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 of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical 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 delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. 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 Lyapunov vectors.

DOI: 10.1103/PhysRevE.95.062214

I. INTRODUCTION correspond to a reasonable physical model, characterized by transport or fixed evolutionary steps? In general, this is not Time-delay systems emerge, for example, in control the- true and the existence of such a well-behaved transformation ory [1], engineering [2], life science [3,4], chaos control [5], depends on the functional properties of the time-varying delay. and synchronization of networks [6]. In all of these fields, In fact, the resulting delay classification implies fundamental effects of time-varying delays are investigated. Introducing differences in the dynamics of systems that are equivalent to time-varying delays can improve the stability of machining systems with constant delay and the remaining systems, as processes [7] and of systems in general via time-delayed demonstrated in [16]. feedback control [8]. Furthermore, in biological models, So, let us consider a general d-dimensional dynamical time-varying delays arise naturally by taking into account system defined by the system of delay differential equations fluctuations of the environment [3,9], and also the effect on (DDE) with time-varying delay τ(t): synchronization is studied [10]. The functional structure of the delay itself influences mathematical properties of general z˙(t) = f (z(t),z(R(t)),t), where R(t):= t − τ(t)(1) delay systems such as causality, minimality [11], and spectral reachability [12]. is called retarded argument. Hence, the results presented below For some systems with time-varying delay it is known hold for all DDEs with one time-varying delay τ(t). In other that they can be transformed to systems with constant delay words, the results are independent of the specific form of f and because the delay is defined by an intrinsic constant delay and apply to autonomous and nonautonomous systems. Although a known time scale transformation [13]. This type of system the present theory holds only for systems with one delay, it pro- arises, for example, in biological models with threshold delays, vides a basis for a more general theory including systems with where the intrinsic constant delay represents the evolutionary multiple delays and, even more general, distributed delays. steps, which have to be passed to reach the state of adulthood In this paper, we focus on the influence of the functional and the corresponding time scale represents the grade of structure of the retarded argument on the state dynamics as evolution [3,14]. Further examples are systems with variable given by Eq. (1) and extend the theory established in [16]. transport delays, where the constant delay is given by the We will show that there is a natural connection between the length of the transport line and the corresponding time scale dynamics of systems with time-varying delay, the dynamics of represents the covered distance [15]. The question arises as to one-dimensional iterated maps, and the spectral properties of whether one can find a transformation for a general system the Koopman operator associated with these maps. This con- with time-varying delay, such that the initial system with nection itself is interesting because it relates well understood time-varying delay is equivalent to a resulting system with and distinct theories and may lead to a better understanding constant delay, or not. In other words, is there a well-behaved of delay systems with time-varying delay. The analysis of transformation in the sense that dynamical quantities are well dynamical systems in the framework of the Koopman operator defined for the new system with constant delay? And, are they is a wide topic of past and recent research [17,18] and is invariant under the transformation and does the new system applied, for example, in the computation of isostables and isochrons [19,20] or in global stability analysis [21]. The paper is organized as follows. In Sec. II we derive the *[email protected] two delay classes as sketched in [16] and give a short overview †[email protected] on the differences in the dynamics. Sections III and IV deal ‡[email protected] with the separation of the dynamics related to the variable

2470-0045/2017/95(6)/062214(16) 062214-1 ©2017 American Physical Society DAVID MÜLLER, ANDREAS OTTO, AND GÜNTER RADONS PHYSICAL REVIEW E 95, 062214 (2017) delay from the dynamics of the delay system. The influence of the delay classes on the Lyapunov spectrum is analyzed in Sec. V and on the Lyapunov vectors in Sec. VI, thereby explaining also the differences in the numerical properties of spectral methods. A

II. DELAY CLASSIFICATION We showed in [16] that two different types of delays can be identified, which result in fundamental differences in the dynamics of the corresponding delay system. In the following, we briefly introduce this classification by the analysis of the existence of a time scale transformation that transforms the system of Eq. (1) to a system with constant delay. Furthermore, we elaborate the differences in the dynamics and numerics of systems belonging to one or the other delay type. FIG. 1. Parameter space of an arbitrary system with time-varying = + A Every system of differential equations is equivalent to delay τ(t) τ0 2π sin(2πt). The white regions correspond to a a large set of systems of similar type that are connected conservative delay, where the system is equivalent to a system with among each other by time scale transformations. Such a constant delay. On the other hand, the black regions correspond to a transformation is easily done by the substitution of the original dissipative delay, where the system is not equivalent to a system with time scale t by an invertible and at least once differentiable constant delay. function t = (t) of the new time scale t. In other words, one introduces the new variable y(t )by conjugacy between one-dimensional iterated maps acting on y = z ◦ , (2) the same interval preserves the rotation number, which is defined by [23] where ◦ denotes function composition and z(t) is the original Rk(t) − t variable. By the substitution of the new variable y(t ) into the ρ = lim =−c. (8) original delay Eq. (1) and by some rearrangements we obtain k→∞ k an equivalent delay system [22] If the retarded argument R(t) defines a map that is topological conjugate to a pure rotation, the constant delay c in Eq. (5)is ˙y(t ) = ˙ (t ) · f ( y(t ), y(Rc(t )),(t )), (3) determined by Eq. (8) and it follows directly from Eq. (4) that where the new retarded argument Rc(t ) is derived from the R(t) fulfills the criterion original retarded argument by qj lim R (t) + pj = t ∀ t, (9) −1 j→∞ Rc =  ◦ R ◦ . (4) where pj is the jth convergent of the continued fraction If we find at least one function (t ) such that qj expansion [24]ofc =−ρ. Circle maps fulfilling this criterion R (t ) = t − c, with c>0(5) c exhibit marginally stable quasiperiodic (periodic) motion since we are able to transform the system with time-varying delay, they are topological conjugate to the pure rotation by an defined by Eq. (1), into a system with constant delay, Eq. (3). irrational (rational) angle c. Invertible maps that do not fulfill Equation (4) takes the structure of a conjugacy equation, the criterion above have stable fixed points or periodic orbits. which is known from the theory of one-dimensional iterated This follows directly from Poincare’s classification [23]. maps [23]. From this point of view, such an equation deter- Thus, the question as to whether a system of DDEs is mines the topological conjugacy between the iterated maps equivalent to a system with constant delay can be reduced to the question of the existence of a topological conjugacy k θk = R(θk−1) = R (θ0), (6) between the access map defined by the retarded argument = = k R(t) and a pure rotation. Equivalence means that the systems θk Rc(θk−1) Rc (θ0). (7) exhibit identical dynamics and that characteristic quantities are Since these maps describe the dynamics of the access to the invariant under the time scale transformation.1 The connection delayed state of our delay system, we call them access maps. to the existence of a circle map conjugacy leads to interesting In the following, we only consider invertible retarded consequences for systems with parameter families of delays. arguments R(t), i.e.,τ ˙(t) < 1, because noninvertible retarded As illustrated in Fig. 1, the parameter regions that are related to arguments cause problems with minimality and causality, and are even considered unphysical [11]. Now, let us assume further that the delay τ(t) is periodic with period 1, i.e., we 1By the iterative method given in [45] a time scale transformation measure time in units of the delay period. Since the related to constant delay can be computed even in the case where the access maps can be reduced to maps acting on the interval [0,1] we are map is not conjugate to a pure rotation. However, in this case the able to apply the theory of circle maps [23] and it follows that transformation is well defined only on finite time intervals but not for some retarded arguments R(t) the corresponding access for infinite times. Hence, asymptotic quantities such as Lyapunov map is topological conjugate to a pure rotation. The topological exponents are not well defined for the resulting systems.

062214-2 FROM DYNAMICAL SYSTEMS WITH TIME-VARYING . . . PHYSICAL REVIEW E 95, 062214 (2017) a system that is equivalent to a system with constant delay form (a) 0 a Cantor set, whereas regions related to systems that are not equivalent to a system with constant delay are characterized 2 by Arnold tongues. n Obviously, the classification depends only on the properties 4 of the access map and not on the specific DDE. In other words, the classification is independent of the specific properties of f . 6 So, if there is any influence of the delay type on the systems exponent dynamics, the influence emerges in all types of time-delay 8 systems and represents a universal property of all systems with time-varying delay. 0 10 20 30 40 50 60 The influence of the delay type on the systems dynamics and its universality can be demonstrated easily by the evolution index n of small perturbations and the related relaxation rates called Lyapunov exponents. The set of all Lyapunov exponents (b) 0 ordered from the largest to the smallest is called Lyapunov 2

spectrum. As already argued in [16], the asymptotic scaling n m2000 behavior of the Lyapunov spectrum depends on the delay 4 m1000 class. In Fig. 2, exemplary Lyapunov spectra of three delay m systems are shown for each delay type, whereby the spectra 6 were computed directly from the systems with time-varying m500 exponent delay without applying any transformation. The systems are 8 given by 10 0 100 200 300 400 z˙(t) = 2z(t)[1 − z(R(t))], (10a) 10z(R(t)) index n z˙(t) = − 5z(t), (10b) + 10 1 z(R(t)) FIG. 2. Comparison of the Lyapunov spectra between systems z¨(t) + z˙(t) + 4π 2z(t) = 8z(R(t)), (10c) with conservative delay (filled) and dissipative delay (empty), computed by the semidiscretization method [2,26]. (a) The Lyapunov where for constant delay Eq. (10a) is the Hutchinson equation spectra are shown for three different systems, where the systems arising from population dynamics [4], Eq. (10b) is the Mackey- fromtoptobottomaregivenbyEq.(10b) (circles), Eq. (10c) Glass equation known as a model for blood-production [25], (triangles), and Eq. (10a) (squares). (b) The convergence properties of and Eq. (10c) arises in the stability analysis of turning the method are illustrated for Eq. (10b), where m denotes the number processes [7]. To illustrate the universality of the following of equidistant points approximating the state of the delay system. As results, i.e., the independence of the specific dynamics of the a guide to the eye, the asymptotic scaling behavior for dissipative system, the parameters and the time-varying delay are chosen delay, which we derive in Sec. V, is represented by the dashed such that Eq. (10a) exhibits a limit cycle, Eq. (10b) chaotic gray line. The jumps in the spectra related to conservative delay dynamics, and Eq. (10c) a stable fixed point, respectively. and the deviation of the spectra from the dashed line for dissipative If the access map of a time-delay system is topological delay, respectively, indicate the bounds on the number of well conjugate to a pure rotation, the dynamics of the system approximated exponents. For all computations, we have chosen the delay τ(t) = τ + 0.9/(2π)sin(2πt) with the mean delay τ = 1.51 equals the dynamics of a system with constant delay. Since 0 0 and 1.54 for dissipative and conservative delay, respectively. the asymptotic scaling behavior of the Lyapunov spectrum for systems with constant delay is known to be logarithmic with respect to the index of the Lyapunov exponents [27], equation in Fig. 2(b) and for the remaining systems in Fig. 10 the scaling behavior of equivalent systems with time-varying in Appendix A. For conservative delays, the number of well delay is logarithmic as well. We call the delays with this approximated exponents increases much faster by increasing property conservative delay, because the related access map the number m of sampling points than for dissipative delay. is a measure preserving system and has no influence on the We see that the classification by the access map’s dynamics scaling behavior of the Lyapunov spectrum. If the access leads to interesting hitherto unknown phenomena such as the map does not fulfill the criterion (9), the specific delay above mentioned influence on the Lyapunov spectra and the system is not equivalent to a system with constant delay numerical properties of their computation. We will discuss this and, consequently, the asymptotic scaling behavior is not in more detail in Sec. VI. In the next section we separate the logarithmic as is shown in Fig. 2(a). Since the Lyapunov influence of the access map’s dynamics from the dynamics of spectrum decreases asymptotically faster than any logarithm, the delay system using a specific operator framework. we call the corresponding delays dissipative delay. Later, we will derive in detail that the scaling behavior is asymptotically III. DELAY DYNAMICS IN OPERATOR FRAMEWORK linear, which was already briefly demonstrated in [16]. The delay type also influences the convergence properties of In this section we give some preliminaries for the separation numerical algorithms as illustrated for the Mackey-Glass of the influence of the access map on the dynamics of the delay

062214-3 DAVID MÜLLER, ANDREAS OTTO, AND GÜNTER RADONS PHYSICAL REVIEW E 95, 062214 (2017) system. The method is derived from the method of steps for To adapt the decomposition of the solution operator of systems with time-varying delay (cf. [22]). The separation will our DDE to the aforementioned notation, we introduce the be used in Secs. IV and V for the explanation of the different restrictions Iˆk and Cˆk of the integration and the Koopman asymptotic scaling behavior of the Lyapunov spectra in Fig. 2. operator, respectively, related to the previously defined time For a given solutionz ˜(t)ofEq.(1), the linearized system intervals. The operator Cˆk is defined by the restriction of the for the dynamics of small perturbations x(t) = z(t) − z˜(t)is Koopman operator Cˆ to the domain of the related state xk(t) d given by living in an appropriate space F([tk−1,tk],R ) of functions d mapping [t − ,t ]toR . We call the family of operators Cˆ x˙ (t) = A(t) x(t) + B(t) x(R(t)), (11) k 1 k k access operators because they describe the access of the delay where A(t) = ∂z(t) f and B(t) = ∂z(R(t)) f denote the Jacobians system to the preceding state: of f with respect to the variable z(t) and the delayed variable ˆ F d → F d z(R(t)) taken at the unperturbed solutionz ˜(t). Consequently, Ck : ([tk−1,tk],R ) ([tk,tk+1],R ): without any loss of generality we can restrict our analysis to Cˆk[xk(t ),t] = xk(R(t)),t∈ [tk,tk+1]. (19) d-dimensional linear delay systems. ˆ ˆ In order to apply the method of steps to our delay system, it The restriction Ik of the integration operator I in our notation F d is convenient to remove the first term on the right-hand side of of labeled time intervals acts on the space ([tk,tk+1],R ) Eq. (11) to reduce the computation of the solution to a simple where our future state xk+1(t) lives and takes the form weighted integration of the system’s history. Hence, we use Iˆ [x + (t ),t] = M(t,t ) x + (t + ) the variation of constants approach and obtain the integral k k 1 k k 1 k 1 t formulation of Eq. (11): + dt M(t,t ) B(t ) xk+1(t ), t t k x(t) = M(t,t0) x(t0) + dt M(t,t ) B(t ) x(R(t )), (12) t ∈ [tk,tk+1]. (20) t0 ˆ where t0 denotes the initial time and M(t,t ) denotes the Since the operator Ik is equivalent to the solution operator fundamental solution of the ordinary differential equation for DDEs with constant delay in terms of the Hale-Krasovkii (ODE) part and solves notation where the time domain of the state is shifted to the time domain of the initial function and the position in the d M(t,t ) = A(t) M(t,t ), M(t ,t ) = Ed , (13) physical time is represented by labeling the state intervals [28], dt we call it constant delay evolution operator. For convenience, where Ed is the d-dimensional identity matrix. The operator we introduce the following short notation: on the right-hand side of Eq. (12) can be decomposed into the operators Iˆ and Cˆ and the integral formulation of our delay Lˆ ·=Lˆ [ · ,t], (21) equation can be represented by the fixed point equation where Lˆ stands for any of the above and below defined x(t) = Iˆ[Cˆ [x(t ),t ],t], (14) operators. Finally, the evolution of a solution of our linear DDE can where Cˆ denotes the Koopman operator also called composi- be expressed in terms of time intervals with finite length, tion operator with the symbol R(t) where the connection between two states on sequential time Cˆ [x(t ),t] = x(R(t)) (15) intervals is given by ˆ and I denotes the integral operator x + (t) = Iˆ Cˆ x (t ) (22) k 1 k k k t −1 Iˆ[x(t ),t] = M(t,t0)x[R (t0)] + dt M(t,t ) B(t ) x(t ). and we have derived a method that separates the dynamics t0 of the delay system with time-varying delay into the access (16) map’s dynamics represented by the access operators Cˆk and the dynamics of a system with constant delay represented The main idea of the method of steps is to divide the time scale into several subintervals, such that the solution of the DDE reduces to the solution of an inhomogeneous (a) (b) ODE. Trivially the intervals are chosen properly if the interval boundaries tk are connected by Ck Ck tk−1 = R(tk), (17) I k I k thus giving further meaning to the access map, Eq. (6). Hence, k  1 k k  1 the temporal evolution of a DDE can be described by the 0 0 0 tk1 tk tk1 iteration of operators acting on functions, which are defined on intervals ordered in time. To be consistent with our preceding FIG. 3. Illustration of the decomposition of the solution operator ˆ definitions we define the initial state x (t) as the initial function into the sequential action of the access operator Ck and the constant 0 ˆ inside the interval [R(t ),t ] and the state x in the time interval delay evolution operator Ik for (a) constant delay and (b) time-varying 0 0 k ˆ after k time steps as the function delay. For constant delay τ0 the access operator Ck reduces to a time shift by τ0 and for time-varying delay the access operator changes the xk(t) = x(t),t∈ [tk−1,tk]. (18) length of the state interval and deforms the state on the time domain.

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ˆ (n) by the constant delay evolution operators Ik. The method is qN of the state after N time steps xN (t)isnowgivenby illustrated in Fig. 3. N−1 (n) = (n) (n) (n) qN I k Ck q0 . (25) IV. DECOMPOSING THE DYNAMICS VIA FINITE k=0 SECTION METHOD Now, let us consider the projection of the dynamics of In the following, we introduce a method to determine our linear delay system defined by Eq. (11)onn-dimensional d the relaxation rate of the evolution of small volumes, called subspaces of F([tk−1,tk],R ) and analyze the evolution of a average divergence, under the dynamics of the DDE (1)ona (n) small n-dimensional initial volume V0 spanned by n linearly finite-dimensional subspace of its phase space. The method, independent vectors on such a subspace of the space of initial d which is a generalization of Farmers approach for systems functions F([t−1,t0],R ). The long term evolution of these vec- with constant delay [27] to systems with time-varying delay, tors and the related volume under the dynamics of Eq. (11)in is based on the method for calculating Lyapunov exponents general are dominated by the n largest Lyapunov exponents λl. given in [29] and uses the presented separation of the solution Thus, the asymptotics of the n-dimensional volume V (n) at time operator in Sec. III. N tN after subsequent projection to the related n-dimensional { } d Let us define the sets φki(t) i∈N to be the bases of each subspace of the underlying space F([t − ,t ],R )isgivenby d N 1 N canonical direction of the function spaces F([tk−1,tk],R ) and (n) n − { } ∼ l=1 λl (tN t0) ψki(t) i∈N to be the corresponding dual sets in the dual spaces VN e . (26) F ∗.2 In other words, let the mentioned sets define biorthogonal Since the long-time evolution of a small volume is in general systems of our spaces F endowed with a suitable inner product independent of the specific subspace, we choose as basis of vectors of the same and of vectors of different spaces F. vectors for our n-dimensional subspace the first m = n/d Then, the state xk(t) can be expanded into a series in terms of members of our basis {φ (t)} ∈ for each of the d canonical the basis vectors of the underlying space, i.e., ki i N directions of the phase space. So, an approximation of the ∞ volume evolution follows directly from the finite section xk(t) = qk,i φki(t). (23) approximation of the evolution operator and is given by = i 0 − V (n) N1 Thus, x (t) is represented as an infinite-dimensional vector N ≈ det I (n)C(n) . (27) k (n) k k qk = cols(qk,0,qk,1,...) composed of the d-dimensional co- V0 k=0 efficient vectors q .Theith vector q is related to the ith k,i k,i Utilizing the factorization of the determinant of the product of basis function φ (t) and its components are related to the ki matrices, we obtain canonical directions of the state xk(t). Furthermore, we define ˆ (n) N−1 N−1 the matrix representations I k and Ck of the operators Ik and V N ≈ (n) · (n) Cˆ , respectively, such that the evolution of the state by Eq. (22) det I k det Ck . (28) k V (n) is represented by 0 k=0 k=0 The codomain of the access operator Cˆ equals the domain of qk+1 = I k Ck qk. (24) k the access operator Cˆk+1. Hence, the matrix product of their Obviously, the infinite-dimensional matrices I k and Ck are finite section approximations can be identified with the finite composed of (d × d) matrices, whereby each is related to section approximation of their product. So, we consider the one of the basis functions φki(t). For the following analysis product of the access operators Cˆk, it is convenient to choose matrix representations, i.e., to N−1 choose the bases and the inner product, in a way such that Dˆ = Cˆ := Cˆ − ...Cˆ Cˆ (29) shift operators are represented by identity matrices. This is a N k N 1 1 0 = quite natural assumption because in our framework the shift k 0 (n) operator is implemented by just switching the underlying and let DN be its n-dimensional finite section approximation. d space F([tk−1,tk],R ). In the case of a constant delay the Furthermore, we assume that the convergence of this approxi- ˆ operators Ck become shift operators and the whole dynamics mation in terms of {φki(t)}i∈N is well behaved in the sense that is sufficiently described by the Iˆk. One possible definition of for some unbounded sequence of N such a matrix representation can be found in Appendix B. − (n) (n) (n) N1 Let I k , Ck , and qk be the n-dimensional approximation (n) ∼ (n) det DN det Ck . (30) of the above mentioned operators and the state at time tk, where k=0 n is an integer multiple of d.Then-dimensional approximation Finally, the volume evolution can be expressed by − V (n) N1 N ≈ det I (n) · det D(n) . (31) 2Note that in the literature, particularly in [29,46], the dynamics (n) k N V0 = of DDEs is represented by a family of solution operators rather on k 0 Rd × F than F because of some mathematical issues defining the Obviously, the volume evolution factorizes into two parts solution operator on Lebesgue spaces, for example. But, as long as where the first part describes the influence of the successive we only consider continuous initial functions we avoid these issues. integrations and the second part describes the influence of the

062214-5 DAVID MÜLLER, ANDREAS OTTO, AND GÜNTER RADONS PHYSICAL REVIEW E 95, 062214 (2017) underlying dynamics of the successive access to the initial As a consequence, the asymptotic scaling behavior of the part state x0(t) by the retarded argument R(t). λI (n) of the Lyapunov spectrum, related to the constant delay More expressive than the explicit volume evolution is the evolution operators, results in related exponential rate, the average divergence δ , which can n 1 be computed by λI (n) ∼− ln(n). (40) τ¯ (n) 1 VN In this section, we have demonstrated that the average di- δn = lim ln = δI (n) + δC(n) (32) N→∞ t − t (n) N 0 V0 vergence and the asymptotic scaling behavior of the Lyapunov spectrum consist of two parts, which are defined by Eqs. (32) and can be separated into the parts and (35), respectively. One part describes the dynamics related N−1 to the constant delay evolution operators and characterizes a = 1 (n) δI (n) lim ln det I k (33) fundamental property of every time-delay system. The second N→∞ t − t N 0 k=0 part describes the characteristic dynamics of the retarded and access, i.e., the deformation of the state space in the time domain. In Sec. V the above presented method is applied to 1 (n) δ (n) = lim ln det D . (34) derive the influence of the dynamics of the retarded access on C →∞ − N N tN t0 the asymptotic scaling of the Lyapunov spectrum. The exponential rate δI (n) describes the volume evolution caused by the successive integrations and the rate δC(n) V. INFLUENCE OF THE DELAY TYPE describes the volume evolution caused by the successive access ON THE LYAPUNOV SPECTRUM to the initial state by the retarded argument R(t). Due to Eq. (26), δn describes the asymptotics of the sum of the n In this section we relate the delay types identified in Sec. II largest Lyapunov exponents. Hence, the asymptotic scaling to the asymptotic scaling behavior of the average divergence behavior of the Lyapunov spectrum can be computed by the and the asymptotic scaling behavior of the Lyapunov spectrum. limit of the difference of the approximations of δn and δn−1 for large n. Just as the average divergence the asymptotic scaling A. Conservative delays of the Lyapunov spectrum splits into two parts Let us begin with a special case of a conservative delay and = − λn ∼ δn − δn−1 = λI (n) + λC(n) . (35) define the retarded argument by R(t) t τ with the constant delay τ.WithEq.(19) in Sec. III we see that the access In the method described above, n has to be a rational multiple operators Cˆ reduce to the forward shift operator, shifting of the dimension d. Thus, Eq. (35) can only be applied k = the state by the delay τ. Since the shift is implemented by directly, if d 1 or if we have found expressions describing the change of the underlying space and is represented by the asymptotics of Eqs. (33) and (34) for large arbitrary n. an identity matrix as defined in Sec. IV, the finite section Otherwise, the asymptotic scaling of the Lyapunov spectrum approximation equals the n-dimensional identity matrix En in can be approximated by any basis:

1 (n) λn ∼ (δn − δn−d ). (36) C = E . (41) d k n

Now, let us investigate the part δI (n) of the average Since the determinant of En is equal to one we obtain (n) = divergence, respectively, the influence of the constant delay det (DN ) 1 for all n. Obviously, the part δC(n) of the evolution operators Iˆk. As already mentioned, the operators Iˆk average divergence equals zero and according to Eq. (40)the can be interpreted as evolution operators of one time step of asymptotic scaling of the Lyapunov spectrum results in the linear DDE (11) where the variable delay τ(t) is substituted λn ∼−C ln(n), (42) by the constant delay τk = tk+1 − tk. The divergence of their finite section approximation by matrices computed by the which is equivalent to the result obtained by Farmer [27]. Euler method can be determined by Farmer’s method [27]. For general conservative delay, let the access map R(t) For simplicity, we assume that ln | det[B(t)]| is integrable for fulfill Eq. (9) in Sec. II with irrational rotation number c = pj all t>t0. Thus, we obtain the asymptotic approximation for limj→∞ . The case of a conservative delay with rational qj large n rotation number is automatically included in the following pj p (n) ≈− + analysis by setting = and leads to the same result. With ln det I k n ln(n) O(n). (37) qj q Eqs. (9) and (15) we obtain With Eq. (33) we obtain for the asymptotic scaling of the part j→∞ δ (n) of the average divergence qj I Cˆ x(t + pj ) −→ x(t). (43) 1 δI (n) ≈− n ln(n) + O(n), (38) Hence, the qj th iteration of the Koopman operator Cˆ converges τ¯ to the temporal shift, which does not change the shape of the whereτ ¯ denotes the arithmetic average, given by state and the length of the state interval. So, it is convenient − to calculate the average divergence after q time steps and 1 N1 j n τ¯ = lim τk. (39) the part δC( ) can be computed by utilizing the finite section N→∞ N ˆ k=0 approximation of the operator Dqj defined by Eq. (29). With

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ˆ state after q iterations of the solution operator is shifted in time Eq. (43) it is clear that the operator Dqj converges to the temporal shift by pj , and since the shift is represented by by p relative to the initial state, the part δC(n) of the average divergence has to be rescaled by p and according to Eqs. (36) the identity matrix, the finite section approximation of Dˆ q j and (40), the asymptotic Lyapunov spectrum is given by converges to the identity matrix (n) = det D(n) lim Dq En. (44) 1 q j→∞ j λ ∼−C ln(n) + ln . (50) n (n−d) dp det Dq Obviously, the determinant det(D(n)) related to the volume qj evolution caused by the access operator tends to one if we send In the further analysis we show that a dissipative delay leads to a linear asymptotic scaling behavior of the Lyapunov j →∞. The corresponding parts δC(n) and λC(n) of the average divergence and the Lyapunov spectrum, respectively, vanish spectrum, which can be made plausible by the following and the Lyapunov spectrum shows the logarithmic scaling property of the eigenfunctions and eigenvalues of the Koopman ξ ˆ behavior characteristic to systems with constant delay operator. So, let (t) be an eigenfunction of the operator Dq with the corresponding eigenvalue a. Since Dˆ maps a function ∼− q λn C ln(n), (45) to a function space whose member’s domain is shifted by p,the which is equivalent to Eq. (42) as one expects due to the terms “eigenvalue” and “eigenfunction” should be interpreted invariance of the Lyapunov exponents under suitable time scale as solutions of transformations. a ξ(t − p) = Dˆ q ξ(t ). (51)

B. Dissipative delays With the short calculation ˆ ξ j = ξ q j = ξ − j = j ξ − j Let us assume R(t) to be the lift of a circle map with rotation Dq ( (t )) [ (R (t))] [a (t p)] a [ (t p)] , p number q that fails the criterion (9). Poincare’s classification (52) for circle maps implies the existence of points t∗ with Rq (t∗) + ξ j p = t∗, in other words, q-periodic orbits [23]. For convenience it follows directly that ( (t)) is also an eigenfunction of ˆ j and without any loss of generality we set the initial time t to the operator Dq but with the corresponding eigenvalue a , 0 ξ = ξ ∗ an arbitrary periodic point t∗, leading to a periodic sequence whereby a constant (t) is always an eigenfunction with of interval lengths for the method of steps. The periodicity corresponding eigenvalue 1. In other words, the eigenfunctions and eigenvalues of a Koopman operator form a semigroup [18]. of the delay τ(t) and the definition of the operators Cˆk imply that they are in some sense periodic with respect to k.The We assume that there is only one 0

From the definitions of the part δC(n) of the average divergence where Dˆ q is defined as in Eq. (29). Due to the factorization of the determinant and the fact that the shift operator is and the part λC(n) of the Lyapunov spectrum, i.e., the part ˆ ≡ αˆ represented by the identity matrix as described for conservative related to Dq in Eq. (50), and with a e follows directly delay, the determinant of the n-dimensional finite section 1 n ˆ δC(n) = n(n − d)ˆα, λC(n) ∼ α.ˆ (54) approximation of Djq simplifies to 2d d (n) = (n) j For an access map with p>1, where the reduced map det D det Dq . (49) jq has again a unique attractive periodic orbit, the argumentation q The corresponding operator Dˆ q can be interpreted as the above has to be slightly modified since R (t) has p attractive restriction of the Koopman operator related to the qth iteration fixed points inside the state interval. We separate the state inter- of the access map R(t) to an interval, whose boundaries are val [tk−1,tk] into the basins of attraction Uk,j = [uk,j−1,uk,j ], q given by two succeeding points of a periodic orbit of R(t). where uk,p−1 = tk, of the fixed points of R (t) as illustrated in q Hence, the symbol of the operator Dˆ q equals R (t). Since the Fig. 4. Therefore, we define the restriction xk,j (t) of the state

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ˆ (a) t1 operators Ck,j by simple time shifts, i.e., U 1,0 −1 Cˆ = Sˆ C˜ + Sˆ , (56) k,j uk+1,j (kp j)modq uk,j u1,1 where the operators Sˆ are shift operators as defined by Eq. (47). +p

 ˆ t In the same way, we represent the operators Dq,j by  q R q−1 u1,0 −1 Dˆ = Sˆ C˜ + Sˆ q,j uq,j (kp j)modq u0,j k=0 t 0 = Sˆ−1 D˜ Sˆ . (57) uq,j q,j u0,j t0 u1,0 u1,1 t1 Since the shift operators only change the domain of the time t members of the domain and codomain of the operators, the ˆ (b) 1 0 1 2 3 spectrum of Ck,j equals the spectrum of the corresponding basic operator and the spectrum of Dˆ q,j equals the spectrum x t 0 C0 x0t C1 C0 x0t of D˜ . With Fig. 4(b) it is easy to see that the operators D˜

 q,j q,j t 

0 are given by x N

D D˜ q,j = C˜j C˜j+1 ...C˜q−1C˜0C˜1 ...C˜j−1, (58)

U0,0 U1,0 U2,0 D˜ + D˜ t such that q,j 1 can be obtained from q,j by switching the 1 order of the operator product between C˜ and the remaining u j 1,1 product of operators. With the fact given in [31] that switching the order of the product of two operators does not change the u1,0

 spectrum of the total operator, except the eigenvalues equal to t  t0 ˆ R zero, together with the fact that the spectrum of Dq,j equals the spectrum of D˜ q,j, we obtain u0,1 u0,0 spec(Dˆ q,j)\{0}=spec(Dˆ q,j )\{0}, ∀ j,j . (59)

Since the Koopman operator maps Uk,j to Uk+1,j and each t1 state interval consists of p basins of attraction, the oper- t u u t u u t u u t 1 0,0 0,1 0 1,0 1,1 1 2,0 2,1 2 ator Dˆq can be represented by the operator valued matrix ˆ ˆ ˆ time t diag(Dq,0,Dq,1,...,Dq,p−1). Hence and with Eq. (59), it is clear that the spectrum of Dˆ q equals the spectrum of all of FIG. 4. (a) Separation of the state intervals into the basins of the Dˆ q,j and each eigenvalue of Dˆ q,j is an eigenvalue of Dˆ q = = attraction Uk,j [uk,j−1,uk,j ], where uk,p−1 tk, of the attractive with multiplicity p. At this point, we expand the restrictions q fixed points of the qth iteration R (t) of an exemplary access map xk,j (t) of the state in terms of some basis functions as done p = 3 R(t) with rotation number q 2 . (b) Evolution of a function x0(t) in Sec. IV, with the difference that the domain of the basis inside the U by the successive application of the access operators k,j functions is now given by the basin of attraction Uk,j .By Cˆ . The basic types of symbols of Cˆ , i.e., segments of R(t), are k k,j doing this, we obtain coefficient vectors qk,j,i corresponding illustrated by the alternating dashed and solid lines. to the ith basis function. The state xk(t) inside the kth state interval now can be represented by an infinite-dimensional vector composed of the dp-dimensional coefficient vectors ˆ ˆ x(t), the restriction Ck,j of the Koopman operator C, and the qk,i = cols(qk,0,i ,qk,1,i ,...,qk,p−1,i ), which correspond to the restriction Dˆ q,j of the operator Dˆ q by ith basis function of each basin of attraction Uk,j inside the kth state interval. Due to the block-diagonal structure of Dˆ q the matrix representation of the operator Dˆ with respect to the x (t) = x(t),t∈ [u − ,u ] (55a) q k,j k,j 1 k,j mentioned expansion of the state is given by a block-diagonal Cˆk,j xk,j (t ) = xk,j (R(t)),t∈ [uk,j−1,uk,j ] (55b) matrix, where the p blocks are the matrix representations of ˆ ˆ = q ∈ the operator and Dq,j, respectively. Consequently, the related Dq,j x0(t ) x0(R (t)),t [uq,j−1,uq,j]. (55c) = (n) n mdp-dimensional finite section approximation Dq , with respect to the first m basis functions of each of the p basins of With Fig. 4(b) it is easy to understand that the Koopman attraction, takes the form operator Cˆ maps the basin of attraction Uk,j to the basin of (n) (md) (md) (md) D = diag D ,D ,...,D − , (60) attraction Uk+1,j and that the restrictions Cˆk,j of the Koopman q q,0 q,1 q,p 1 operator are basically given by q different Koopman operators where the D(md) are the md-dimensional finite section shifted in time, which correspond to the symbols given by q,j ˆ q different segments of R(t), repeating with period q in k approximations of Dq,j. With the assumption that the operators and j. Thus, we can simplify our analysis by introducing q Dˆ q,j are compact, together with Eq. (59) and the results basic operators C˜0,C˜1,...,C˜q−1, which are connected to the for the case p = 1, it follows that spec(Dˆ q ) = spec(Dˆ q,j) =

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{1,a,a2,...}∪{0}, where a denotes the unique slope of Rq (t) (a) (b) at the attractive fixed points and all eigenvalues of Dˆ q occur 2 1 )

with multiplicity dp. With the assumption that the determinant t 1 + A (n) = ( of Dq is now given by the product of the largest n mdp ˆ eigenvalues of Dq counting multiplicity, we obtain ) 1 mod t ( (1 − A) t delay m−1 R 0 dp − 1 n(n−dp) 1 (n) = dp k = 2 m(m 1) = 2dp det Dq a a a . (61) 0 1 2 0 1 k=0 time t time t αˆ Finally, with a ≡ e we obtain for the part δC(n) of the average divergence and for the part λC(n) of the asymptotic scaling of FIG. 5. (a) Sawtooth-shaped delay and (b) corresponding reduced the Lyapunov spectrum access map (solid line) and bisectrix (dashed line).

= 1 − ∼ n δC(n) n(n dp)ˆα, λC(n) α,ˆ (62) = 2dp2 dp2 If we set the initial time t0 0, the boundaries tk of our state intervals related to the method of steps and given by the inverse respectively. Hence, the asymptotically linear scaling behavior iteration of the access map are equal to the natural numbers of the Lyapunov spectrum is directly connected to the and the interval length is equal to the delay’s period. So, the semigroup property of the Koopman eigenfunctions. Note that access operators Cˆk are given by for general access maps R(t) possessing multiple attractive q-periodic orbits, Eq. (59) does not apply and there is no Cˆkxk(t ) = xk[(1 − A)(t − k) + k − 1],t∈ [k,k + 1] q unique slope a of the attractive fixed points of R (t). So, (65) there are many ways to combine the al corresponding to ∗ the fixed points t by multiplying powers of the related and the constant delay evolution operators Iˆk are given by l eigenfunctions ξ l(t) using their semigroup property. Thus, t −1 the part λC(n) of the Lyapunov spectrum and its asymptotic Iˆkxk+1(t ) = xk+1[k + (1 − A) ] + xk+1(t ) dt . (66) k slopeα ˆ should depend on all of the al. In general the operator Dˆ q is also not compact. Even the exemplary access map in Since the length of the state intervals is equal to the period Fig. 4 leads to a noncompact Koopman operator due to the of the delay, the solution operator IˆkCˆk is a monodromy q repulsive fixed points of R (t) at the boundaries of Uk,j . operator known from Floquet theory and all information However, the semigroup property of the eigenfunctions and about the dynamics can be obtained by analyzing one time eigenvalues of the Koopman operator is valid for all maps and step. In the following, we derive a matrix representation of the method described above was successfully applied at DDEs the monodromy operator utilizing the method introduced in with sawtooth-shaped delay [16], which leads to a compact Secs. III and IV together with the definition of the matrix operator Dˆ q in the space of analytic functions and can be seen representation in Appendix B. A convenient choice of the as the limiting case where the slope at the repulsive fixed points biorthogonal bases needed for the finite section approximation of Rq (t) is infinite. A further discussion about this problem is are the monomial basis and the corresponding adjoint basis made at the end of this section. consisting of the ith derivatives δ(i)(t) of the delta distribution In the following, we quantitatively analyze the average δ(t), i.e., we expand the state xk(t) into a Taylor series around divergence and the asymptotic scaling behavior of the Lya- t = k. So, we set punov spectrum for delay systems with dissipative delay. i Since the contribution of the constant delay evolution operator φki(t) = (t − k + 1) , (67a) was already investigated in Sec. IV, we concentrate on the − i ( 1) (i) contribution by the access operator or Koopman operator, ψki(t) = δ (t − k + 1), (67b) respectively. So, it is sufficient to consider the simplest delay i! system given by w(t) = 1. (67c) x˙(t) = x(R(t)). (63) The n-dimensional finite section approximation with respect to the monomial basis of the constant delay evolution operators First, we analyze a simple exemplary system with dissi- Iˆk is computed by Eq. (B8) in Appendix B and results in pative delay where the asymptotic scaling behavior of the ⎛ ⎞ 1 1 1 Lyapunov spectrum can be computed analytically and which 1 − − 2 ... − n−1 ⎜ (1 A) (1 A) (1 A) ⎟ is a special case of the systems with sawtooth-shaped delay ⎜10 0 ... 0 ⎟ ⎜ ⎟ analyzed in the Supplemental Material of [16]. We consider ⎜ 1 ⎟ (n) = ⎜0 0 ... 0 ⎟ the DDE (63) with a sawtooth-shaped delay. The definitions I k ⎜ 2 ⎟, (68) ⎜. . . . . ⎟ of the delay τ(t) and the retarded argument R(t) read as ⎝...... ⎠ = + −  ˙ = − ∈ 1 τ(t) 1 A(t t ), R(t) 1 A, if t/N. (64) 0 ... 0 n−1 0 With Fig. 5(b) it is easy to see that for 0 linear map possessing an attractive fixed point at the origin. the solution x(t). The lower rows are the matrix representation

062214-9 DAVID MÜLLER, ANDREAS OTTO, AND GÜNTER RADONS PHYSICAL REVIEW E 95, 062214 (2017) of the integration in the monomial basis. In our simple (a) (b) example, the n-dimensional finite section approximation of the access operators Cˆk computed utilizing Eq. (B9) in Appendix B reduces to a diagonal matrix and is given by (n) = − − 2 − n−1 Ck diag(1,1 A,(1 A) ,...,(1 A) ). (69) Due to the periodicity of the system and the solution operator, (n) (n) I k and Ck are time independent and the two parts of the average divergence δn reduce to the logarithm of the determinant of the above mentioned operators. The part δI (n) of the average divergence related to the constant delay evolution FIG. 6. (a) Triangle delay characterized by two slopes A and B operator is now given by with (b) corresponding reduced access map (solid line) and bisectrix = (n) (dashed line). δI (n) ln det I k =−ln[(n − 1)!] − (n − 1) ln(1 − A). (70) of the parameters A and B and measured the asymptotic slope Utilizing Stirling’s approximation of the factorial leads to α of the Lyapunov spectra. The measured slopes are illustrated in Fig. 7(a). It is easy to see that the asymptotic slope of the ≈− − − + − − − − δI (n) (n 1) ln(n 1) n 1 (n 1) ln(1 A). Lyapunov spectrum depends on both parameters of the delay (71) A and B, which justifies our statement that it depends on

The part δC(n) of the average divergence, which represents the contribution by the access operator or the Koopman operator, (a) respectively, results in = (n) = 1 − − δC(n) ln det Ck 2 n(n 1) ln(1 A), (72) which is equivalent to Eq. (54). Thus, we obtain for the asymptotic scaling of the Lyapunov spectrum

λn ∼−ln n + n ln(1 − A). (73) So, we have shown by a straightforward calculation that the Lyapunov spectrum of our simple delay system with dissipative sawtooth-shaped delay is asymptotically linear in the limit of large Lyapunov indices. Furthermore, it has to be (b) 0.0 noticed that in this special case the asymptotic slope of the Lyapunov spectrum equals the logarithm of the slope of the access map at the attractive fixed point, i.e., it equals the access 0.2 map’s Lyapunov exponent. In the following, we analyze the same simple system (63) with a more general delay. It is a known fact that sufficiently 0.4 smooth iterated maps with the same structure of hyperbolic q-periodic orbits are topological conjugate, where the slope of 0.0 q 0.6 the qth iteration R (t)ofthemapR(t) at the periodic points 0.1 via Koopman operator is invariant under differentiable conjugacies [32]. This leads 0.2 to the conjecture that the asymptotic slope of the Lyapunov 0.3 0.8 spectrum is only influenced by the slopes of Rq (t)atthe 0.4 0.5 periodic points of the access map. Hence, we choose a slope triangular-shaped delay as shown in Fig. 6(a), where the slopes 0.6 1.0 0.60.50.40.30.20.1 0.0 at the attractive and repulsive fixed points can be specified by the parameters A and B, respectively. For the further analysis 1.0 0.8 0.6 0.4 0.2 0.0 the mean delay is set to one. Thus, the reduced access map has asymptotic slope of Lyap. spectrum = 1 an attractive fixed point at t 2 and repulsive fixed points at the interval boundaries, which are identified due to the modulo FIG. 7. (a) Asymptotic slope of the Lyapunov spectrum vs delay operation. parameters A and B and (b) comparison of the asymptotic slope from The Koopman operator corresponding to this access map the finite section approximation of the Koopman operator using an is not compact due to the presence of both attractive and orthonormal basis (empty circles) and the Lyapunov basis (dots) vs repulsive fixed points. Thus, the results above cannot be direct measurement at the Lyapunov spectrum (line), for all A and B applied directly and therefore we perform the following from (a). The inset shows the same analysis, whereby the Lyapunov = + As numerical analysis. First, we have computed the Lyapunov basis was used, for sine-shaped delay τ(t) 1 2π sin(2πt) under spectra via the semidiscretization method for different values variation of As .

062214-10 FROM DYNAMICAL SYSTEMS WITH TIME-VARYING . . . PHYSICAL REVIEW E 95, 062214 (2017) both slopes of the fixed points of the symbol Rq (t)ofthe subspace of the infinite-dimensional state space, the spectral operator Dˆ q . Additionally, we computed the asymptotic slope properties of the Koopman operator are connected to the αˆ of the part λC(n) of the asymptotic Lyapunov spectrum using asymptotic scaling behavior of the Lyapunov spectrum, as we the approach presented in Sec. IV, together with the definition have shown before. But, there are still open questions on the of the matrix representations of the involved operators given separate calculation of the spectrum of the Koopman operator. in Appendix B. For the finite section approximation of the The spectrum of a Koopman operator depends on the corresponding Koopman√ operator√ we choose the Fourier dynamics of the underlying map and can be computed analyt- cosine basis φi (t) ∈{1, 2 cos(πt), 2 cos(2πt),...} on the ically, for example, for maps with a unique (attractive) fixed one hand as a representative example of an orthonormal basis, point, as mentioned before, and for purely expanding circle and on the other hand we choose the left eigenfunctions maps [33]. In our case of invertible access maps there are only of the monodromy operator corresponding to Eq. (63)for two possible types of dynamics, which determine our delay ψki(t) and the corresponding right eigenfunctions or the types, conservative and dissipative delay. If the access map Lyapunov vectors, respectively, for φki(t), which gives us preserves some measure, the corresponding Koopman operator an appropriate biorthogonal system, which we call Lyapunov is unitary with respect to this measure (cf. [17,34]). Roughly basis.InFig.7(b), the asymptotic slopesα ˆ computed by the speaking, on the attractor of the map the Koopman operator finite section approach are plotted versus the related slopes just mixes up the values of the observable that are assigned α resulting from the direct measurement at the Lyapunov to the attractor manifold. For conservative delay, we have spectrum. So, the values resulting from the finite section implicitly shown that the Koopman operator corresponding approximation of the Koopman operator are correct if the to the access map is unitary with respect to some nonsingular points are located at the bisectrix. The approach leads to measure because the limit operator of some special sequence the correct results if the finite section approximation of the of iterations of the related Koopman operator is the identity. In Koopman operator is made with respect to the Lyapunov basis the case of dissipative delay, the attractor only consists of the corresponding to the specific delay and access map. However, periodic points or fixed points of the access map. Indeed, the the finite section approach fails for orthonormal bases similar corresponding Koopman operator is unitary with respect to the to the Fourier basis due to specific structural properties of the Dirac measure but the related space is only a finite-dimensional Lyapunov vectors, which are discussed in Sec. VI. subspace of the infinite-dimensional state space of our DDE, Note that the above mentioned results are not restricted hence less useful. Except for the spectral analysis of the adjoint to access maps that possess only fixed points. The method of the Koopman operators corresponding to circle maps in [35], is constructed to relate the Koopman operator corresponding there is a lack of literature about this special topic. However, to the qth iteration of the access map, where q denotes considering the Koopman operator acting on the space of ∞ the period of the periodic orbits of the access map, to the bounded functions L as the adjoint of the Frobenius-Perron 1 part δC(n) of the average divergence and the part λC(n) of the operator acting on L is not the right framework for the present asymptotic Lyapunov spectrum. Consequently, all dissipative problem and leads to wrong results in the case of dissipative delays that are related to an invertible retarded argument lead to delays. There is no need to assume the eigenfunctions of the a linear asymptotic scaling behavior of the Lyapunov spectrum. Koopman operator to be bounded. By doing this, one restricts Furthermore, the method is not restricted to simple systems like the spectrum of the Koopman operator to the unit circle as Eq. (63). The specific dynamics of the system only influences shown in [35], even in the case of dissipative delays. The part the properties of the constant delay evolution operator Iˆk, δC(n) of the average divergence would be equal to one, which whereby the access operator Cˆk and the corresponding Koop- contradicts the analytical and numerical results we derived man operator only depends on the delay. In the case of a system above. admitting more complicated, especially chaotic, dynamics one If a dynamical system possesses hyperbolic fixed points and has to take into account that the Lyapunov base changes for is sufficiently smooth, the related Koopman operator is well each state interval since the direction of the Lyapunov vectors described around these fixed points by the Koopman operator changes for each state interval. Hence, the computation of the corresponding to the linear approximation of the system since finite section approximation of the Koopman operator gets there is a topological conjugacy between the nonlinear system slightly more complex than for the simple system analyzed and its linear approximation, which is known as Hartman- above. Grobman theorem (cf. [36,37]). The authors of [38] extend the Hartman-Grobman theorem and show that the aforementioned conjugacy exists in the whole basin of attraction if the system C. Discussion is sufficiently smooth and its linear approximation has only As we have shown by analyzing the average divergence and eigenvalues with modulus smaller (or all greater) than one. the asymptotic scaling behavior of the Lyapunov spectrum, the In this case, the eigenfunctions of the Koopman operator dynamics of time-delay systems is not only influenced by the can be constructed using generalized Laplace averages as integral operator, but also influenced by the Koopman operator, shown in [39], and its spectrum consists of powers of the which describes a dynamical system from the observables eigenvalues of the system’s linearization due to the semigroup point of view [18]. In the case of delay systems with time- property of the eigenfunctions and eigenvalues of the Koopman varying delay the dynamical system of interest is given by the operator. These results substantiate our conjecture about the access map and the observable is given by the state of the linear scaling behavior of the part λC(n) of the Lyapunov delay system inside a proper interval of the method of steps. spectrum in the case of dissipative delays, which can be proven Due to the evolution of small volumes on a finite-dimensional analytically for our simple linear DDE with a sawtooth-shaped

062214-11 DAVID MÜLLER, ANDREAS OTTO, AND GÜNTER RADONS PHYSICAL REVIEW E 95, 062214 (2017) delay that is related to an access map with an attractive fixed (a) point. However, this is not the whole truth since invertible 0.2 one-dimensional iterated maps possess one repulsive fixed point or periodic orbit per each attractive fixed point or periodic 0.1 dissipative )) t orbit, and the linearization around an attractive fixed point ( n will always be a bad approximation near the corresponding x repulsive fixed point. Due to the lack of an analytic approach Re( 0.0 in this general case, we have done some numerical analysis and are able to justify that the Koopman operator causes the −0.1 conservative linear scaling behavior of the Lyapunov spectrum. The cause of the deviations of the finite section approach for orthonormal −1.5 −1.0 −0.5 0.0 bases is explained in Sec. VI. time t

(b) VI. LYAPUNOV VECTORS 50 For a more detailed analysis of the differences between the two delay classes, we present some qualitative and quantitative properties of the covariant Lyapunov vectors, which can be 10

computed by the method given in [40]. Note that in the IPR 5 following we always consider covariant Lyapunov vectors even if we drop the term “covariant”. Due to the equivalence of delay systems with conservative delay to systems with 1 constant delay, which where extensively studied in the past 0.5 and recent literature, we concentrate especially on the case of 0 10 20 30 40 50 60 70 systems with dissipative delay. We are interested in structural properties of the Lyapunov vectors that are connected to index n the spectral properties of the Koopman operator following from the dynamics of the underlying access map. Hence, FIG. 8. (a) Lyapunov vectors (real part), i.e., eigenvectors of the the following analysis is intended to be independent of the monodromy operator, and (b) inverse participation ratio (IPR) of dynamics of the delay system and concomitant questions such the periodic part pn(t) of the Floquet decomposition for exemplary as stability and orbit structure. So, let us again consider the conservative (dots) and dissipative (triangles) delays of Eq. (63) with τ(t) = τ0 + (0.9/2/π)sin(2πt). For conservative delay, the mean simplest DDE given by Eq. (63). = = Note that in this simple case of a delay system with periodic delay was set to τ0 1.51 and for dissipative delay τ0 1.54. The IPR is plotted versus the index of the corresponding Lyapunov delay, the Lyapunov vectors are equal to the eigenvectors exponent ordered from the largest to the smallest. (λn+ıωn)t xn(t) = pn(t) e of the monodromy operator, where pn(t) is a periodic function with the same period as the delay’s period T and λn equals the nth Lyapunov exponent ordered from the largest to the smallest. In Fig. 8, some exemplary In the following, we use the aforementioned results for Lyapunov vectors for dissipative and conservative delay and the Lyapunov vectors related to dissipative delay to develop the inverse participation ratio (IPR) of the periodic prefactors a simple analytic model that describes the characteristic pn(t) given by [41] properties and gives us an idea how the structure of the Lyapunov vectors is connected to the scaling behavior of the T | |4 Lyapunov spectrum and the related average divergence. So, 0 dt pn(t) IPR[pn(t)] = (74) let us consider our simple delay system given by Eq. (63). We T | |2 2 0 dt pn(t) assume the reduced access map corresponding to the retarded argument R(t) to possesses a repulsive fixed point at t = 0 related to the first Lyapunov vectors are shown. In Fig. 8, one − 1 1 and attractive fixed points at 2 and 2 , i.e., the fixed points can see that the exemplary Lyapunov vector corresponding t∗ fulfill R(t∗) = t∗ − 1. Consequently, the initial function of to the case of dissipative delay shows sharper peaks than − 1 1 our delay system is defined in the interval [ 2 , 2 ] whereto the the vector corresponding to conservative delay. A qualitative time intervals are mapped due to the Hale-Krasovkii notation, difference can be noticed by comparing the IPR of the which was already mentioned in Sec. III. Since this system periodic part pn(t) for the two delay types. Obviously, is periodic, the solution operator mapping one state interval the IPR for dissipative delay increases much faster than the to the next state interval, which is given by the method steps, IPR for conservative delay. For dissipative delays, the IPR equals the monodromy operator. Hence, the Lyapunov vectors seams to increase exponentially, which indicates that the peak are equal to the eigenfunctions x(t) of the monodromy operator width decreases exponentially with the index of the Lyapunov and are solutions of the eigenvalue equation exponent and the corresponding Lyapunov vector. In this case, the strongly localized peaks of the Lyapunov vectors are t located at the members of the repulsive periodic orbits or at es x(t) = x(1/2) + dt x(R˜(t)),t∈ [−1/2,1/2] (75) the repulsive fixed points. − 1 2

062214-12 FROM DYNAMICAL SYSTEMS WITH TIME-VARYING . . . PHYSICAL REVIEW E 95, 062214 (2017) whereby the Lyapunov exponents are the real parts of the (a) Floquet exponents s and R˜(t) = R(t) + 1. Since the Lyapunov vectors are localized at the members of the repulsive periodic orbits or fixed points, in our simple case they are localized at t = 0. Hence, the integration in Eq. (75) is dominated by f X the values of the integrand in a small neighborhood of t = 0. Roughly speaking, we assume that the system “sees” a linear retarded argument R˜(t) ≈ bt for t ≈ 0. Furthermore, we assume the value of the Lyapunov vectors at the end of the state intervals to be negligible compared to the strongly localized peak around t = 0 in the limit of small Lyapunov exponents, i.e., x(1/2) ≈ 0, which implies x(−1/2) ≈ 0 due f to Floquet theory. Hence, we approximate Eq. (75)by (b) t es x(t) = dt x(bt),t∈ [−1/2,1/2] (76) − 1 2 n which denotes an eigenvalue equation of an operator acting on periodic functions. Since the corresponding eigenfunctions are expected to be strongly localized, it is reasonable to approximate their periodic extension by a sum of localized functions defined on the whole real line and shifted by integer multiples of the period 1. The Fourier transform of this periodic approximation is given by the product of a train of delta functions and an envelope, which is analog to the “form factor” n of diffraction patterns of crystals in solid state physics related to the structure of the crystals basis. Hence, the envelope FIG. 9. (a) Comparison of the real part (dots) and imaginary part (triangles) of the Fourier amplitudes of a Lyapunov vector (index of the Fourier transform of the approximate eigenfunctions n = 29) with the corresponding envelope of the Fourier transform connects the localized structure of the Lyapunov vectors to the ∝ s X(f ) X1(f ) (lines), with X1(f )givenbyEq.(81). (b) Lyapunov eigenvalues e and we simplify Eq. (76) by its expansion to spectrum from semidiscretization (empty circles) compared with the whole real line “reducing the crystal to one atom,” i.e., modeled spectrum (dots) computed by fitting Eq. (81) to the Fourier t amplitudes of the Lyapunov vectors. The system (63) with R(t) = s = − 0.1 − e x(t) dt x(bt ). (77) t 2π sin(2πt) 1 was used. −∞ This expression can be interpreted as solution operator of the − rearrangements pantograph DDE x˙(t) = e s x(bt) corresponding to solutions that exist on the whole real line, whereby the equation is named 1 γX(f ) = X(f/b), where γ := ı2πbes . (80) by the mechanical part of a current collection system of an f electric locomotive [42]. It is easy to see that this equation possesses solutions for arbitrary s. So, the assumptions we With Eq. (80) we have derived a linear first-order q-difference made lead to the loss of the information that the Lyapunov equation (cf. [43]), which is solved by any linear combination spectrum is discrete, which is connected to the discreteness of the solutions X1(f ) and X2(f ) given by of the Floquet exponents for periodic delay equations. Not − 1 (ln f +ln γ + ln b )2 surprising because we built our model to approximate the X1/2(f ) = (±f ) e 2lnb 2 . (81) Lyapunov vectors of a delay system with any access map of any period that has a repulsive fixed point. We omit the derivation of the solution above because it is The right-hand side of Eq. (77) can be expressed by a a simple straightforward calculation by transforming the q- convolution using the Heaviside step function (t). Hence, difference equation into a linear ordinary difference equation we write and does not contribute to a better understanding. In Fig. 9(a), our approximation of the envelope of the s e x(t) = (t) ∗ x(bt). (78) Fourier transform (81) corresponding to a given small Lya- punov exponent of an exemplary system is compared to the Now, its Fourier transform is trivially given by Fourier amplitudes of the corresponding Lyapunov vector, 1 1 whereby b was set to the slope at the repulsive fixed point es X(f ) = δ(f ) + X(f/b), (79) 2b ıπf of the access map and s was set to the numerically computed Floquet exponent. Additionally, we computed the Lyapunov where X(f ) denotes the Fourier transform of x(t) with respect spectrum of the system by fitting a suitable linear combination to the frequency f . For simplicity, we assume the integral of Eq. (81) to the eigenfunctions of the monodromy operator over the solutions of Eq. (77) to be equal to zero, i.e., we by varying γ . The result is shown in Fig. 9(b). As desired, neglect the delta distribution in Eq. (79) and obtain after some our simple model connects the structure of the Lyapunov

062214-13 DAVID MÜLLER, ANDREAS OTTO, AND GÜNTER RADONS PHYSICAL REVIEW E 95, 062214 (2017) vectors to the corresponding Lyapunov exponents and holds have the potential to extend the view on the dynamics of asymptotically for small Lyapunov exponents. Consequently, systems with time-varying delay and can be a nice foundation our model indicates that the localization is caused by the for further research. repulsive fixed points or periodic orbits of the reduced access Using our above mentioned framework, we have analyzed map and the spectral properties of the corresponding Koopman the influence of the two fundamental classes of delays on the operator. Lyapunov spectrum and the evolution of small volumes on One interesting property of the Lyapunov vectors in the finite-dimensional subspaces on the infinite-dimensional state case of dissipative delay can be directly derived from Eq. (80). space of the delay system. The classification is determined by A solution X(f ) can be rescaled on the frequency domain the question as to whether a system with time-varying delay to obtain other solutions. This means, if X(f ) is a solution is equivalent to a system with constant delay or not, which with eigenvalue γ,X(βf ) is a solution with eigenvalue βγ depends directly on the existence of a topological conjugacy for arbitrary β. Consequently, the envelope of the Fourier between the access map and a pure rotation. We have shown transform of the Lyapunov vectors exponentially broadens that the asymptotic scaling behavior of the Lyapunov spectrum with increasing index of the Lyapunov exponent and it follows can be separated into a logarithmic part related to the constant that the Lyapunov vectors localize exponentially on the time delay evolution operator known from systems with constant domain as already shown in Fig. 8(b). This also has conse- delay and a part related to the Koopman operator depending on quences for the finite section method introduced in Sec. IV the dynamics of the underlying access map. If the access map is and other algorithms for the computation of the Lyapunov topological conjugate to a pure rotation, the delay system with spectrum. If the Lyapunov spectrum λn asymptotically scales time-varying delay is equivalent to a system with constant de- linear with the index n, the number of Fourier modes or lay, the part of the Lyapunov spectrum related to the Koopman similar basis vectors that are needed to obtain a good approxi- operator vanishes, and we call the related delays conservative mation of the nth Lyapunov vector increases exponentially. delay because the corresponding access map is a conservative Consequently, the number of well-approximated Lyapunov system. Contrarily, if the conjugacy does not exist, the delay exponents and Lyapunov vectors increases logarithmically system is not equivalent to a system with constant delay and with the dimension of the finite section approximation, which the asymptotic scaling behavior of the Lyapunov spectrum is is worse for numerical analysis and also explains the slow dominated by the part related to the Koopman operator, which convergence of the Lyapunov spectra for the systems with is linear with respect to the index of the Lyapunov exponent. dissipative delay in Fig. 2(b) in Sec. II. In other words, the Since the related delays lead to an additional dissipation by complexity of numerical algorithms increases exponentially. the Koopman operator, we call them dissipative delays. We Moreover, the question arises as to whether the finite section obtain a similar dependence on the delay type for the evolution approximations with respect to common orthonormal bases of small volumes on finite-dimensional subspaces of the state of the operators have anything to do with the exact operator space and their mean relaxation rate (average divergence). Due because the ratio between the number of well-approximated to the volume-preserving retarded access by the unitary Koop- exponents and computed exponents or vectors tends to zero. man operator, conservative delays lead to the same scaling Since the determinant is the product over all eigenvalues of a behavior of the average divergence with respect to the dimen- matrix, the determinant approach to compute the asymptotic sion of the subspace as known from systems with constant scaling behavior of the Lyapunov spectrum will fail for typical delay, and dissipative delays lead to an additional contribution orthonormal bases in the case of systems with dissipative delay. to the average divergence that describes the volume evolution That is why we considered the Lyapunov basis in Sec. V for under the action of the nonunitary Koopman operator. the numerical analysis. Furthermore, we have analyzed the influence of the delay type on the structure of the Lyapunov vectors and briefly investigated the consequences on numerical algorithms, par- VII. CONCLUSION ticularly those including finite section approximations of the In this paper, we have investigated the influence of the solution operator. The Lyapunov vectors of delay systems with access map defined by the delay on the dynamics of delay dissipative delays localize exponentially, which leads to an systems with time-varying delay. In particular, we have exponential broadening of the envelope of the coefficients of separated the dynamics due to the variable delay from the their expansion in terms of common orthonormal bases. We dynamics of the whole delay system by an operator theoretic have demonstrated that for dissipative delay, the number of approach, where the solution operator is decomposed into well-approximated Lyapunov exponents or Floquet exponents two different operators. Hence, the time evolution of delay raises only logarithmically with the dimension of the finite systems with time-varying delay is characterized by the action section approximation of the solution operator. So, maybe it of the Koopman operator corresponding to the access map is worth to check whether this and similar algorithms can and by the action of the constant delay evolution operator, be improved by special bases, such as wavelet bases, which which is similar to the solution operator of delay systems with take into account the exponential localization of the Lyapunov constant delay. The decomposition of the dynamics into the vectors or eigenvectors of the monodromy operator in the case deformation of the state by the Koopman operator and the of periodic systems. Indeed, the inverse Fourier transform of integration by the constant delay solution operator connects the function appearing in Eq. (81) can serve itself as a mother the theory of DDEs with the theory of circle maps and the wavelet [44], and is nowadays also known as Log-Gabor Koopman operator framework of the analysis of dynamical wavelet. The consequences of this fact will be explored in systems called “Koopmanism”. This connection itself should a future publication.

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∗ (a) 0 in the dual spaces F , as defined in Sec. IV. The mentioned sets define biorthogonal systems of our spaces F endowed −2 with a weighted inner product such that n −4 tk −6 dt w (t) ψ (t) φ (t) = δ , (B1) m=2000 k ki kj ij −8 tk−1 m exponent −10 m=1000 where the tk are connected to the initial time t0 by Eq. (17)in m=500 Sec. III. For a simpler computation of the matrix representation −12 d of operators between the spaces F([tk−1,tk],R ) with different { } 0 100 200 300 400 k 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- index n tion. Hence, we set

(b) 0 φ(k+1)i (t) = φki[hk(t)], (B2) −2 ψ(k+1)i (t) = ψki[hk(t)], (B3) n −4 tk − tk−1 wk+1(t) = wk[hk(t)], (B4) −6 tk+1 − tk m=2000 −8 whereby exponent m m=1000 − −10 = tk tk−1 − + m=500 hk(t) (t tk) tk−1 (B5) tk+1 − tk 0 100 200 300 400 maps [tk,tk+1]to[tk−1,tk] by shifting and linear rescaling. This index n leads to the advantage that “switching” between the spaces by the linear transformation hk(t) is represented by the identity FIG. 10. Comparison of the convergence properties of the Lya- matrix, i.e., punov spectra shown in Fig. 2(a) in Sec. II between systems with t + conservative delay (filled) and dissipative delay (empty), computed k 1 dt w + (t) ψ + (t) φ [h (t)] = δ . (B6) by the semidiscretization method [2,26] for (a) the turning model, k 1 (k 1)i kj k ij tk given by Eq. (10c) (triangles) and (b) the Hutchinson equation, given by Eq. (10a) (squares). The parameter m denotes the number of The state xk(t) can be represented as an infinite-dimensional equidistant points approximating the state of the delay system. As a vector qk composed of the d-dimensional coefficient vectors guide to the eye, the asymptotic scaling behavior for dissipative delay, qk,i, which are given by which is derived in Sec. V, is represented by the dashed gray line. tk qk,i = dt wk(t) ψki(t) xk(t). (B7) tk−1 APPENDIX A: CONVERGENCE OF LYAPUNOV SPECTRA In the same way we represent the operators Iˆ and Cˆ by In this section, the convergence properties of the Lyapunov k k infinite-dimensional matrices composed of (d × d) matrices. spectra of the turning model given by Eq. (10c) and the Thus, the submatrix {I } of the matrix representation of the Hutchinson equation given by Eq. (10a) are analyzed as k ij ˆ done for the Mackey-Glass equation in Sec. II.InFig.10, constant delay evolution operator Ik is given by the Lyapunov spectra of these systems for the two delay tk+1 classes, conservative and dissipative delay, are shown under {I k}ij = dt wk+1(t) ψ(k+1)i (t)[Iˆk φ(k+1)j (t )], (B8) variation of the number m of equidistant points approximating tk the state, whereby the same system parameters and delay whereby each component of the matrix valued operator Iˆ parameters are used as in Sec. II. As in the case of the Mackey- k is applied to the scalar basis function φ(k+1)j (t) and each Glass equation, for dissipative delay the number of well- component of the result is projected subsequently to the initial approximated Lyapunov exponents increases much slower basis by the weighted scalar product with ψ(k+1)i (t). The than for conservative delay. This is clearly a consequence of { } submatrix Ck ij of the matrix representation of the access the localization properties of the Lyapunov vectors as derived ˆ in Sec. VI. operator Ck is computed by tk+1 {Ck}ij = dt wk+1(t) ψ(k+1)i (t)[Cˆk φkj (t )], (B9) APPENDIX B: MATRIX REPRESENTATIONS tk OF THE OPERATORS where the computation is done as above with the difference Let us consider the bases {φki(t)}i∈N of the spaces that Cˆk changes the underlying space of the state. Hence, we d F([tk−1,tk],R ) and the corresponding dual sets {ψki(t)}i∈N utilize the orthogonality relation (B6).

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