stable quasiperiodic (periodic) motion sincethey are topological conjugate to the pure rotation by anirrational (rational) angle c. Invertible maps that do not fulfillthe criterion above have stable fixed points or periodic orbits.This follows directly from Poincare’s classification [23].

Thus, the question as to whether a system of DDEs isequivalent to a system with constant delay can be reducedto the question of the existence of a topological conjugacybetween the access map defined by the retarded argumentR(t) and a pure rotation. Equivalence means that the systemsexhibit identical dynamics and that characteristic quantities areinvariant under the time scale transformation.1 The connectionto the existence of a circle map conjugacy leads to interestingconsequences for systems with parameter families of delays.As illustrated in Fig. 1, the parameter regions that are related to

1By the iterative method given in [45] a time scale transformationto constant delay can be computed even in the case where the accessmap is not conjugate to a pure rotation. However, in this case thetransformation is well defined only on finite time intervals but notfor infinite times. Hence, asymptotic quantities such as Lyapunovexponents are not well defined for the resulting systems.

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FROM DYNAMICAL SYSTEMS WITH TIME-VARYING . . . PHYSICAL REVIEW E 95, 062214 (2017)

a system that is equivalent to a system with constant delay forma Cantor set, whereas regions related to systems that are notequivalent to a system with constant delay are characterizedby Arnold tongues.

Obviously, the classification depends only on the propertiesof the access map and not on the specific DDE. In other words,the classification is independent of the specific properties of f .So, if there is any influence of the delay type on the systemsdynamics, the influence emerges in all types of time-delaysystems and represents a universal property of all systemswith time-varying delay.

The influence of the delay type on the systems dynamicsand its universality can be demonstrated easily by the evolutionof small perturbations and the related relaxation rates calledLyapunov exponents. The set of all Lyapunov exponentsordered from the largest to the smallest is called Lyapunovspectrum. As already argued in [16], the asymptotic scalingbehavior of the Lyapunov spectrum depends on the delayclass. In Fig. 2, exemplary Lyapunov spectra of three delaysystems are shown for each delay type, whereby the spectrawere computed directly from the systems with time-varyingdelay without applying any transformation. The systems aregiven by

z(t) = 2z(t)[1 − z(R(t))], (10a)

z(t) = 10z(R(t))1 + z(R(t))10

− 5z(t), (10b)

z(t) + z(t) + 4π2z(t) = 8z(R(t)), (10c)

where for constant delay Eq. (10a) is the Hutchinson equationarising from population dynamics [4], Eq. (10b) is the Mackey-Glass equation known as a model for blood-production [25],and Eq. (10c) arises in the stability analysis of turningprocesses [7]. To illustrate the universality of the followingresults, i.e., the independence of the specific dynamics of thesystem, the parameters and the time-varying delay are chosensuch that Eq. (10a) exhibits a limit cycle, Eq. (10b) chaoticdynamics, and Eq. (10c) a stable fixed point, respectively.

If the access map of a time-delay system is topologicalconjugate to a pure rotation, the dynamics of the systemequals the dynamics of a system with constant delay. Sincethe asymptotic scaling behavior of the Lyapunov spectrumfor systems with constant delay is known to be logarithmicwith respect to the index of the Lyapunov exponents [27],the scaling behavior of equivalent systems with time-varyingdelay is logarithmic as well. We call the delays with thisproperty conservative delay, because the related access mapis a measure preserving system and has no influence on thescaling behavior of the Lyapunov spectrum. If the accessmap does not fulfill the criterion (9), the specific delaysystem is not equivalent to a system with constant delayand, consequently, the asymptotic scaling behavior is notlogarithmic as is shown in Fig. 2(a). Since the Lyapunovspectrum decreases asymptotically faster than any logarithm,we call the corresponding delays dissipative delay. Later, wewill derive in detail that the scaling behavior is asymptoticallylinear, which was already briefly demonstrated in [16]. Thedelay type also influences the convergence properties ofnumerical algorithms as illustrated for the Mackey-Glass

(a)

0 10 20 30 40 50 60

�8

�6

�4

�2

0

index n

expo

nent

n

(b)

m m�1000

m�2000

m�500

0 100 200 300 400�10

�8

�6

�4

�2

0

index n

expo

nent

n

FIG. 2. Comparison of the Lyapunov spectra between systemswith conservative delay (filled) and dissipative delay (empty),computed by the semidiscretization method [2,26]. (a) The Lyapunovspectra are shown for three different systems, where the systemsfrom top to bottom are given by Eq. (10b) (circles), Eq. (10c)(triangles), and Eq. (10a) (squares). (b) The convergence properties ofthe method are illustrated for Eq. (10b), where m denotes the numberof equidistant points approximating the state of the delay system. Asa guide to the eye, the asymptotic scaling behavior for dissipativedelay, which we derive in Sec. V, is represented by the dashedgray line. The jumps in the spectra related to conservative delayand the deviation of the spectra from the dashed line for dissipativedelay, respectively, indicate the bounds on the number of wellapproximated exponents. For all computations, we have chosen thedelay τ (t) = τ0 + 0.9/(2π ) sin(2π t) with the mean delay τ0 = 1.51and 1.54 for dissipative and conservative delay, respectively.

equation in Fig. 2(b) and for the remaining systems in Fig. 10in Appendix A. For conservative delays, the number of wellapproximated exponents increases much faster by increasingthe number m of sampling points than for dissipative delay.

We see that the classification by the access map’s dynamicsleads to interesting hitherto unknown phenomena such as theabove mentioned influence on the Lyapunov spectra and thenumerical properties of their computation. We will discuss thisin more detail in Sec. VI. In the next section we separate theinfluence of the access map’s dynamics from the dynamics ofthe delay system using a specific operator framework.

III. DELAY DYNAMICS IN OPERATOR FRAMEWORK

In this section we give some preliminaries for the separationof the influence of the access map on the dynamics of the delay

062214-3


system. The method is derived from the method of steps forsystems with time-varying delay (cf. [22]). The separation willbe used in Secs. IV and V for the explanation of the differentasymptotic scaling behavior of the Lyapunov spectra in Fig. 2.

For a given solution z(t) of Eq. (1), the linearized systemfor the dynamics of small perturbations x(t) = z(t) − z(t) isgiven by

x(t) = A(t) x(t) + B(t) x(R(t)), (11)

where A(t) = ∂z(t) f and B(t) = ∂z(R(t)) f denote the Jacobiansof f with respect to the variable z(t) and the delayed variablez(R(t)) taken at the unperturbed solution z(t). Consequently,without any loss of generality we can restrict our analysis tod-dimensional linear delay systems.

In order to apply the method of steps to our delay system, itis convenient to remove the first term on the right-hand side ofEq. (11) to reduce the computation of the solution to a simpleweighted integration of the system’s history. Hence, we usethe variation of constants approach and obtain the integralformulation of Eq. (11):

x(t) = M(t,t0) x(t0) +∫ t

t0

dt ′ M(t,t ′) B(t ′) x(R(t ′)), (12)

where t0 denotes the initial time and M(t,t ′) denotes thefundamental solution of the ordinary differential equation(ODE) part and solves

d

dtM(t,t ′) = A(t) M(t,t ′), M(t ′,t ′) = Ed , (13)

where Ed is the d-dimensional identity matrix. The operatoron the right-hand side of Eq. (12) can be decomposed into theoperators I and C and the integral formulation of our delayequation can be represented by the fixed point equation

x(t) = I [C[x(t ′′),t ′],t], (14)

where C denotes the Koopman operator also called composi-tion operator with the symbol R(t)

C[x(t ′),t] = x(R(t)) (15)

and I denotes the integral operator

I [x(t ′),t] = M(t,t0)x[R−1(t0)] +∫ t

t0

dt ′ M(t,t ′) B(t ′) x(t ′).

(16)

The main idea of the method of steps is to divide thetime scale into several subintervals, such that the solutionof the DDE reduces to the solution of an inhomogeneousODE. Trivially the intervals are chosen properly if the intervalboundaries tk are connected by

tk−1 = R(tk), (17)

thus giving further meaning to the access map, Eq. (6). Hence,the temporal evolution of a DDE can be described by theiteration of operators acting on functions, which are defined onintervals ordered in time. To be consistent with our precedingdefinitions we define the initial state x0(t) as the initial functioninside the interval [R(t0),t0] and the state xk in the time intervalafter k time steps as the function

xk(t) = x(t), t ∈ [tk−1,tk]. (18)

To adapt the decomposition of the solution operator ofour DDE to the aforementioned notation, we introduce therestrictions Ik and Ck of the integration and the Koopmanoperator, respectively, related to the previously defined timeintervals. The operator Ck is defined by the restriction of theKoopman operator C to the domain of the related state xk(t)living in an appropriate space F([tk−1,tk],Rd ) of functionsmapping [tk−1,tk] to Rd . We call the family of operators Ck

access operators because they describe the access of the delaysystem to the preceding state:

Ck : F([tk−1,tk],Rd ) → F([tk,tk+1],Rd ) :

Ck[xk(t ′),t] = xk(R(t)), t ∈ [tk,tk+1]. (19)

The restriction Ik of the integration operator I in our notationof labeled time intervals acts on the space F([tk,tk+1],Rd )where our future state xk+1(t) lives and takes the form

Ik[xk+1(t ′),t] = M(t,tk) xk+1(tk+1)

+∫ t

tk

dt ′ M(t,t ′) B(t ′) xk+1(t ′),

t ∈ [tk,tk+1]. (20)

Since the operator Ik is equivalent to the solution operatorfor DDEs with constant delay in terms of the Hale-Krasovkiinotation where the time domain of the state is shifted to thetime domain of the initial function and the position in thephysical time is represented by labeling the state intervals [28],we call it constant delay evolution operator. For convenience,we introduce the following short notation:

L · = L[ · ,t], (21)

where L stands for any of the above and below definedoperators.

Finally, the evolution of a solution of our linear DDE canbe expressed in terms of time intervals with finite length,where the connection between two states on sequential timeintervals is given by

xk+1(t) = Ik Ck xk(t ′) (22)

and we have derived a method that separates the dynamicsof the delay system with time-varying delay into the accessmap’s dynamics represented by the access operators Ck andthe dynamics of a system with constant delay represented

(a) (b)

�k �1� 0 k 0 �k �1� 0

Ck

I k

tk�1 tk tk�1

Ck

I k

FIG. 3. Illustration of the decomposition of the solution operatorinto the sequential action of the access operator Ck and the constantdelay evolution operator Ik for (a) constant delay and (b) time-varyingdelay. For constant delay τ0 the access operator Ck reduces to a timeshift by τ0 and for time-varying delay the access operator changes thelength of the state interval and deforms the state on the time domain.

062214-4


by the constant delay evolution operators Ik . The method isillustrated in Fig. 3.

IV. DECOMPOSING THE DYNAMICS VIA FINITESECTION METHOD

In the following, we introduce a method to determinethe relaxation rate of the evolution of small volumes, calledaverage divergence, under the dynamics of the DDE (1) on afinite-dimensional subspace of its phase space. The method,which is a generalization of Farmers approach for systemswith constant delay [27] to systems with time-varying delay,is based on the method for calculating Lyapunov exponentsgiven in [29] and uses the presented separation of the solutionoperator in Sec. III.

Let us define the sets {φki(t)}i∈N to be the bases of eachcanonical direction of the function spaces F([tk−1,tk],Rd ) and{ψki(t)}i∈N to be the corresponding dual sets in the dual spacesF∗.2 In other words, let the mentioned sets define biorthogonalsystems of our spacesF endowed with a suitable inner productof vectors of the same and of vectors of different spaces F .Then, the state xk(t) can be expanded into a series in terms ofthe basis vectors of the underlying space, i.e.,

xk(t) =∞∑i=0

qk,i φki(t). (23)

Thus, xk(t) is represented as an infinite-dimensional vectorqk = cols(qk,0,qk,1, . . . ) composed of the d-dimensional co-efficient vectors qk,i . The ith vector qk,i is related to the ithbasis function φki(t) and its components are related to thecanonical directions of the state xk(t). Furthermore, we definethe matrix representations Ik and Ck of the operators Ik andCk , respectively, such that the evolution of the state by Eq. (22)is represented by

qk+1 = Ik Ck qk. (24)

Obviously, the infinite-dimensional matrices Ik and Ck arecomposed of (d × d) matrices, whereby each is related toone of the basis functions φki(t). For the following analysisit is convenient to choose matrix representations, i.e., tochoose the bases and the inner product, in a way such thatshift operators are represented by identity matrices. This is aquite natural assumption because in our framework the shiftoperator is implemented by just switching the underlyingspace F([tk−1,tk],Rd ). In the case of a constant delay theoperators Ck become shift operators and the whole dynamicsis sufficiently described by the Ik . One possible definition ofsuch a matrix representation can be found in Appendix B.

Let I (n)k , C (n)

k , and q(n)k be the n-dimensional approximation

of the above mentioned operators and the state at time tk , wheren is an integer multiple of d. The n-dimensional approximation

2Note that in the literature, particularly in [29,46], the dynamicsof DDEs is represented by a family of solution operators rather onRd × F than F because of some mathematical issues defining thesolution operator on Lebesgue spaces, for example. But, as long aswe only consider continuous initial functions we avoid these issues.

q(n)N of the state after N time steps xN (t) is now given by

q(n)N =

N−1∏k=0

I (n)k C (n)

k q(n)0 . (25)

Now, let us consider the projection of the dynamics ofour linear delay system defined by Eq. (11) on n-dimensionalsubspaces of F([tk−1,tk],Rd ) and analyze the evolution of asmall n-dimensional initial volume V

(n)0 spanned by n linearly

independent vectors on such a subspace of the space of initialfunctionsF([t−1,t0],Rd ). The long term evolution of these vec-tors and the related volume under the dynamics of Eq. (11) ingeneral are dominated by the n largest Lyapunov exponents λl .Thus, the asymptotics of the n-dimensional volume V

(n)N at time

tN after subsequent projection to the related n-dimensionalsubspace of the underlying space F([tN−1,tN ],Rd ) is given by

V(n)N ∼ e

∑nl=1 λl (tN −t0). (26)

Since the long-time evolution of a small volume is in generalindependent of the specific subspace, we choose as basisvectors for our n-dimensional subspace the first m = n/d

members of our basis {φki(t)}i∈N for each of the d canonicaldirections of the phase space. So, an approximation of thevolume evolution follows directly from the finite sectionapproximation of the evolution operator and is given by

V(n)N

V(n)

0

≈∣∣∣∣∣det

(N−1∏k=0

I (n)k C (n)

k

)∣∣∣∣∣. (27)

Utilizing the factorization of the determinant of the product ofmatrices, we obtain

V(n)N

V(n)

0

≈(

N−1∏k=0

∣∣ det(I (n)

k

)∣∣) ·∣∣∣∣∣det

(N−1∏k=0

C (n)k

)∣∣∣∣∣. (28)

The codomain of the access operator Ck equals the domain ofthe access operator Ck+1. Hence, the matrix product of theirfinite section approximations can be identified with the finitesection approximation of their product. So, we consider theproduct of the access operators Ck ,

DN =N−1∏k=0

Ck := CN−1 . . . C1C0 (29)

and let D(n)N be its n-dimensional finite section approximation.

Furthermore, we assume that the convergence of this approxi-mation in terms of {φki(t)}i∈N is well behaved in the sense thatfor some unbounded sequence of N

det(

D(n)N

) ∼ det

(N−1∏k=0

C (n)k

). (30)

Finally, the volume evolution can be expressed by

V(n)N

V(n)

0

≈(

N−1∏k=0

∣∣ det(I (n)

k

)∣∣) · ∣∣ det(

D(n)N

)∣∣. (31)

Obviously, the volume evolution factorizes into two partswhere the first part describes the influence of the successiveintegrations and the second part describes the influence of the

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underlying dynamics of the successive access to the initialstate x0(t) by the retarded argument R(t).

More expressive than the explicit volume evolution is therelated exponential rate, the average divergence δn, which canbe computed by

δn = limN→∞

1

tN − t0ln

(V

(n)N

V(n)

0

)= δI (n) + δC (n) (32)

and can be separated into the parts

δI (n) = limN→∞

1

tN − t0

N−1∑k=0

ln∣∣ det

(I (n)

k

)∣∣ (33)

and

δC (n) = limN→∞

1

tN − t0ln

∣∣ det(

D(n)N

)∣∣. (34)

The exponential rate δI (n) describes the volume evolutioncaused by the successive integrations and the rate δC (n)

describes the volume evolution caused by the successive accessto the initial state by the retarded argument R(t). Due toEq. (26), δn describes the asymptotics of the sum of the n

largest Lyapunov exponents. Hence, the asymptotic scalingbehavior of the Lyapunov spectrum can be computed by thelimit of the difference of the approximations of δn and δn−1 forlarge n. Just as the average divergence the asymptotic scalingof the Lyapunov spectrum splits into two parts

λn ∼ δn − δn−1 = λI (n) + λC (n) . (35)

In the method described above, n has to be a rational multipleof the dimension d. Thus, Eq. (35) can only be applieddirectly, if d = 1 or if we have found expressions describingthe asymptotics of Eqs. (33) and (34) for large arbitrary n.Otherwise, the asymptotic scaling of the Lyapunov spectrumcan be approximated by

λn ∼ 1

d(δn − δn−d ). (36)

Now, let us investigate the part δI (n) of the averagedivergence, respectively, the influence of the constant delayevolution operators Ik . As already mentioned, the operators Ik

can be interpreted as evolution operators of one time step ofthe linear DDE (11) where the variable delay τ (t) is substitutedby the constant delay τk = tk+1 − tk . The divergence of theirfinite section approximation by matrices computed by theEuler method can be determined by Farmer’s method [27].For simplicity, we assume that ln | det[B(t)]| is integrable forall t > t0. Thus, we obtain the asymptotic approximation forlarge n

ln∣∣ det

(I (n)

k

)∣∣ ≈ −n ln(n) + O(n). (37)

With Eq. (33) we obtain for the asymptotic scaling of the partδI (n) of the average divergence

δI (n) ≈ − 1

τn ln(n) + O(n), (38)

where τ denotes the arithmetic average, given by

τ = limN→∞

1

N

N−1∑k=0

τk. (39)

As a consequence, the asymptotic scaling behavior of the partλI (n) of the Lyapunov spectrum, related to the constant delayevolution operators, results in

λI (n) ∼ − 1

τln(n). (40)

In this section, we have demonstrated that the average di-vergence and the asymptotic scaling behavior of the Lyapunovspectrum consist of two parts, which are defined by Eqs. (32)and (35), respectively. One part describes the dynamics relatedto the constant delay evolution operators and characterizes afundamental property of every time-delay system. The secondpart describes the characteristic dynamics of the retardedaccess, i.e., the deformation of the state space in the timedomain. In Sec. V the above presented method is applied toderive the influence of the dynamics of the retarded access onthe asymptotic scaling of the Lyapunov spectrum.

V. INFLUENCE OF THE DELAY TYPEON THE LYAPUNOV SPECTRUM

In this section we relate the delay types identified in Sec. IIto the asymptotic scaling behavior of the average divergenceand the asymptotic scaling behavior of the Lyapunov spectrum.

A. Conservative delays

Let us begin with a special case of a conservative delay anddefine the retarded argument by R(t) = t − τ with the constantdelay τ . With Eq. (19) in Sec. III we see that the accessoperators Ck reduce to the forward shift operator, shiftingthe state by the delay τ . Since the shift is implemented bythe change of the underlying space and is represented byan identity matrix as defined in Sec. IV, the finite sectionapproximation equals the n-dimensional identity matrix En inany basis:

C (n)k = En. (41)

Since the determinant of En is equal to one we obtaindet (D(n)

N ) = 1 for all n. Obviously, the part δC (n) of theaverage divergence equals zero and according to Eq. (40) theasymptotic scaling of the Lyapunov spectrum results in

λn ∼ −C ln(n), (42)

which is equivalent to the result obtained by Farmer [27].For general conservative delay, let the access map R(t)

fulfill Eq. (9) in Sec. II with irrational rotation number c =limj→∞

pj

qj. The case of a conservative delay with rational

rotation number is automatically included in the followinganalysis by setting pj

qj= p

qand leads to the same result. With

Eqs. (9) and (15) we obtain

Cqj x(t ′ + pj )j→∞−→ x(t). (43)

Hence, the qj th iteration of the Koopman operator C convergesto the temporal shift, which does not change the shape of thestate and the length of the state interval. So, it is convenientto calculate the average divergence after qj time steps andthe part δC (n) can be computed by utilizing the finite sectionapproximation of the operator Dqj

defined by Eq. (29). With

062214-6


Eq. (43) it is clear that the operator Dqjconverges to the

temporal shift by pj , and since the shift is represented bythe identity matrix, the finite section approximation of Dqj

converges to the identity matrix

limj→∞

D(n)qj

= En. (44)

Obviously, the determinant det(D(n)qj

) related to the volumeevolution caused by the access operator tends to one if we sendj → ∞. The corresponding parts δC (n) and λC (n) of the averagedivergence and the Lyapunov spectrum, respectively, vanishand the Lyapunov spectrum shows the logarithmic scalingbehavior characteristic to systems with constant delay

λn ∼ −C ln(n), (45)

which is equivalent to Eq. (42) as one expects due to theinvariance of the Lyapunov exponents under suitable time scaletransformations.

B. Dissipative delays

Let us assume R(t) to be the lift of a circle map with rotationnumber p

qthat fails the criterion (9). Poincare’s classification

for circle maps implies the existence of points t∗ with Rq(t∗) +p = t∗, in other words, q-periodic orbits [23]. For convenienceand without any loss of generality we set the initial time t0 toan arbitrary periodic point t∗, leading to a periodic sequenceof interval lengths for the method of steps. The periodicityof the delay τ (t) and the definition of the operators Ck implythat they are in some sense periodic with respect to k. Theoperator Ck+q is almost identical to the operator Ck with thedifference that the domain of the elements of the underlyingspace is shifted by p relative to the domain of the elements ofthe spaces where Ck lives. Hence, Ck+q is connected to Ck by

Ck+q = S−1p Ck Sp, (46)

where the temporal displacement is represented by the shiftoperator

Sp xk(t ′) = xk(t + p). (47)

Since the length of the state intervals repeats after q time steps,it will be convenient to compute the average divergence afterj q time steps. Therefore, we compute the operator Djq usingour definitions above and obtain

Djq =j−1∏l=0

S−1lp

(q−1∏k=0

Ck

)Slp =

j−1∏l=0

S−1lp Dq Slp, (48)

where Dq is defined as in Eq. (29). Due to the factorizationof the determinant and the fact that the shift operator isrepresented by the identity matrix as described for conservativedelay, the determinant of the n-dimensional finite sectionapproximation of Djq simplifies to

det(

D(n)jq

) = det(

D(n)q

)j. (49)

The corresponding operator Dq can be interpreted as therestriction of the Koopman operator related to the qth iterationof the access map R(t) to an interval, whose boundaries aregiven by two succeeding points of a periodic orbit of R(t).Hence, the symbol of the operator Dq equals Rq(t). Since the

state after q iterations of the solution operator is shifted in timeby p relative to the initial state, the part δC (n) of the averagedivergence has to be rescaled by p and according to Eqs. (36)and (40), the asymptotic Lyapunov spectrum is given by

λn ∼ −C ln(n) + 1

d pln

∣∣∣∣∣ det(

D(n)q

)det

(D(n−d)

q

)∣∣∣∣∣. (50)

In the further analysis we show that a dissipative delayleads to a linear asymptotic scaling behavior of the Lyapunovspectrum, which can be made plausible by the followingproperty of the eigenfunctions and eigenvalues of the Koopmanoperator. So, let ξ (t) be an eigenfunction of the operator Dq

with the corresponding eigenvalue a. Since Dq maps a functionto a function space whose member’s domain is shifted by p, theterms “eigenvalue” and “eigenfunction” should be interpretedas solutions of

a ξ (t − p) = Dqξ (t ′). (51)

With the short calculation

Dq (ξ (t ′))j = [ξ (Rq(t))]j = [a ξ (t − p)]j = aj [ξ (t − p)]j ,

(52)

it follows directly that (ξ (t))j is also an eigenfunction ofthe operator Dq but with the corresponding eigenvalue aj ,whereby a constant ξ (t) = ξ ∗ is always an eigenfunction withcorresponding eigenvalue 1. In other words, the eigenfunctionsand eigenvalues of a Koopman operator form a semigroup [18].We assume that there is only one 0 < a < 1 defining thewhole spectrum by the semigroup property, which holds ifthe Koopman operator Dq is compact and a equals the slopeof the symbol Rq(t) of Dq at the sole attractive fixed point t∗inside the state interval [30], where “fixed point” means in ourcase a point fulfilling t∗ = Rq(t∗) + p. Thus, we first considerfor simplicity that p = 1 and that the reduced access map R(t)mod 1 has a unique attractive q-periodic orbit. The operator Dq

acts independently and identically at each of the d componentsof the initial state x0(t), which are approximated in termsof the first m members of the basis {φki(t)}i∈N. Hence, alleigenvalues occur with multiplicity d and with the assumptionthat the determinant of D(n)

q is now given by the product of the

largest n = m d eigenvalues of Dq counting multiplicity, weobtain

det(

D(n)q

) =m−1∏k=0

ad k = ad2 m(m−1) = a

12d

n(n−d). (53)

From the definitions of the part δC (n) of the average divergenceand the part λC (n) of the Lyapunov spectrum, i.e., the partrelated to Dq in Eq. (50), and with a ≡ eα follows directly

δC (n) = 1

2dn(n − d) α, λC (n) ∼ n

dα. (54)

For an access map with p > 1, where the reduced maphas again a unique attractive periodic orbit, the argumentationabove has to be slightly modified since Rq(t) has p attractivefixed points inside the state interval. We separate the state inter-val [tk−1,tk] into the basins of attraction Uk,j = [uk,j−1,uk,j ],where uk,p−1 = tk , of the fixed points of Rq(t) as illustrated inFig. 4. Therefore, we define the restriction xk,j (t) of the state

062214-7


(a)U1,0

t0 u1,0 u1,1 t1

t0

u1,0

u1,1

t1

time t

Rq �t�+p

(b)x0�t� C0 x0�t� C1 C0 x0�t�

U0,0 U1,0 U2,0

t�1 u0,0u0,1 t0 u1,0 u1,1 t1 u2,0u2,1 t2

t�1

u0,0

u0,1

t0

u1,0

u1,1

t1

�1 0 1 2 3

time t

R�t�

DNx 0�t�

FIG. 4. (a) Separation of the state intervals into the basins ofattraction Uk,j = [uk,j−1,uk,j ], where uk,p−1 = tk , of the attractivefixed points of the qth iteration Rq (t) of an exemplary access mapR(t) with rotation number p

q= 3

2 . (b) Evolution of a function x0(t)inside the Uk,j by the successive application of the access operatorsCk . The basic types of symbols of Ck,j , i.e., segments of R(t), areillustrated by the alternating dashed and solid lines.

x(t), the restriction Ck,j of the Koopman operator C, and therestriction Dq,j of the operator Dq by

xk,j (t) = x(t), t ∈ [uk,j−1,uk,j ] (55a)

Ck,j xk,j (t ′) = xk,j (R(t)), t ∈ [uk,j−1,uk,j ] (55b)

Dq,j x0(t ′) = x0(Rq(t)), t ∈ [uq,j−1,uq,j ]. (55c)

With Fig. 4(b) it is easy to understand that the Koopmanoperator C maps the basin of attraction Uk,j to the basin ofattraction Uk+1,j and that the restrictions Ck,j of the Koopmanoperator are basically given by q different Koopman operatorsshifted in time, which correspond to the symbols given byq different segments of R(t), repeating with period q in k

and j . Thus, we can simplify our analysis by introducing q

basic operators C0,C1, . . . ,Cq−1, which are connected to the

operators Ck,j by simple time shifts, i.e.,

Ck,j = S−1uk+1,j

C(k p+j ) mod q Suk,j, (56)

where the operators S are shift operators as defined by Eq. (47).In the same way, we represent the operators Dq,j by

Dq,j = S−1uq,j

(q−1∏k=0

C(k p+j ) mod q

)Su0,j

= S−1uq,j

Dq,j Su0,j. (57)

Since the shift operators only change the domain of themembers of the domain and codomain of the operators, thespectrum of Ck,j equals the spectrum of the correspondingbasic operator and the spectrum of Dq,j equals the spectrumof Dq,j . With Fig. 4(b) it is easy to see that the operators Dq,j

are given by

Dq,j = Cj Cj+1 . . . Cq−1C0C1 . . . Cj−1, (58)

such that Dq,j+1 can be obtained from Dq,j by switching theorder of the operator product between Cj and the remainingproduct of operators. With the fact given in [31] that switchingthe order of the product of two operators does not change thespectrum of the total operator, except the eigenvalues equal tozero, together with the fact that the spectrum of Dq,j equalsthe spectrum of Dq,j , we obtain

spec(Dq,j )\{0} = spec(Dq,j ′ )\{0}, ∀ j,j ′. (59)

Since the Koopman operator maps Uk,j to Uk+1,j and eachstate interval consists of p basins of attraction, the oper-ator Dq can be represented by the operator valued matrixdiag(Dq,0,Dq,1, . . . ,Dq,p−1). Hence and with Eq. (59), it isclear that the spectrum of Dq equals the spectrum of all ofthe Dq,j and each eigenvalue of Dq,j is an eigenvalue of Dq

with multiplicity p. At this point, we expand the restrictionsxk,j (t) of the state in terms of some basis functions as donein Sec. IV, with the difference that the domain of the basisfunctions is now given by the basin of attraction Uk,j . Bydoing this, we obtain coefficient vectors qk,j,i correspondingto the ith basis function. The state xk(t) inside the kth stateinterval now can be represented by an infinite-dimensionalvector composed of the dp-dimensional coefficient vectorsqk,i = cols(qk,0,i ,qk,1,i , . . . ,qk,p−1,i), which correspond to theith basis function of each basin of attraction Uk,j inside the kthstate interval. Due to the block-diagonal structure of Dq thematrix representation of the operator Dq with respect to thementioned expansion of the state is given by a block-diagonalmatrix, where the p blocks are the matrix representations ofthe operator and Dq,j , respectively. Consequently, the relatedn = mdp-dimensional finite section approximation D(n)

q , withrespect to the first m basis functions of each of the p basins ofattraction, takes the form

D(n)q = diag

(D(md)

q,0 ,D(md)q,1 , . . . ,D(md)

q,p−1

), (60)

where the D(md)q,j are the md-dimensional finite section

approximations of Dq,j . With the assumption that the operatorsDq,j are compact, together with Eq. (59) and the resultsfor the case p = 1, it follows that spec(Dq) = spec(Dq,j ) =

062214-8


{1,a,a2, . . . } ∪ {0}, where a denotes the unique slope of Rq(t)at the attractive fixed points and all eigenvalues of Dq occurwith multiplicity dp. With the assumption that the determinantof D(n)

q is now given by the product of the largest n = mdp

eigenvalues of Dq counting multiplicity, we obtain

det(

D(n)q

) =m−1∏k=0

adp k = adp

2 m(m−1) = a1

2dpn(n−dp)

. (61)

Finally, with a ≡ eα we obtain for the part δC (n) of the averagedivergence and for the part λC (n) of the asymptotic scaling ofthe Lyapunov spectrum

δC (n) = 1

2dp2n(n − dp) α, λC (n) ∼ n

dp2α, (62)

respectively. Hence, the asymptotically linear scaling behaviorof the Lyapunov spectrum is directly connected to thesemigroup property of the Koopman eigenfunctions. Note thatfor general access maps R(t) possessing multiple attractiveq-periodic orbits, Eq. (59) does not apply and there is nounique slope a of the attractive fixed points of Rq(t). So,there are many ways to combine the al corresponding tothe fixed points t∗l by multiplying powers of the relatedeigenfunctions ξ l(t) using their semigroup property. Thus,the part λC (n) of the Lyapunov spectrum and its asymptoticslope α should depend on all of the al . In general the operatorDq is also not compact. Even the exemplary access map inFig. 4 leads to a noncompact Koopman operator due to therepulsive fixed points of Rq(t) at the boundaries of Uk,j .However, the semigroup property of the eigenfunctions andeigenvalues of the Koopman operator is valid for all maps andthe method described above was successfully applied at DDEswith sawtooth-shaped delay [16], which leads to a compactoperator Dq in the space of analytic functions and can be seenas the limiting case where the slope at the repulsive fixed pointsof Rq(t) is infinite. A further discussion about this problem ismade at the end of this section.

In the following, we quantitatively analyze the averagedivergence and the asymptotic scaling behavior of the Lya-punov spectrum for delay systems with dissipative delay.Since the contribution of the constant delay evolution operatorwas already investigated in Sec. IV, we concentrate on thecontribution by the access operator or Koopman operator,respectively. So, it is sufficient to consider the simplest delaysystem given by

x(t) = x(R(t)). (63)

First, we analyze a simple exemplary system with dissi-pative delay where the asymptotic scaling behavior of theLyapunov spectrum can be computed analytically and whichis a special case of the systems with sawtooth-shaped delayanalyzed in the Supplemental Material of [16]. We considerthe DDE (63) with a sawtooth-shaped delay. The definitionsof the delay τ (t) and the retarded argument R(t) read as

τ (t) = 1 + A(t − t�), R(t) = 1 − A, if t /∈ N. (64)

With Fig. 5(b) it is easy to see that for 0 < A < 1, the reducedaccess map corresponding to the retarded argument is a simplelinear map possessing an attractive fixed point at the origin.

(t)

(a)

0 1 21

1 + A

2

time t

dela

y

(b)

(1 − A) t

0 10

1

time t

R(t)

mod

1

FIG. 5. (a) Sawtooth-shaped delay and (b) corresponding reducedaccess map (solid line) and bisectrix (dashed line).

If we set the initial time t0 = 0, the boundaries tk of our stateintervals related to the method of steps and given by the inverseiteration of the access map are equal to the natural numbersand the interval length is equal to the delay’s period. So, theaccess operators Ck are given by

Ckxk(t ′) = xk[(1 − A)(t − k) + k − 1], t ∈ [k,k + 1]

(65)

and the constant delay evolution operators Ik are given by

Ikxk+1(t ′) = xk+1[k + (1 − A)−1] +∫ t

k

xk+1(t ′) dt ′. (66)

Since the length of the state intervals is equal to the periodof the delay, the solution operator IkCk is a monodromyoperator known from Floquet theory and all informationabout the dynamics can be obtained by analyzing one timestep. In the following, we derive a matrix representation ofthe monodromy operator utilizing the method introduced inSecs. III and IV together with the definition of the matrixrepresentation in Appendix B. A convenient choice of thebiorthogonal bases needed for the finite section approximationare the monomial basis and the corresponding adjoint basisconsisting of the ith derivatives δ(i)(t) of the delta distributionδ(t), i.e., we expand the state xk(t) into a Taylor series aroundt = k. So, we set

φki(t) = (t − k + 1)i , (67a)

ψki(t) = (−1)i

i!δ(i)(t − k + 1), (67b)

w(t) = 1. (67c)

The n-dimensional finite section approximation with respectto the monomial basis of the constant delay evolution operatorsIk is computed by Eq. (B8) in Appendix B and results in

I (n)k =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 1(1−A)

1(1−A)2 . . . 1

(1−A)n−1

1 0 0 . . . 0

0 12 0 . . . 0

.... . .

. . .. . .

...

0 . . . 0 1n−1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, (68)

where the first row represents the summation of the value of thestate at the interval endpoint xk(k), ensuring the continuity ofthe solution x(t). The lower rows are the matrix representation

062214-9


of the integration in the monomial basis. In our simpleexample, the n-dimensional finite section approximation of theaccess operators Ck computed utilizing Eq. (B9) in Appendix Breduces to a diagonal matrix and is given by

C (n)k = diag(1,1 − A,(1 − A)2, . . . ,(1 − A)n−1). (69)

Due to the periodicity of the system and the solution operator,I (n)

k and C (n)k are time independent and the two parts of

the average divergence δn reduce to the logarithm of thedeterminant of the above mentioned operators. The part δI (n) ofthe average divergence related to the constant delay evolutionoperator is now given by

δI (n) = ln∣∣ det

(I (n)

k

)∣∣= − ln[(n − 1)!] − (n − 1) ln(1 − A). (70)

Utilizing Stirling’s approximation of the factorial leads to

δI (n) ≈ −(n − 1) ln(n − 1) + n − 1 − (n − 1) ln(1 − A).

(71)

The part δC (n) of the average divergence, which represents thecontribution by the access operator or the Koopman operator,respectively, results in

δC (n) = ln∣∣ det

(C (n)

k

)∣∣ = 12n(n − 1) ln(1 − A), (72)

which is equivalent to Eq. (54). Thus, we obtain for theasymptotic scaling of the Lyapunov spectrum

λn ∼ − ln n + n ln(1 − A). (73)

So, we have shown by a straightforward calculation thatthe Lyapunov spectrum of our simple delay system withdissipative sawtooth-shaped delay is asymptotically linear inthe limit of large Lyapunov indices. Furthermore, it has to benoticed that in this special case the asymptotic slope of theLyapunov spectrum equals the logarithm of the slope of theaccess map at the attractive fixed point, i.e., it equals the accessmap’s Lyapunov exponent.

In the following, we analyze the same simple system (63)with a more general delay. It is a known fact that sufficientlysmooth iterated maps with the same structure of hyperbolicq-periodic orbits are topological conjugate, where the slope ofthe qth iteration Rq(t) of the map R(t) at the periodic pointsis invariant under differentiable conjugacies [32]. This leadsto the conjecture that the asymptotic slope of the Lyapunovspectrum is only influenced by the slopes of Rq(t) at theperiodic points of the access map. Hence, we choose atriangular-shaped delay as shown in Fig. 6(a), where the slopesat the attractive and repulsive fixed points can be specified bythe parameters A and B, respectively. For the further analysisthe mean delay is set to one. Thus, the reduced access map hasan attractive fixed point at t = 1

2 and repulsive fixed points atthe interval boundaries, which are identified due to the modulooperation.

The Koopman operator corresponding to this access mapis not compact due to the presence of both attractive andrepulsive fixed points. Thus, the results above cannot beapplied directly and therefore we perform the followingnumerical analysis. First, we have computed the Lyapunovspectra via the semidiscretization method for different values

(a) (b)

FIG. 6. (a) Triangle delay characterized by two slopes A and B

with (b) corresponding reduced access map (solid line) and bisectrix(dashed line).

of the parameters A and B and measured the asymptotic slopeα of the Lyapunov spectra. The measured slopes are illustratedin Fig. 7(a). It is easy to see that the asymptotic slope of theLyapunov spectrum depends on both parameters of the delayA and B, which justifies our statement that it depends on

�1.0 �0.8 �0.6 �0.4 �0.2 0.0

�1.0

�0.8

�0.6

�0.4

�0.2

0.0

asymptotic slope of Lyap. spectrum

slop

evi

a K

oopm

an o

pera

tor

�0.6�0.5�0.4�0.3�0.2�0.1 0.0�0.6

�0.5

�0.4

�0.3

�0.2

�0.1

0.0

(a)

(b)

FIG. 7. (a) Asymptotic slope of the Lyapunov spectrum vs delayparameters A and B and (b) comparison of the asymptotic slope fromthe finite section approximation of the Koopman operator using anorthonormal basis (empty circles) and the Lyapunov basis (dots) vsdirect measurement at the Lyapunov spectrum (line), for all A and B

from (a). The inset shows the same analysis, whereby the Lyapunovbasis was used, for sine-shaped delay τ (t) = 1 + As

2πsin(2π t) under

variation of As .

062214-10


both slopes of the fixed points of the symbol Rq(t) of theoperator Dq . Additionally, we computed the asymptotic slopeα of the part λC (n) of the asymptotic Lyapunov spectrum usingthe approach presented in Sec. IV, together with the definitionof the matrix representations of the involved operators givenin Appendix B. For the finite section approximation of thecorresponding Koopman operator we choose the Fouriercosine basis φi(t) ∈ {1,

√2 cos(π t),

√2 cos(2π t), . . . } on the

one hand as a representative example of an orthonormal basis,and on the other hand we choose the left eigenfunctionsof the monodromy operator corresponding to Eq. (63) forψki(t) and the corresponding right eigenfunctions or theLyapunov vectors, respectively, for φki(t), which gives usan appropriate biorthogonal system, which we call Lyapunovbasis. In Fig. 7(b), the asymptotic slopes α computed by thefinite section approach are plotted versus the related slopesα resulting from the direct measurement at the Lyapunovspectrum. So, the values resulting from the finite sectionapproximation of the Koopman operator are correct if thepoints are located at the bisectrix. The approach leads tothe correct results if the finite section approximation of theKoopman operator is made with respect to the Lyapunov basiscorresponding to the specific delay and access map. However,the finite section approach fails for orthonormal bases similarto the Fourier basis due to specific structural properties of theLyapunov vectors, which are discussed in Sec. VI.

Note that the above mentioned results are not restrictedto access maps that possess only fixed points. The methodis constructed to relate the Koopman operator correspondingto the qth iteration of the access map, where q denotesthe period of the periodic orbits of the access map, to thepart δC (n) of the average divergence and the part λC (n) of theasymptotic Lyapunov spectrum. Consequently, all dissipativedelays that are related to an invertible retarded argument lead toa linear asymptotic scaling behavior of the Lyapunov spectrum.Furthermore, the method is not restricted to simple systems likeEq. (63). The specific dynamics of the system only influencesthe properties of the constant delay evolution operator Ik ,whereby the access operator Ck and the corresponding Koop-man operator only depends on the delay. In the case of a systemadmitting more complicated, especially chaotic, dynamics onehas to take into account that the Lyapunov base changes foreach state interval since the direction of the Lyapunov vectorschanges for each state interval. Hence, the computation of thefinite section approximation of the Koopman operator getsslightly more complex than for the simple system analyzedabove.

C. Discussion

As we have shown by analyzing the average divergence andthe asymptotic scaling behavior of the Lyapunov spectrum, thedynamics of time-delay systems is not only influenced by theintegral operator, but also influenced by the Koopman operator,which describes a dynamical system from the observablespoint of view [18]. In the case of delay systems with time-varying delay the dynamical system of interest is given by theaccess map and the observable is given by the state of thedelay system inside a proper interval of the method of steps.Due to the evolution of small volumes on a finite-dimensional

subspace of the infinite-dimensional state space, the spectralproperties of the Koopman operator are connected to theasymptotic scaling behavior of the Lyapunov spectrum, as wehave shown before. But, there are still open questions on theseparate calculation of the spectrum of the Koopman operator.

The spectrum of a Koopman operator depends on thedynamics of the underlying map and can be computed analyt-ically, for example, for maps with a unique (attractive) fixedpoint, as mentioned before, and for purely expanding circlemaps [33]. In our case of invertible access maps there are onlytwo possible types of dynamics, which determine our delaytypes, conservative and dissipative delay. If the access mappreserves some measure, the corresponding Koopman operatoris unitary with respect to this measure (cf. [17,34]). Roughlyspeaking, on the attractor of the map the Koopman operatorjust mixes up the values of the observable that are assignedto the attractor manifold. For conservative delay, we haveimplicitly shown that the Koopman operator correspondingto the access map is unitary with respect to some nonsingularmeasure because the limit operator of some special sequenceof iterations of the related Koopman operator is the identity. Inthe case of dissipative delay, the attractor only consists of theperiodic points or fixed points of the access map. Indeed, thecorresponding Koopman operator is unitary with respect to theDirac measure but the related space is only a finite-dimensionalsubspace of the infinite-dimensional state space of our DDE,hence less useful. Except for the spectral analysis of the adjointof the Koopman operators corresponding to circle maps in [35],there is a lack of literature about this special topic. However,considering the Koopman operator acting on the space ofbounded functions L∞ as the adjoint of the Frobenius-Perronoperator acting on L1 is not the right framework for the presentproblem and leads to wrong results in the case of dissipativedelays. There is no need to assume the eigenfunctions of theKoopman operator to be bounded. By doing this, one restrictsthe spectrum of the Koopman operator to the unit circle asshown in [35], even in the case of dissipative delays. The partδC (n) of the average divergence would be equal to one, whichcontradicts the analytical and numerical results we derivedabove.

If a dynamical system possesses hyperbolic fixed points andis sufficiently smooth, the related Koopman operator is welldescribed around these fixed points by the Koopman operatorcorresponding to the linear approximation of the system sincethere is a topological conjugacy between the nonlinear systemand its linear approximation, which is known as Hartman-Grobman theorem (cf. [36,37]). The authors of [38] extend theHartman-Grobman theorem and show that the aforementionedconjugacy exists in the whole basin of attraction if the systemis sufficiently smooth and its linear approximation has onlyeigenvalues with modulus smaller (or all greater) than one.In this case, the eigenfunctions of the Koopman operatorcan be constructed using generalized Laplace averages asshown in [39], and its spectrum consists of powers of theeigenvalues of the system’s linearization due to the semigroupproperty of the eigenfunctions and eigenvalues of the Koopmanoperator. These results substantiate our conjecture about thelinear scaling behavior of the part λC (n) of the Lyapunovspectrum in the case of dissipative delays, which can be provenanalytically for our simple linear DDE with a sawtooth-shaped

062214-11


delay that is related to an access map with an attractive fixedpoint. However, this is not the whole truth since invertibleone-dimensional iterated maps possess one repulsive fixedpoint or periodic orbit per each attractive fixed point or periodicorbit, and the linearization around an attractive fixed pointwill always be a bad approximation near the correspondingrepulsive fixed point. Due to the lack of an analytic approachin this general case, we have done some numerical analysisand are able to justify that the Koopman operator causes thelinear scaling behavior of the Lyapunov spectrum. The causeof the deviations of the finite section approach for orthonormalbases is explained in Sec. VI.

VI. LYAPUNOV VECTORS

For a more detailed analysis of the differences between thetwo delay classes, we present some qualitative and quantitativeproperties of the covariant Lyapunov vectors, which can becomputed by the method given in [40]. Note that in thefollowing we always consider covariant Lyapunov vectorseven if we drop the term “covariant”. Due to the equivalenceof delay systems with conservative delay to systems withconstant delay, which where extensively studied in the pastand recent literature, we concentrate especially on the case ofsystems with dissipative delay. We are interested in structuralproperties of the Lyapunov vectors that are connected tothe spectral properties of the Koopman operator followingfrom the dynamics of the underlying access map. Hence,the following analysis is intended to be independent of thedynamics of the delay system and concomitant questions suchas stability and orbit structure. So, let us again consider thesimplest DDE given by Eq. (63).

Note that in this simple case of a delay system with periodicdelay, the Lyapunov vectors are equal to the eigenvectorsxn(t) = pn(t) e(λn+ıωn)t of the monodromy operator, wherepn(t) is a periodic function with the same period as the delay’speriod T and λn equals the nth Lyapunov exponent orderedfrom the largest to the smallest. In Fig. 8, some exemplaryLyapunov vectors for dissipative and conservative delay andthe inverse participation ratio (IPR) of the periodic prefactorspn(t) given by [41]

IPR[pn(t)] =∫ T

0 dt |pn(t)|4( ∫ T

0 dt |pn(t)|2)2 (74)

related to the first Lyapunov vectors are shown. In Fig. 8, onecan see that the exemplary Lyapunov vector correspondingto the case of dissipative delay shows sharper peaks thanthe vector corresponding to conservative delay. A qualitativedifference can be noticed by comparing the IPR of theperiodic part pn(t) for the two delay types. Obviously,the IPR for dissipative delay increases much faster than theIPR for conservative delay. For dissipative delays, the IPRseams to increase exponentially, which indicates that the peakwidth decreases exponentially with the index of the Lyapunovexponent and the corresponding Lyapunov vector. In this case,the strongly localized peaks of the Lyapunov vectors arelocated at the members of the repulsive periodic orbits or atthe repulsive fixed points.

(a)

conservative

dissipative

−1.5 −1.0 −0.5 0.0

−0.1

0.0

0.1

0.2

time t

Re(x n

(t))

(b)

0 10 20 30 40 50 60 700.5

1

510

50

index n

IPR

FIG. 8. (a) Lyapunov vectors (real part), i.e., eigenvectors of themonodromy operator, and (b) inverse participation ratio (IPR) ofthe periodic part pn(t) of the Floquet decomposition for exemplaryconservative (dots) and dissipative (triangles) delays of Eq. (63) withτ (t) = τ0 + (0.9/2/π ) sin(2π t). For conservative delay, the meandelay was set to τ0 = 1.51 and for dissipative delay τ0 = 1.54.The IPR is plotted versus the index of the corresponding Lyapunovexponent ordered from the largest to the smallest.

In the following, we use the aforementioned results forthe Lyapunov vectors related to dissipative delay to developa simple analytic model that describes the characteristicproperties and gives us an idea how the structure of theLyapunov vectors is connected to the scaling behavior of theLyapunov spectrum and the related average divergence. So,let us consider our simple delay system given by Eq. (63). Weassume the reduced access map corresponding to the retardedargument R(t) to possesses a repulsive fixed point at t = 0and attractive fixed points at − 1

2 and 12 , i.e., the fixed points

t∗ fulfill R(t∗) = t∗ − 1. Consequently, the initial function ofour delay system is defined in the interval [− 1

2 , 12 ] whereto the

time intervals are mapped due to the Hale-Krasovkii notation,which was already mentioned in Sec. III. Since this systemis periodic, the solution operator mapping one state intervalto the next state interval, which is given by the method steps,equals the monodromy operator. Hence, the Lyapunov vectorsare equal to the eigenfunctions x(t) of the monodromy operatorand are solutions of the eigenvalue equation

es x(t) = x(1/2) +∫ t

− 12

dt ′ x(R(t ′)), t ∈ [−1/2,1/2] (75)

062214-12


whereby the Lyapunov exponents are the real parts of theFloquet exponents s and R(t) = R(t) + 1. Since the Lyapunovvectors are localized at the members of the repulsive periodicorbits or fixed points, in our simple case they are localizedat t = 0. Hence, the integration in Eq. (75) is dominated bythe values of the integrand in a small neighborhood of t =0. Roughly speaking, we assume that the system “sees” alinear retarded argument R(t) ≈ b t for t ≈ 0. Furthermore,we assume the value of the Lyapunov vectors at the end ofthe state intervals to be negligible compared to the stronglylocalized peak around t = 0 in the limit of small Lyapunovexponents, i.e., x(1/2) ≈ 0, which implies x(−1/2) ≈ 0 dueto Floquet theory. Hence, we approximate Eq. (75) by

es x(t) =∫ t

− 12

dt ′ x(b t ′), t ∈ [−1/2,1/2] (76)

which denotes an eigenvalue equation of an operator actingon periodic functions. Since the corresponding eigenfunctionsare expected to be strongly localized, it is reasonable toapproximate their periodic extension by a sum of localizedfunctions defined on the whole real line and shifted by integermultiples of the period 1. The Fourier transform of this periodicapproximation is given by the product of a train of deltafunctions and an envelope, which is analog to the “form factor”of diffraction patterns of crystals in solid state physics relatedto the structure of the crystals basis. Hence, the envelopeof the Fourier transform of the approximate eigenfunctionsconnects the localized structure of the Lyapunov vectors to theeigenvalues es and we simplify Eq. (76) by its expansion tothe whole real line “reducing the crystal to one atom,” i.e.,

es x(t) =∫ t

−∞dt ′ x(b t ′). (77)

This expression can be interpreted as solution operator of thepantograph DDE x(t) = e−sx(b t) corresponding to solutionsthat exist on the whole real line, whereby the equation is namedby the mechanical part of a current collection system of anelectric locomotive [42]. It is easy to see that this equationpossesses solutions for arbitrary s. So, the assumptions wemade lead to the loss of the information that the Lyapunovspectrum is discrete, which is connected to the discretenessof the Floquet exponents for periodic delay equations. Notsurprising because we built our model to approximate theLyapunov vectors of a delay system with any access map ofany period that has a repulsive fixed point.

The right-hand side of Eq. (77) can be expressed by aconvolution using the Heaviside step function �(t). Hence,we write

es x(t) = �(t) ∗ x(b t). (78)

Now, its Fourier transform is trivially given by

es X(f ) = 1

2b

[δ(f ) + 1

ıπf

]X(f/b), (79)

where X(f ) denotes the Fourier transform of x(t) with respectto the frequency f . For simplicity, we assume the integralover the solutions of Eq. (77) to be equal to zero, i.e., weneglect the delta distribution in Eq. (79) and obtain after some

n

(a)

(b)

n

f

Xf

FIG. 9. (a) Comparison of the real part (dots) and imaginary part(triangles) of the Fourier amplitudes of a Lyapunov vector (indexn = 29) with the corresponding envelope of the Fourier transformX(f ) ∝ X1(f ) (lines), with X1(f ) given by Eq. (81). (b) Lyapunovspectrum from semidiscretization (empty circles) compared withmodeled spectrum (dots) computed by fitting Eq. (81) to the Fourieramplitudes of the Lyapunov vectors. The system (63) with R(t) =t − 0.1

2πsin(2π t) − 1 was used.

rearrangements

γ X(f ) = 1

fX(f/b), where γ := ı2πb es. (80)

With Eq. (80) we have derived a linear first-order q-differenceequation (cf. [43]), which is solved by any linear combinationof the solutions X1(f ) and X2(f ) given by

X1/2(f ) = �(±f ) e− 12 ln b

(ln f +ln γ+ ln b2 )2

. (81)

We omit the derivation of the solution above because it isa simple straightforward calculation by transforming the q-difference equation into a linear ordinary difference equationand does not contribute to a better understanding.

In Fig. 9(a), our approximation of the envelope of theFourier transform (81) corresponding to a given small Lya-punov exponent of an exemplary system is compared to theFourier amplitudes of the corresponding Lyapunov vector,whereby b was set to the slope at the repulsive fixed pointof the access map and s was set to the numerically computedFloquet exponent. Additionally, we computed the Lyapunovspectrum of the system by fitting a suitable linear combinationof Eq. (81) to the eigenfunctions of the monodromy operatorby varying γ . The result is shown in Fig. 9(b). As desired,our simple model connects the structure of the Lyapunov

062214-13


vectors to the corresponding Lyapunov exponents and holdsasymptotically for small Lyapunov exponents. Consequently,our model indicates that the localization is caused by therepulsive fixed points or periodic orbits of the reduced accessmap and the spectral properties of the corresponding Koopmanoperator.

One interesting property of the Lyapunov vectors in thecase of dissipative delay can be directly derived from Eq. (80).A solution X(f ) can be rescaled on the frequency domainto obtain other solutions. This means, if X(f ) is a solutionwith eigenvalue γ,X(βf ) is a solution with eigenvalue βγ

for arbitrary β. Consequently, the envelope of the Fouriertransform of the Lyapunov vectors exponentially broadenswith increasing index of the Lyapunov exponent and it followsthat the Lyapunov vectors localize exponentially on the timedomain as already shown in Fig. 8(b). This also has conse-quences for the finite section method introduced in Sec. IVand other algorithms for the computation of the Lyapunovspectrum. If the Lyapunov spectrum λn asymptotically scaleslinear with the index n, the number of Fourier modes orsimilar basis vectors that are needed to obtain a good approxi-mation of the nth Lyapunov vector increases exponentially.Consequently, the number of well-approximated Lyapunovexponents and Lyapunov vectors increases logarithmicallywith the dimension of the finite section approximation, whichis worse for numerical analysis and also explains the slowconvergence of the Lyapunov spectra for the systems withdissipative delay in Fig. 2(b) in Sec. II. In other words, thecomplexity of numerical algorithms increases exponentially.Moreover, the question arises as to whether the finite sectionapproximations with respect to common orthonormal basesof the operators have anything to do with the exact operatorbecause the ratio between the number of well-approximatedexponents and computed exponents or vectors tends to zero.Since the determinant is the product over all eigenvalues of amatrix, the determinant approach to compute the asymptoticscaling behavior of the Lyapunov spectrum will fail for typicalorthonormal bases in the case of systems with dissipative delay.That is why we considered the Lyapunov basis in Sec. V forthe numerical analysis.

VII. CONCLUSION

In this paper, we have investigated the influence of theaccess map defined by the delay on the dynamics of delaysystems with time-varying delay. In particular, we haveseparated the dynamics due to the variable delay from thedynamics of the whole delay system by an operator theoreticapproach, where the solution operator is decomposed intotwo different operators. Hence, the time evolution of delaysystems with time-varying delay is characterized by the actionof the Koopman operator corresponding to the access mapand by the action of the constant delay evolution operator,which is similar to the solution operator of delay systems withconstant delay. The decomposition of the dynamics into thedeformation of the state by the Koopman operator and theintegration by the constant delay solution operator connectsthe theory of DDEs with the theory of circle maps and theKoopman operator framework of the analysis of dynamicalsystems called “Koopmanism”. This connection itself should

have the potential to extend the view on the dynamics ofsystems with time-varying delay and can be a nice foundationfor further research.

Using our above mentioned framework, we have analyzedthe influence of the two fundamental classes of delays on theLyapunov spectrum and the evolution of small volumes onfinite-dimensional subspaces on the infinite-dimensional statespace of the delay system. The classification is determined bythe question as to whether a system with time-varying delayis equivalent to a system with constant delay or not, whichdepends directly on the existence of a topological conjugacybetween the access map and a pure rotation. We have shownthat the asymptotic scaling behavior of the Lyapunov spectrumcan be separated into a logarithmic part related to the constantdelay evolution operator known from systems with constantdelay and a part related to the Koopman operator depending onthe dynamics of the underlying access map. If the access map istopological conjugate to a pure rotation, the delay system withtime-varying delay is equivalent to a system with constant de-lay, the part of the Lyapunov spectrum related to the Koopmanoperator vanishes, and we call the related delays conservativedelay because the corresponding access map is a conservativesystem. Contrarily, if the conjugacy does not exist, the delaysystem is not equivalent to a system with constant delay andthe asymptotic scaling behavior of the Lyapunov spectrum isdominated by the part related to the Koopman operator, whichis linear with respect to the index of the Lyapunov exponent.Since the related delays lead to an additional dissipation bythe Koopman operator, we call them dissipative delays. Weobtain a similar dependence on the delay type for the evolutionof small volumes on finite-dimensional subspaces of the statespace and their mean relaxation rate (average divergence). Dueto the volume-preserving retarded access by the unitary Koop-man operator, conservative delays lead to the same scalingbehavior of the average divergence with respect to the dimen-sion of the subspace as known from systems with constantdelay, and dissipative delays lead to an additional contributionto the average divergence that describes the volume evolutionunder the action of the nonunitary Koopman operator.

Furthermore, we have analyzed the influence of the delaytype on the structure of the Lyapunov vectors and brieflyinvestigated the consequences on numerical algorithms, par-ticularly those including finite section approximations of thesolution operator. The Lyapunov vectors of delay systems withdissipative delays localize exponentially, which leads to anexponential broadening of the envelope of the coefficients oftheir expansion in terms of common orthonormal bases. Wehave demonstrated that for dissipative delay, the number ofwell-approximated Lyapunov exponents or Floquet exponentsraises only logarithmically with the dimension of the finitesection approximation of the solution operator. So, maybe itis worth to check whether this and similar algorithms canbe improved by special bases, such as wavelet bases, whichtake into account the exponential localization of the Lyapunovvectors or eigenvectors of the monodromy operator in the caseof periodic systems. Indeed, the inverse Fourier transform ofthe function appearing in Eq. (81) can serve itself as a motherwavelet [44], and is nowadays also known as Log-Gaborwavelet. The consequences of this fact will be explored ina future publication.

062214-14


(a)

m m=1000

m=2000

m=500

0 100 200 300 400

−12

−10

−8

−6

−4

−2

0

index n

expo

nent

n

(b)

mm=1000

m=2000

m=500

0 100 200 300 400

−10

−8

−6

−4

−2

0

index n

expo

nent

n

FIG. 10. Comparison of the convergence properties of the Lya-punov spectra shown in Fig. 2(a) in Sec. II between systems withconservative delay (filled) and dissipative delay (empty), computedby the semidiscretization method [2,26] for (a) the turning model,given by Eq. (10c) (triangles) and (b) the Hutchinson equation, givenby Eq. (10a) (squares). The parameter m denotes the number ofequidistant points approximating the state of the delay system. As aguide to the eye, the asymptotic scaling behavior for dissipative delay,which is derived in Sec. V, is represented by the dashed gray line.

APPENDIX A: CONVERGENCE OF LYAPUNOV SPECTRA

In this section, the convergence properties of the Lyapunovspectra of the turning model given by Eq. (10c) and theHutchinson equation given by Eq. (10a) are analyzed asdone for the Mackey-Glass equation in Sec. II. In Fig. 10,the Lyapunov spectra of these systems for the two delayclasses, conservative and dissipative delay, are shown undervariation of the number m of equidistant points approximatingthe state, whereby the same system parameters and delayparameters are used as in Sec. II. As in the case of the Mackey-Glass equation, for dissipative delay the number of well-approximated Lyapunov exponents increases much slowerthan for conservative delay. This is clearly a consequence ofthe localization properties of the Lyapunov vectors as derivedin Sec. VI.

APPENDIX B: MATRIX REPRESENTATIONSOF THE OPERATORS

Let us consider the bases {φki(t)}i∈N of the spacesF([tk−1,tk],Rd ) and the corresponding dual sets {ψki(t)}i∈N

in the dual spaces F∗, as defined in Sec. IV. The mentionedsets define biorthogonal systems of our spaces F endowedwith a weighted inner product such that∫ tk

tk−1

dt wk(t) ψki(t) φkj (t) = δij , (B1)

where the tk are connected to the initial time t0 by Eq. (17) inSec. III. For a simpler computation of the matrix representationof operators between the spaces F([tk−1,tk],Rd ) with differentk it will be convenient to define the bases {φki(t)}i∈N and{ψki(t)}i∈N to be connected by a linear time scale transforma-tion. Hence, we set

φ(k+1)i(t) = φki[hk(t)], (B2)

ψ(k+1)i(t) = ψki[hk(t)], (B3)

wk+1(t) = tk − tk−1

tk+1 − tkwk[hk(t)], (B4)

whereby

hk(t) = tk − tk−1

tk+1 − tk(t − tk) + tk−1 (B5)

maps [tk,tk+1] to [tk−1,tk] by shifting and linear rescaling. Thisleads to the advantage that “switching” between the spaces bythe linear transformation hk(t) is represented by the identitymatrix, i.e.,∫ tk+1

tk

dt wk+1(t) ψ(k+1)i(t) φkj [hk(t)] = δij . (B6)

The state xk(t) can be represented as an infinite-dimensionalvector qk composed of the d-dimensional coefficient vectorsqk,i , which are given by

qk,i =∫ tk

tk−1

dt wk(t) ψki(t) xk(t). (B7)

In the same way we represent the operators Ik and Ck byinfinite-dimensional matrices composed of (d × d) matrices.Thus, the submatrix {Ik}ij of the matrix representation of theconstant delay evolution operator Ik is given by

{Ik}ij =∫ tk+1

tk

dt wk+1(t) ψ(k+1)i(t) [Ik φ(k+1)j (t ′)], (B8)

whereby each component of the matrix valued operator Ik

is applied to the scalar basis function φ(k+1)j (t) and eachcomponent of the result is projected subsequently to the initialbasis by the weighted scalar product with ψ(k+1)i(t). Thesubmatrix {Ck}ij of the matrix representation of the accessoperator Ck is computed by

{Ck}ij =∫ tk+1

tk

dt wk+1(t) ψ(k+1)i(t) [Ck φkj (t ′)], (B9)

where the computation is done as above with the differencethat Ck changes the underlying space of the state. Hence, weutilize the orthogonality relation (B6).

062214-15


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