Spatio-temporal representation of long-delayed systems: an alternative approach
SSpatio-temporal representation of long-delayed systems: an alternative approach
Francesco Marino and Giovanni Giacomelli CNR - Istituto Nazionale di Ottica, largo E. Fermi 6, I-50125 Firenze, Italy CNR - Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy (Dated: July 7, 2020)Dynamical systems with long delay feedback can exhibit complicated temporal phenomena, whichonce re-organized in a two-dimensional space are reminiscent of spatio-temporal behavior. In thisframework, normal forms description have been developed to reproduce the dynamics and the op-portunity to treat the corresponding variables as true space and time has been since established.However, recently an alternative approach has been proposed in Ref. [20] with a different interpre-tation of the variables involved, which takes better into account their physical character and allowsfor an easier determination of the normal forms. In this paper, we extend such idea and apply it to anumber of paradigmatic examples, paving the way to a re-thinking of the concept of spatio-temporalrepresentation of long-delayed systems.
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I. INTRODUCTION
A long-delayed dynamical system is characterized bya feedback action, which acts by re-injecting far-in-the-past information from the system itself. Notably, thetime interval between the ”present” and the ”past” isassumed to be much longer than any other characteris-tic time-scale of the system without feedback ( long-delaylimit ). Such a condition, apparently quite specific, ap-pears naturally in disparate phenomena and the topichas attracted considerable attention in the last years (fora review, see e.g. [1]). Dynamical systems with longdelayed feedback can display a rich variety of complexphenomena [2]. Such richness derives from the high di-mensional phase space, and it is witnessed by scaling re-lations for extensive quantities analogous to those foundin one-dimensional (1D) setups [3].The standard approach to spatio-temporal modeling oflong-delayed systems stems from the proposal introducedin Ref. [4]. There, a two-dimensional (2D) coordinatesystem was used where a continuous variable σ ranged ina delay interval played the role of a pseudo-space variable.The correspondent pseudo-time variable was the discreteindex θ , numbering the sequence of consecutive, disjointdelay intervals in the time series. This procedure, called spatio-temporal representation (STR) amounts to expressthe time variable t as t = σ + θT , (1)where T is the delay time. While this mapping is alwaysfeasible independently of the delay value, it is under spe-cific circumstances that it shows its usefulness. One ofthe most important is that the system is actually operat-ing in the long-delay limit: it is indeed in this case thatthe system evolves on two well-separated timescales, σ and θ , which thus effectively act as mutually indepen-dent variables [1].The above representation was quite successful and al-lowed to disclose many relevant features, common to thedelayed and spatially-extended systems [5]. From the 1990s up to now, a number of experiments have been re-alized, in particular in the field of optics, demonstratingdifferent kinds of equivalent spatiotemporal phenomenahidden in the temporal dynamics. These include e.g. de-fect propagation [6], domain coarsening and nucleation[7–9], front pinning and localized structures[10, 12–14],chimera states [15] and critical phase transitions [16]. Re-cently, even generalizations involving two, hierarchicallylong delays have been considered leading to the evidenceof spiral defects and defects turbulence [17], 2D chimerasand dissipative solitons [18] and excitable waves [19]. Theemergence of such a wealth of pattern structures con-firms the role played by the multiple timescales in thelong-delayed dynamics, supporting their natural identifi-cation as the main independent variables of the system.However, recently a critical analysis of this approachhas been reported [20], introducing an alternative repre-sentation for the data generated from long-delayed sys-tems and suggesting a different spatiotemporal interpre-tation. In this framework the bulk dynamics is describedin terms of a new rule, the so-called dynamical represen-tation (DR), employing the opposite definition of pseudo-space and -time variables with respect to the STR. Theanalysis in [20] has suggested that, while the two rep-resentations are equally effective in evidencing patternstructures, a physical description in terms of a spatiotem-poral model is more properly obtained in the DR.In this work, we extend these results and compare thetwo representations in several respects, both on the ba-sis of general arguments and with the help of a few ofparadigmatic examples.The plan of the paper is the following. In Sec.II werecall the main features of the representations remarkingthe conditions in which they are valid. In Sec. III, DRand STR are compared with respect to causality, first interms of validity of Kramers-Kr¨onig relations, and thenevaluating the comoving Lyapunov exponents. In thesubsequent sections we analyze the two representationsin the framework of two specific examples. In Sec.IV,we focus on the delayed Adler equation, describing the a r X i v : . [ n li n . PS ] J u l phase dynamics of an optically-injected laser system withfeeedback [12, 21, 22]. In Sec. V, we will treat a modelof passively mode-locked external-cavity surface-emittinglaser recently introduced in [23, 24]. In Sec. VI, we willdiscuss show how to move between the representations onthe basis of parity arguments. We will draw our conclu-sions and present some perspectives in the final section. II. DEFINITION OF THE REPRESENTATIONS
In this Section, we start by recalling the concepts atthe basis of STR and DR. Without any loss of generalitywe consider the scalar system u t = F ( u, u d ) , (2)where u d ( t ) = u ( t − T ) is the delayed variable and T is the delay time. The model (2) has to be accompaniedwith an initial condition specified on an interval of length T , e.g. u ( t ) = u ( t ) , t ∈ [ − T, . (3)As mentioned before, for a meaningful spatio-temporalrepresentation of (2-3) the delay time T should be longerthan any other timescale of (2) without delay. Such con-dition is necessary but not sufficient: we should also con-sider an observation time t T OT (cid:29) T for an appropriatedefinition of the thermodynamic limit T → ∞ (4) S = (cid:2) t T OT T (cid:3) → ∞ , (5)where [ . ] stands for the integer part.For the sake of clarity we introduce different sets ofnames for the variables involved in the two representa-tions. We write (1) in the form t = x + yT , (6)where we refer to x as the fast time and to y as the slow time and we define the field Φ( x, y ) = u ( t ).In the limit T → ∞ , the time derivative can be ex-pressed as ddt = ∂ x + 1 T ∂ y → ∂ x . (7)This condition holds in the absence of the so-calledanomalous Lyapunov exponent [25] or, equivalently inthe weak-chaos regime [26], and amounts to state thatthe variations of Φ( x, y ) along the y direction are neg-ligible asymptotically. In simple terms, Φ( x, y ) shouldexhibits small variations between two successive delayunits. Accordingly, the integer variable y will be embed-ded into a real domain. Such an assert implies that thereexist a correlation length L y of the pattern along the slowtime such that L y >
1. The field Φ( x, y ) does not vary significantly on a scale ∆ y = 1 along y , which results intoa smooth pattern since several discrete points fall withina correlation length L y .Notably, the above requirements stay at the basis ofboth representations. They indicate whether the twotimescales behave as mutually independent variables andthus can be used to parametrize a 2D smooth pattern.On the basis of these considerations, it is clear that theabove re-organization of data in itself does not provideany constraint on the physical role of the variables in gen-erating the dynamics. These are actually introduced inthe framework of the two representations, where the orig-inal model (2) is re-written in terms of the new variables x and y , in order to build a suitable two-dimensional rule.Setting x = σ and y = θ , and defining U ( σ, θ ) = u ( t ),the model (2) reads ∂ σ U = F ( U, U ( σ, θ − , σ ∈ [0 , T ] , (8)together with the boundary conditions U ( σ, −
1) = u ( σ ) , σ ∈ [0 , T ] U ( σ + T, θ ) = U ( σ, θ + 1) . (9)Eqs. (8)-(9) correspond to the standard spatio-temporal description of the delay model in the STR: thevariable σ is interpreted as the pseudo-space and θ as thepseudo-time. In particular, the smoothness of the pat-tern along θ allows to approximate U ( σ, θ + 1) ≈ U ( σ, θ ),leading to U ( σ + T, θ ) ≈ U ( σ, θ ), similarly to the periodicboundary conditions for a 1D spatially extended system.In order to provide an effective mapping of the de-layed dynamics, the next step is to employ Eq. (8) toderive an explicit rule for the pseudo-time evolution (i.e.along the θ direction). This can be achieved by means ofdifferent methods. As we have seen, the pseudo-spatialand pseudo-temporal variables are related the multipletimescales of the system, the fast time and slow time. Amultiscale approach separating such scales into differentperturbation orders is often very convenient, and allowsto derive a partial-differential equation (PDE) able to re-produce the delayed dynamics in the ( σ, θ ) domain, obvi-ously within some degree of approximation [5, 17, 27–30].The DR is an alternative approach to the STR, pro-posed in [20]. It considers the opposite dynamical rolefor the two variables. In this scheme, we name x = τ asthe pseudo-time and y = ξ as the pseudo-space, defininga new field variable Z ( ξ, τ ) = u ( t ). The evolution rulederived from Eq.(2) is now written as ∂ τ Z = F ( Z, Z NL ) , (10)where the delayed term translates into the non-local asymmetric spatial coupling Z NL ( ξ, τ ) = Z ( ξ − , τ )and the temporal evolution is along the former pseudo-space. Eq.(10) should also be complemented with suit-able boundary conditions. Here we consider spatially-periodic boundaries conditions Z ( ξ,
0) = z ( ξ ) , ξ ∈ [0 , S ] Z ( ξ + S, τ ) = Z ( ξ, τ ) , (11)in the thermodynamic limit. We remark that from astrict mathematical point of view the correct solution ofthe original delay problem would be obtained only forone choice of the initial and boundary conditions (gen-erally different from the periodic ones here used). Weexpect however that in the thermodynamic limit even anarbitary choice would produce patterns well approximat-ing the delayed dynamics. In particular, we will see thatthis is indeed the case whenever conditions (11) hold.The topology of the variable domains associated tothe two representations is illustrated in Fig. 1, evidenc-ing different global manifolds. The dashed circular linesmark the initial conditions, the cylinder axis defines thedirection of evolution (pseudo-time axis) and the cross-sectional circumference corresponds to the size of the spa-tial cell. The patterns produced in either one of the tworepresentions can be readily identified looking at the loca-tion of the initial conditions and/or spatial boundaries.On the other hand, in the bulk region we will observeessentially the same dynamics since the rule generatingthe pattern far from space-time boundaries remains thesame.Interestingly, the representation (10) also allows for astraightforward expansion of the non-local coupling interms of spatial derivatives, leading eventually to a nor-mal form description through standard PDEs Z ( ξ − , τ ) ≈ Z ( ξ, τ ) − Z ξ ( ξ, τ ) + 12 Z ξξ ( ξ, τ ) − .. , (12)where Z ξ = ∂ ξ Z, Z ξξ = ∂ ξξ Z, .. , obtaining the PDE Z τ = F ( Z, Z ξ , Z ξξ , .. ) . (13)As discussed before, the validity of the formal expan-sion (12) relies on the assumption that the pattern ex-hibits small variations along ξ , i.e. that the correlationalong ξ decays over a length L ξ (cid:29)
1. In this case, a PDEmodel (PDEM) where the time derivative is explicitlywritten in terms of the spatial derivatives can be obtaineddirectly, expanding the non-local term up to a given orderand approximating the delayed dynamics with arbitraryprecision. This represents an advantage with respect tothe STR, in which the derivation of a PDE model oftenrequires long calculations and the vicinity to a bifurca-tion. In the specific case of a linear delay term, eachorder of the expansion can be associated to a specificphysical effect: the zero-order is a renormalization of thelocal force, the first provides the advection (that can beremoved with a suitable choice of a comoving referenceframe), the second is diffusion, the third corresponds todispersion, etc.We conclude this section remarking that any differentchoice of the reference frame in the plane ( x, y ) can bechosen to accordingly rewrite the bulk rule (2) and repro-duce the pattern Φ. Such models will be mathematicallyequivalent, all sharing a non-local coupling arising fromthe delayed feedback and a fairly good independence onthe boundary conditions in their correspondent thermo-dynamic limit. However, they might give rise to physical s q x t FIG. 1: Pictorial view of the long delay pattern embeddedinto the STR (left) and DR (right) manifolds. The dashedcircular lines mark the initial conditions. The curved andstraight arrows indicate respectively the periodic boundaryconditions and the direction of evolution. inconsistencies. In the next section we will compare theSTR and DR with respect to causality.
III. REPRESENTATIONS AND CAUSALITY
The core point at the basis of a dynamically correct,spatio-temporal representation is whether the resultingmathematical model not only is capable to produce theembedding pattern Φ( x, y ), but its variables can play therole of well-behaving space and time coordinates. Whileno special constraints can be assumed on the spatial vari-able, the temporal one must satisfy causality, i.e. theevolution along it must depend on its previous valuesonly (the past ). Our aim here is to investigate the causalstructure of the two representations and thus of their as-sociated spatiotemporal PDE models.
A. Susceptibility
For a general linear system the notion of causality isequivalent to satisfy Kramers-Kr¨onig relations relatingthe real and imaginary parts of the complex susceptibilityfunction [31]. We thus consider the linear long-delayedequation ˙ X = AX + BX d , (14)where X is a vectorial variable, A and B the matrices ofcoefficients and X d = X ( t − T ) is the delayed vector.We begin our analysis writing the above equation inone of the two representations, say the STR, and evalu-ating the system response to an external spatiotemporalperturbation Y , X σ = AX + BX ( σ, θ −
1) +
Y . (15)We then look for solutions in the Laplace domain forboth variables after the transient related to the initialconditions.Denoting with ( s σ , s θ ) the Laplace-conjugate variablesof ( σ, θ ) and with ˜ X and ˜ Y the transformed variables, wefind ( s σ I − A − e − s θ I B ) ˜ X = ˜ Y , (16)where I is the identity matrix. We thus obtain the re-sponse of the system to the stimulus Y in Laplace space˜ X = χ ( s σ , s θ ) ˜ Y , (17)where we have defined the susceptibility matrix as χ ( s σ , ¯ θ ) = ( s σ I − A − e − s θ I B ) − . (18)Since the function (18) represents the system responseto a unit impulse, it must satisfy Kramers-Kr¨onig rela-tions to obey causality (no response before the impulseis applied) [31].The Kramers-Kr¨onig relations are valid for any func-tion which is analytic in the upper-half complex planeand vanishes as 1 / | s | or faster as | s | → ∞ , where s isthe Laplace-conjugate variable relative to the directionunder consideration. One can readily verify that this isactually the case when considering the variable σ : indeedfor | s σ | → ∞ and s θ = const , i.e. along the σ direction,the susceptibility displays the asymptotic behavior χ (cid:39) s − σ I. (19)On the other hand, along the θ direction, i.e. for | s θ | →∞ and s σ = const , we find χ (cid:39) ( s σ I − A ) − . (20)A finite susceptibility (for each spatial frequency s σ ) atinfinity along the s θ axis has a precise physical meaning:the system equally responds at all temporal frequenciesup to infinity. In the time domain, this would imply anunphysical instantaneous coupling (i.e. at the same σ point) between a delay and the successive.We can thus conclude that, in the susceptibility of thefull problem (i.e. without any approximation) there ex-ists a forbidden direction along the θ variable where thecausality falls. As a consequence, one should consider theopportunity to use such a variable as equivalent to thephysical time. In the next subsection, we support thisinterpretation by the analysis of the comoving Lyapunovexponent. B. Comoving Lyapunov Exponent
The (maximum) comoving Lyapunov exponent (CLE)is an useful tool to characterize how a localized spatio-temporal disturbance propagates in different directions[32]. In particular, it allows to determine how informa-tion is transmitted along lines in the domain of a pattern,shading light on their possible physical interpretation ascausal routes. In the following, we calculate it explicitly for the lineardelay model ˙ z ( t ) = − z ( t ) + ηz ( t − T ) , (21)using the method of chronotopic Lyapunov analysis [33].To this aim, we rewrite Eq. (21) in the STR ∂ σ Z ( σ, θ ) = − Z ( σ, θ ) + ηZ ( σ, θ − , (22)and in the DR as ∂ τ Y ( ξ, τ ) = − Y ( ξ, τ ) + ηY ( ξ − , τ ) . (23)and we look for solutions of the type Y ( ξ, τ ) = Y exp(¯ µξ + ¯ λτ ) , (24)where ¯ λ = λ + iω, ¯ µ = µ + iκ (a similar ansatz can beused for the STR). Substituting the above solution into(23) and separating real and imaginary part, we obtain λ = − ηe − µ cos( κ ) (25) ω = − ηe − µ sin( κ ) . (26)The maximum LE for both the STR and DR is foundat ω = 0 or, equivalently, κ = 0. In the STR, the propa-gation velocity of the disturbance is V ST R = − dµdλ = 11 + λ , (27)and the corresponding CLE isΛ ST R ( V ST R ) = µ + λV ST R = 1 − V ST R + log( ηV ST R ) , (28)as reported in [5].In the case of the DR, the velocity is given by V DR = − dλdµ = ηe − µ , (29)and the CLE byΛ DR ( V DR ) = λ + µV DR = − V DR − V DR log( V DR η ) . (30)The velocities in the two representations are thus re-lated by V DR = ηe − µ = 1 + λ = 1 V ST R , (31)which can be interpreted geometrically (see Fig.2) interms of the relation between complementary propaga-tion angles V DR = tan β = 1tan α = 1 V ST R . (32) q,x b s,t a FIG. 2: Geometric representation of the propagation anglesin the STR and DR reference frame.
The CLE are related as well byΛ
ST R ( V ST R ) = 1 V DR Λ DR ( V DR ) , (33)or, equivalentlyΛ DR ( V DR ) = 1 V ST R Λ ST R ( V ST R ) . (34)The above formulas relate the rates for a perturbationmeasured in the tangent space of the two representationsfor an arbitrary velocity, i.e. for a certain propagationdirection of the perturbation.For instance, a spatio-temporal perturbation withcharacteristic width ∆ ξ in the DR space-time ( ξ , τ ) prop-agates in an interval ∆ τ = ∆ ξ/V DR . Writing V ST R = ∆ σ/ ∆ θ = ∆ τ / ∆ ξ = 1 /V DR , (35)we get Λ ST R ∆ θ = Λ DR ∆ τ . (36)Eq. 36 expresses the absolute spreading (or shrinking)of a perturbation as measured in the two representationsalong the vertical and horizontal directions, which resultsas an invariant.We plot the CLE for the DR and STR in Fig.3, for dif-ferent values of the feedback gain η as a function of theirvelocities. The macroscopic observables represented bythe correlation directions, defined by zeros of the CLE[11], are the same. Indeed, if Λ ST R ( ¯ V ST R ) = 0 for V ST R = ¯ V ST R , then Λ DR ( V DR = 1 / ¯ V ST R ) = 0 and vicev-ersa. In both representations the negative velocities arenot allowed, indicating the presence of a causality bound-ary. As already discussed in [5], the STR curves displaysa logarithmic divergence in zero, while the non-local cou-pling of the DR is mapped to infinity (thus removing thelogarithmic divergence). In the STR, this in fact cor-responds to instantaneous coupling between consecutivedelays related the non-analytic (i.e. non-causal) suscep-tibility in the θ domain (20).As discussed in Sec. II a PDEM can be obtained ex-panding the non-local term up to a given order to ap-proximate the delayed dynamics. We can thus evaluate DR -3-2-101 Λ DR η = 0.8η = 1η = 1.2Λ AD STR -3-2-101 Λ STR
FIG. 3: CLE for the DR and STR models in the stable ( η =0 . η = 1 . η = 1it is also shown the exponent for the AD model as a functionof V DR (see text). the CLE for the various orders of a spatio-temporal ap-proximation.We start defining ψ ( µ ) = e − µ and rewrite λ ψ = − ηψ ( µ ) , (37)leading to the velocity V ψ = − dλ ψ dµ = − ηψ (cid:48) ( µ ) . (38)The CLE is thenΛ ψ ( V ψ ) = λ ψ + µV ψ = − ηψ ( µ ) − µV ψ . (39)One can therefore use the (37) and (38) to eliminatethe auxiliary variables { λ ψ , µ } to obtain eventually Λ ψ =Λ ψ ( V ψ ) for the chosen function ψ . In particular, we cantreat in this way different orders of expansion of the non-local term.For a second-order expansion of the DR model, whichcorresponds to an advection-diffusion (AD) term, ψ ( µ ) = e − µ ≈ − µ + µ . As a consequence, the velocity is givenby V AD = − dλdµ = η (1 − µ ) , (40)and the CLE readsΛ AD ( V AD ) = λ + µV AD (41)= − V AD + η V AD η − − V AD ( V AD η − . In Fig. 3 (left panel), we compare the above exponentat the bifurcation point η = 1 with the CLE in the DR.The horizontal variable is evaluated by V AD = η (1 +log | V DR η | ). As seen from the plot, the advection-diffusionmodel is already a good approximation of the systemaround the maximum in terms of the CLE.We finally remark that only the non-local or delayedterm may induce problems with causality. Every finite-order PDE model is free from that, and it is increasinglycorrect at higher orders around the comoving direction(the location of maximum of the CLE). IV. THE DELAYED ADLER EQUATION
We now investigate and discuss the two representationsin the framework of the so-called delayed Adler’s equa-tion. The model describes the evolution of the phase ofthe optical field in optically-injected laser systems withtime-delay feedback and accounts for the formation andinteraction of topological localized states [12] (homoclinic2 π -kink solutions) very similar to those found in the Sine-Gordon equation. The model reads˙ φ = ∆ − sin φ + χ sin( φ d − φ − ψ ) , (42)where φ is the phase of the optical field, ∆ is proportionalto the detuning between the injection and the laser fre-quency, χ is the normalized feedback strength and ψ isrelated to the feedback phase. For the purposes of thiswork, Eq. (42) just provide a non-trivial scalar systemwhere the delayed feedback is nonlinear, thus leading tosignificant differences in the spatio-temporal representa-tion with respect to models considered in [20].According to the DR, we obtain φ τ = ∆ − sin φ + χ sin (cid:0) φ NL − φ − ψ (cid:1) , (43)which together to suitable boundary conditions for φ along the ξ domain, which we take periodic, representsthe essence of our approach.A normal form approximation of (43) can be readilyobtained by expanding the non-local term up to a givenorder. At the second-order we obtain φ τ + φ ξ χ cos ψ = ∆ − χ sin ψ − sin φ + 12 χ sin ψ φ ξ + 12 χ cos ψ φ ξξ . (44)The comparison between the models (42), (43) and (44)is reported in Fig.4. Starting from a rectangular initialcondition, we identify two propagating regimes as thefeedback strenght is varied: at low values of χ , a singlelocalized state propagating with constant velocity and,for higher values of the parameter, two pulses propagat-ing at different speeds. The corresponding spatiotempo-ral patterns are shown in the insets of Fig.4(a) where weplot the sinus of the phase variable φ . These coexixtinglocalized states correspond to coarsening of kink-antikink solutions while the single pulse regime at low values of χ corresponds to the propagation of a single kink (phase-slip).In 4(a), we plot the velocities of the kinks for a decaderange of the feedback gain parameter χ . We observe anexcellent agreement between the delayed (42) and thenon-local model (43), not only at the level of propa-gation speeds, but also in the transverse profiles of thesolutions, as reported in Fig.4(b). Although this couldappear somehow expected as the two models share thesame bulk rule, we remark that they strongly differ atthe boundaries. This supports our initial hypothesis thatin the thermodynamic limit (i.e. in the bulk region) thenon-local model well approximates the delayed dynamics,independetly from the choice of the boundary conditions.The second-order normal form (44) captures most ofthe phenomenology of the delayed and non-local models,reproducing the qualitative behavior of the velocities asa function of χ (see Fig.4a) and also providing a goodapproximation of the profiles of the kinks (Fig.4b). Onthe other hand, it does not display the transition to thesingle-kink regime at low values of χ , and a noticeabledifference in the magnitude of the velocities is observed.As a peculiar benefit of the DR, we can improve thequality of the normal form approximation by simply in-creasing the order of the expansion of the non-local term.We report in Fig.4 the results obtained by integrating athird-order expansion normal form, obtained by addingto the r.h.s. of (44) the terms16 [( χ cos ψ φ ξ − φ ξξξ ) − χ sin ψ φ ξ φ ξξ ] . (45)Even in this case, we the model is unable to reproducethe transition from the double to the single pulse regime.However, the agreement between both the velocities andthe spatial profiles is substantially improved and barelydistinguishable from those obtained from the full modelsEqs. (42) and (43). Here, the introduction of higherorders in the normal form breaks the parity symmetry ofthe solutions around the comoving direction, witnessedby the presence of only even terms in (44) besides thedrift term, thus leading to a better approximation of theoriginal, asymmetric profiles.We finally observe that, writing Eq. (44) in the co-moving reference frame corresponding to the velocity v = χ cos ψ to remove the advection term in the l.h.s.,and rescaling the space by ξ → ξ (cid:114) χ cos ψ (46)we eventually get φ τ = sin ¯ φ − sin φ + φ ξξ + φ ξ tan ψ , (47)where sin ¯ φ = ∆ − χ sin ψ .The model (47) is now formally identical to the second-order normal form equation obtained in Ref. [12] in theSTR [cf their Eq. (3)], and represents an alternative FIG. 4: The delayed Adler model and its spatio-temporaldescriptions. Top: Pulse propagation velocities in the ( σ, θ )plane for the delayed (red) and the non-local (black dots)model together with its second-order (green) and third-order(blue) approximations in the DR. In the insets the 1-pulse (a)and two-pulses (b) solutions of the delayed Adler model areplotted in the DR for χ = 2 and χ = 4, respectively (herewe plot the sinus of φ ). Bottom: Transverse cuts along thespatial ξ direction for the different models: the color codeis the same used for the top panel. Other parameters are∆ = 0 . T = 2 × and ψ = 0. mathematical description of the system. However, in (47)the role of time and space is exchanged: indeed for the ad-vection velocity associated to the feedback term we findthe value χ cos ψ that is the inverse of what reported in[12]. We will turn back to this issue in Sec. VI, wherewe will discuss the connection between the two represen-tations based on general arguments. V. AN OPTICAL DELAYED MODEL WITHDISPERSION
In this section we deal with a rather interesting frame-work, formalized by a delayed differential equation withan algebric constraint, which is thus intermediate be-tween a scalar and a vectorial case. The model has beenfirst introduced in [23] for the study of dispersive instabil-ities of pulse trains in mode-locked semiconductor lasers.Here, we specifically refer to the single-mode version in [24] [see also our Eq. (62)], from which an equivalentPDE in the STR has been derived by means of multiple-scale analysis.
A. The linear case
Before discussing the full model (62) we first examinea linear prototype system, in which already most of thetopic can be elucidated,˙ E = − E + hY (48) Y = η ( E d − Y d ) . Here, we are interested in studying the above model initself, regardless its physical meaning; notice however,that it could be obtained from (62) by eliminating thecarrier dynamics (i.e. neglecting all nonlinear terms).Setting u = E and w = E − Y the model can berewritten as ˙ u = ( h − u − hw (49) w = u − ηw d . (50)In the framework of the DR, the above equation takesthe form ∂ τ u = ( h − u − hw (51) u = u − ηw NL . (52)Fourier transforming we can derive the exact disper-sion relation p ( q ) = h − − h ηe − iq , (53)where we associate the frequency p ( q ) and the wavevec-tor q to the time derivative and spatial shift operator,respectively, i.e. ∂ τ → p ( q ) and S → e − iq .Expanding the exponential term for small wavevectors,e.g. up to the second order we get p ( q ) = ( η − h − η + 1 − ηh ( η + 1) iq (54) − η ( η − h η + 1) ( iq ) , which corresponds in the direct spacetime ( ξ , τ ) to thenormal form Z τ = ( η − h − η + 1 Z − ηh ( η + 1) Z ξ − η ( η − h η + 1) Z ξξ . (55)Most of the interest for this model comes from theobservation that for high reflectivities ( η → − ) the co-efficient of the second-order spatial derivative, i.e. thediffusion, vanishes. In this limit, we are left with a dom-inant role of the dispersive effects related to the thirdorder term. To study this regime, we set h = 2 (cor-responding to the Gires-Tournois interferometer regime[23, 35]) and η = 1 − ε with ε = o (1), to obtain p ( q ) = − ε − iq + 18 ε ( iq ) + 124 ( iq ) + ... (56)Truncating the expansion at the third-order, the cor-responding normal form writes as U τ = − εU − U ξ + 18 εU ξ + 124 U ξ . (57)In the limit ε = 0 we get the dispersive-advection equa-tion U τ + 12 U ξ = 124 U ξ . (58)Eq. 58 has the form of the linear Korteweg - De Vries(KdV) equation [36], although with a positive third-ordercoefficient. In optics, this corresponds to an anoma-lous dispersion term implying that the higher spatial-frequency waves travel faster than the lower frequencywaves. The integration of model (58) is in good agree-ment with the original delayed system (49). The spa-tiotemporal plots also highlight the different boundaryconditions of the two models. This is evidenced also look-ing at the maxima of the profiles that are found at differ-ent pseudospatial positions. Both the delay and the spa-tially extended model display anomalous dispersion ef-fects with high-frequency components of the wavepacketpropagating faster than the lower ones (see Fig. 5).On the other hand, the KdV is often written with anegative third order coefficient, leading to the normaldispersion phenomena with lower frequency waves trav-elling faster. This is what we find in the STR description.Reversing spatial and temporal variables q ( p ) = − log (cid:16) ipη ( h − − ip ) (cid:17) , (59)and expanding for small p up to the third-order we getthe normal form for h = 2 U θ = log( η ) U − U σ − U σ . (60)Eq. (60) corresponds to the normal form obtained in[23] by means of functional mapping method [34]. Byrescaling θ → θ/ σ → σ and in the limit ε = 0, weobtain U θ = − U σ − U σ , (61)which is the same PDEM found for the DR but with theopposite sign for the dispersion term.Remarkably, we have found that the same pattern canbe generated with normal or anomalous dispersion whenobserved in the STR and in the DR, respectively. As (a) (b) (c) (d) x t x t x x FIG. 5: (a,b) Numerical integration of the delayed model (49)and (c,d) of the spatially-extended system (58) for h = 2 and T = 200. In (a,c) the spatiotemporal patterns are shown inthe DR spacetime ( ξ, τ ). In (b,d) the profiles are transversecuts along ξ evaluated at fixed τ , as indicated by the dashedlines). The vertical arrows indicate the pseudospace positionsof the profiles maxima. a consequence, the very same bulk phenomena are ex-pected to arise in the two PDEMs well approximatingthe original ones, but with opposite symmetry with re-spect to the diagonal axis. In the next section we will an-alyze these dispersive phenomena in the fully non-linearproblem. B. The nonlinear dispersive model
We now consider the full model discussed in Ref. [24],which describes the dynamics of the intracavity field E and of the population inversions N i ( i = 1 ,
2) of apassively mode-locked integrated external-cavity surface-emitting laser˙ E = (cid:0) (1 − iα ) N + (1 − iα ) N − (cid:1) E + hY (62) Y = η (cid:0) E ( t − T ) − Y ( t − T ) (cid:1) ˙ N = γ ( J − N ) − | E | N ˙ N = γ ( J − N ) − s | E | N . Here, Y is the field in the external cavity, α i , J i and γ i are the linewidth enhancement factors, the bias andrecovery time relative to the gain ( i = 1) and absorbersection ( i = 2), respectively, and s is ratio of the gainand absorber saturation intensities.We rewrite (62) in the DR, where, as usual, the delayedterm becomes nonlocal in space and the standard timederivatives transform into derivatives with respect to theDR time τ : ∂ τ E = (cid:0) (1 − iα ) N + (1 − iα ) N − (cid:1) E + hY (63) Y = η (cid:0) E NL − Y NL (cid:1) ∂ τ N = γ ( J − N ) − | E | N ∂ τ N = γ ( J − N ) − s | E | N . The Y variable can be eliminated using the secondequation in the Fourier domain¯ Y = ¯ E ηe − iq ηe − iq , (64)where the bar indicates the Fourier transform and q thespatial wavevector.Expanding up to the third order and reverting to spa-tial variables we get Y = (cid:0) η η − η (1 + η ) ∂ ξ + η (1 − η )2(1 + η ) ∂ ξ (65)+ η (4 η − − η )6(1 + η ) ∂ ξ + .. (cid:1) E .
Substituting into the full model, we obtain ∂ τ E = (cid:0) (1 − iα ) N + (1 − iα ) N (cid:1) E (66)+( hη η − E − hη (1 + η ) ∂ ξ E + hη (1 − η )2(1 + η ) ∂ ξ E + hη (4 η − − η )6(1 + η ) ∂ ξ E∂ τ N = γ ( J − N ) − | E | N ∂ τ N = γ ( J − N ) − s | E | N . In Fig. 6 we compare the results obtained by numeri-cal integration of the delay system (62), the correspond-ing nonlocal DR model (63) and the third-order PDEM(66). The spatiotemporal plot shows the propagation ofa single pulse in the DR, the dynamics of which havebeen analyzed in detail in Ref. [24]. The agreement be-tween the delayed and DR model is excellent. On theother hand, while well reproducing the phenomenology,the third-order approximation exhibits quantitive differ-ences, both in the propagation velocity and in the trans-verse profiles (see Fig. 6b).We now compare our PDEM with the results of [24],where a third-order model in the STR has been derived.Rescaling the pseudo-space ξ by (1 + η ) /hη and thepseudo-time τ by hη/ (1 + η ) , we have ∂ ξ n → (1 + η ) n ( hη ) n ∂ ξ n , n = 1 , , ∂ τ → hη/ (1 + η ) ∂ τ , (68) and we eventually obtain ∂ τ E = (cid:0) (1 − iα ) N + (1 − iα ) N (69)+ hη η − (cid:1) hη (1 + η ) E − (1 + η ) hη ∂ ξ E + (1 − η )2 (cid:0) ηhη (cid:1) ∂ ξ E + 4 η − − η η ) (1 + η ) ( hη ) ∂ ξ E∂ τ N = γ ( J − N ) − | E | N ∂ τ N = γ ( J − N ) − s | E | N . It is interesting to note that apart from the usual ex-change between space and time, the drift (first-order) anddiffusion (second-order) terms are equal to those found inthe STR model in [24] (see their Eq. (11) for the modeland Eqs. (14-15) for the drift and diffusion coefficients,respectively). On the other hand, the coefficient of thethird-order derivative is d DR = 4 η − − η η ) (1 + η ) ( hη ) = (70)(4 η − − η )(1 + η )2(1 + η ) ‘ × (1 + η )3 (cid:0) ηhη (cid:1) = Q ( η ) × (1 + η )3 (cid:0) ηhη (cid:1) = − Q ( η ) d , where d is the dispersion constant in [24] [cf their Eq.(16)]. The two dispersions thus differ both in sign andabsolute value. Interestingly however, since η = 1 − ε inthe high reflectivity limit ε (cid:28) Q ( η ) ≈ − ε (71)Hence, up to second order corrections, or for the idealcase of perfect reflectivity ε = 0, the two coefficients willonly differ in sign as found in the case of the linear model.We also notice that when η = 2 − √ ≈ . d DR = 0while d is finite. As such, for this value of the reflectiv-ity, the DR model (69) is dispersionless while the STRmodel in [24] remains dispersive. We remark that at thenonlocal level, including i.e. all infinite orders of the ex-pansion, both the STR and DR models must coincideand reproduce the delayed dynamics in the thermody-namic limit. On the other hand, the rate at which thetwo representations converge towards the solution of thenonlocal problem is generally different and depends onthe specific details of the system under consideration. Inthis case and for this value of reflectivity, higher-orderderivatives are necessary for the DR model to capturedispersive effects.0 t x t |E| FIG. 6: a) Spatio-temporal plot in the DR of for a single propagating pulse obtained by numerical integration of the full delaymodel (62) (yellow pattern), the nonlocal DR model (63) (black solid line) and its third-order approximation (66) (green solidline). b) Transverse cuts of the above patterns at fixed ξ indicated by the vertical dashed lines. Parameters: α , =0, J = 0 . J = − . γ = 3 × − , γ = 0 . η = 0 . h = 2 and s = 1. In the delay model T = 2000. VI. SWAPPING SPACE AND TIME: FROM STRTO DR AND BACK
As we have seen in the previous sections, an effectiveapproach to describe a long-delayed systems is to derivePDE’s from the two representations. The task is to ex-plicitly rule the evolution of the field variable along thetemporal-like direction in terms of the space-like deriva-tives of it. Such a scheme can be very convenient, bothfrom a conceptual and practical point of view. Indeed,whenever the model is obtained by means a suitable ex-pansion, a few terms could be sufficient to well approxi-mate the dynamics.Since the role of pseudo-space and time is exchangedin the two representations, the function expressing thetime derivative in terms of the space derivatives can bedifferent in the two cases. As a consequence, the relatedPDEMs would differ as well, at least from a certain orderon. In the following we will consider a specific class ofPDEMs allowing to easily switch between the representa-tions. In particular, we will find the conditions in whichis possible to obtain the same reduced PDEM, obviouslylimited to some order in the space derivatives.We start from the n -order description obtained fromthe delay system in one of the two representationsΦ ( n ) y = F ( n ) (Φ ( n ) , Φ ( n ) x , Φ ( n ) x , .., Φ ( n ) x n ) . (72)We now move to the comoving reference frame corre-sponding to the diagonal of the 2D domain of the (suit-ably rescaled) variables. A pictorial view of these tworeference frames is illustrated in Fig. 7.¯ x = x − y (73)¯ y = y . y = y x x FIG. 7: Representation of the original and comoving referencesystems related by transformation (73).
Accordingly, Eq. (72) rewrites asΦ ( n )¯ y = Φ ( n )¯ x + F ( n ) (Φ ( n ) , Φ ( n )¯ x , Φ ( n )¯ x , .., Φ ( n )¯ x n ) . (74)The parity symmetry transformation ¯ x ↔ − ¯ x whichleaves invariant the above equation corresponds to x → y as seen by the comoving reference frame (73) and repre-sents the commuting rule between the two representa-tions.We now consider the class of model for which the co-moving term, expressed by the linear first-order deriva-tive, disappears from the equation: in the very commoncase of n = 2 the equation (74) is thus invariant under(73). Therefore the transformed equation, which corre-sponds to the PDEM in the other representation is for-mally the same and admits the same solutions. In thiscase the parity transformation maps one representation,and its related solutions, into the other.As a paradigmatic example in this class of second-order systems, we mention the Delayed Complex Landau(DCL) model˙ u = µu − (1 + iβ ) | u | u + ηu d , (75)where u is complex. Once re-written in the DR andexpanded up to the second-order yields Z τ = ( µ + η ) Z − ηZ ξ + η Z ξξ − (1 + iβ ) | Z | Z , (76)i.e. a Complex Ginzburg-Landau (CGL) equation withdrift η and diffusion η/ ηZ θ = µ Z − Z σ + 12 η Z σσ − (1 + iβ ) | Z | Z , (77)(see their Eq.(17) when reported in the original coor-dinate system). Eqs.(76) and (77) are identical setting µ + η = µ and ∂ τ → η∂ θ (78) ∂ ξ → η ∂ σ . As seen, Eq.(78) corresponds to swap space and timebetween the two models and thus passing from STR toDR, with the proper units assured by the presence of the”velocity” 1 /η .We remark how, as long as the normal forms (76) and(77) derived from the two representations can be ex-changed making use of the (78), the patterns obtainedfrom the integration of the two are expected to be anal-ogous and close to the one produced by the DCL model.This is true close to the Hopf bifurcation, but movingaway from that this is no more the case as shown in [20].Indeed, the inclusion of a further term in the expansion(76) allowed to better approximate the original dynamicsincluding deviations from the parity symmetry observedmoving away from the Hopf bifurcation.The bistable system with delay [7, 9], with e.g. a quar-tic potential with asymmetry a ˙ u = − u ( u − u + 1 + a ) + gu d = − U (cid:48) ( u ) + gu d (79)also belongs to this class of models. Again, it is straight-forward to write the expansion in the DR at any order.At the second-order, we have the reaction-diffusion sys-tem Z τ = − U (cid:48) ( Z ) + gZ − gZ ξ + 12 gZ ξξ , (80)and thus we set a correspondent STR model by using therescaling (78) adopted for the DCL (with η = g ) gZ θ = − U (cid:48) ( Z ) + gZ − Z σ + 12 g Z σσ . (81)At the same order, the simplest nonlinear case occurswhen the first order derivatives appear at second orderpower. This is indeed what we have found for the Adlermodel discussed in Sec. IV where a term Φ x is presentand again the same PDEM is obtained in the two repre-sentations.More complicate situations can arise for higher orders.For e.g. n = 3, we can associate different functions forthe two representation only differing by the sign of thethird order derivative:Φ (3) x = F (3) (Φ (3) , Φ (3) x , Φ (3) x , Φ (3) x ) , (82)for the first representation andΦ (3) y = F (3) (Φ (3) , Φ (3) y , Φ (3) y , − Φ (3) y ) (83):= ¯ F (3) (Φ (3) , Φ (3) y , Φ (3) y , Φ (3) y ) (84)for the second. In this way, the switch x → y has to beaccompanied by F → ¯ F . This is what we have found2in the Sec. V, where the two PDEMs only differ for thethird order coefficient sign.We finally remark that the PDEMs for the two repre-sentations are generally different also in the very simplecases. For instance, we consider again the linear equation(21). Once written according to the DR in the Laplace-domain we obtain s τ = − − s ξ ) , (85)where ( s ξ , s τ ) the Laplace-conjugate variables of ( ξ, τ ).Expanding up to the third order we have s τ ≈ − s ξ + 12 s ξ − s ξ , (86)which leads to the normal formΦ τ = − Φ ξ + 12 Φ ξξ −
16 Φ ξξξ . (87)In the STR, we obtain instead for the correspondingconjugate variables s σ = − − s θ ) , (88)and expanding s θ = − log(1 + s σ ) ≈ − s σ + 12 s σ − s σ , (89)we eventually get the normal formΦ θ = − Φ σ + 12 Φ σσ −
13 Φ σσσ . (90)We report in Fig. 8 the numerical integration of thetwo models (87) and (90), in both cases using periodicboundary conditions and a gaussian initial function. Asseen in Fig. 8a-b, the spatiotemporal patterns plotted intheir respective spatiotemporal domains are quite simi-lar. In Fig. 8c we compare the temporal and spatial pro-files of system (90). Due to dispersion, the two profilesare clearly asymmetric and different one from each other.A similar situation is found in model (87), although witha weaker asymmetry owing to the lower dispersion coeffi-cient. On the other hand, a remarkable agreement, up toalmost three decades, is observed when we compare thetwo models along the corresponding directions, ( θ → ξ )and ( σ → τ ), thus demonstrating the equivalence of thetwo representations. As an example, we plot in Fig. 8dthe profile along θ shown in Fig. 8c, and a transversecut along ξ of the pattern in Fig. 8b. The residual de-viations can be associated to the order of the expansionused in the two PDEs. Indeed, the two representationsdo not uniformly converge towards the solution of thedelayed model and the inclusion of suitably different or-ders would be necessary to compensate the discrepancyin the profiles. In Fig. 8d we also plot the solution ofthe delayed system (21) for comparison. Incidentally, inthis specific case the DR model (87) shows already atthe third-order an excellent agreement with the originaldelay equation. However, deviations are eventually ex-pected, as any finite-order expansion cannot capture theintrinsically nonlocal nature of the delay problem. s t x q -1 -2 -3 -5 -4 -1 -2 -3 -5 -4
0 200 400 600 0 200 400 600 s ,q x ,q (a) (b) (c) (d)
FIG. 8: Spatiotemporal plots of the linear models (a) (90) and(b) (87) for a gaussian initial condition of unitary amplitudeand width. (c) Profiles along the σ (black line) and θ direc-tions (red line) as obtained from model (90). (d) Comparisonbetween the profile along θ as obtained from (90) (red line),the profile along ξ as obtained from (87) (black dots) and thesolution of the delay model (21) (green line). The profilesare displayed for the same value of σ ↔ τ . The spatially ex-tended models have been integrated using periodic boundaryconditions. VII. CONCLUSIONS
The study of long delayed dynamical systems stronglybenefits of a spatio-temporal description whenever it ispossible. Such a mapping, besides realizing a bridge be-tween different high-dimensional systems allows for sim-ple conceptual interpretation of complicated phenomenaotherwise hidden in the temporal series of a delayed sys-tem. As such, the success of the now widespread STRis explained and justified. However, some practical dif-ficulties arise in the derivation of normal forms in theSTR, as the implicit non-locality in time leads to involvedmathematical derivations often requiring vicinity to a bi-furcation. Moreover, both the evaluation of the comov-ing Lyapunov exponent and analytical considerations inlinear models indicates that the choice of the slow-timevariable as the pseudo-time in the representation couldnot be the most appropriate. In the spirit to better un-derstand and describe the spatio-temporal equivalence,recently the new DR has been introduced. According tothis new approach, the role of space and time is reversedin the mapping, aiming to describe a far from boundaries(bulk) evolution. While mathematically speaking the DRadmits the very same solution of the original delay prob-lem only for a specific choice of the boundary conditions,in the thermodynamic limit it is shown that the DR pro-vides a very good approximation of the dynamics.In this work, we have supported the preliminary argu-ments and evidences of the validity of DR over the STRby the analysis of new systems (with a nontrivial struc-ture of the delayed feedback) and discussed in details thenovelties and the peculiarities of this new approach. In3particular, the easy derivation of PDEM at any order al-lows for a straightforward application of the method todescribe the bulk dynamics of any long delayed systems.We believe that with the new examples and enlighten-ment carried by this work our approach could representa significant advance in the area of long-delayed dynam-ical systems. In particular, one can expect this to hap- pen in the relevant cases of conceptual description andquantitative evaluation of bulk behaviours and quantitiesanalogous to those of spatio-temporal systems.Further investigations remain to be carried out, to pre-cise the limit of application and the a-priori degree ofapproximation one could expect for a specific expansion. [1] S. Yanchuk and G. Giacomelli, Journal of Physics A:Mathematical and Theoretical , 103001 (2017).[2] For a recent review: T. Erneux, J. Javaloyes, M. Wolfrumand S. Yanchuk, Chaos , 114201 (2017).[3] J. D. Farmer, Physica D: Nonlinear Phenomena , 366(1982).[4] F. T. Arecchi, G. Giacomelli, A. Lapucci, R. Meucci,Phys. Rev. A , 4225 (1992).[5] G. Giacomelli and A. Politi, Phys. Rev. Lett. , 2686(1996).[6] G. Giacomelli, R. Meucci, A. Politi, and F. T. Arecchi,Phys. Rev. Lett. , 1099 (1994).[7] G. Giacomelli, F. Marino, M. A. Zaks, and S. Yanchuk,EPL (Europhys. Lett.) , 58005 (2012).[8] J. Javaloyes, T. Ackemann, and A. Hurtado, Phys. Rev.Lett. , 203901 (2015).[9] G. Giacomelli, F. Marino, M. A. Zaks, and S. Yanchuk,Phys. Rev. E , 062920 (2013).[10] F. Marino, G. Giacomelli, and S. Barland, Phys. Rev.Lett. , 103901 (2014); ibid Phys. Rev. E , 052204(2017).[11] G Giacomelli, R Hegger, A Politi, M Vassalli, Phys. Rev.Lett. , 3616 (2000).[12] B. Garbin, J. Javaloyes, G. Tissoni, and S. Barland, Nat.Commun. 6, 5915 (2015).[13] B. Romeira, R. Av´o, J. M. L. Figueiredo, S. Barland,J. Javaloyes, Scientific Reports , 19510 (2016).[14] F. Marino and G. Giacomelli, Chaos , 114302 (2017).[15] L. Larger and B. Penkovsky, Y. Maistrenko Phys. Rev.Lett. , 054103 (2013); L. Larger, B. Penkovsky, andY. L. Maistrenko, Nature Commun. 6, 7752 (2015).[16] M. Faggian, F. Ginelli, F. Marino and G. Giacomelli,Phys. Rev. Lett. , 173901 (2018).[17] S. Yanchuk and G. Giacomelli, Phys. Rev. Lett. ,174103 (2014); ibid. Phys. Rev. E , 042 903 (2015).[18] D. Brunner, B. Penkovsky, R. Levchenko, E. Schoell, L. Larger, Y. Maistrenko, Chaos , 103106 (2018).[19] F. Marino and G. Giacomelli, Phys. Rev. Lett. ,174102 (2019).[20] F. Marino, and G. Giacomelli, Phys. Rev. E , 060201(2018).[21] S. Yanchuk, S. Ruschel, J. Sieber and M. Wolfrum, Phys.Rev. Lett. , 053901 (2019).[22] L. Munsberg, J. Javaloyes, S. V. Gurevich,arXiv:2001.07556.[23] C. Schelte, P. Camelin, M. Marconi, A. Garnache, G.Huyet, G. Beaudoin, I. Sagnes, M. Giudici, J. Javaloyes,and S. V. Gurevich, Phys. Rev. Lett. , 043902 (2019).[24] C. Schelte, D. Hessel, J. Javaloyes, S. V. Gurevich, Phys.Rev. Applied , 054050 (2020).[25] G. Giacomelli, S. Lepri, A. Politi, Phys. Rev. E , 234102 (2011).[27] G. Giacomelli and A. Politi, Physica D , 26 (1998).[28] S. A. Kashchenko, Computational Mathematics andMathematical Physics
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