aa r X i v : . [ n li n . C D ] J un Chaotic fluctuations in graphs with amplification
Stefano Lepri a,b a Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10 I-50019 Sesto Fiorentino, Italy b Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1 I-50019, Sesto Fiorentino, Italy
Abstract
We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of theinterval. We investigate the possibility of having power-law tails in the invariant measure by approximate solution ofthe Perron-Frobenius equation and discuss the connection with the generalized Lyapunov exponents L ( q ). We thenconsider the case of open maps where trajectories escape and demonstrate that stationary power-law distributions occurwhen L ( q ) = r , with r being the escape rate. The proposed system is a toy model for coupled active chaotic cavities orlasing networks and allows to elucidate in a simple mathematical framework the conditions for observing L´evy statisticalregimes and chaotic intermittency in such systems. Keywords:
Chaotic map, Power-law distributions, Diffusion and amplification on graphs, Generalized Lyapunovexponents
1. Introduction
Dynamical systems defined on graphs are subject ofcurrent research, due to the many applications to modelcomplex interacting units with non uniform connectivity[1]. Also, one of the features of complex systems is thepossibility of display non-Gaussian fluctuations that makelarge rare events very relevant. The distributions of the ob-servables can have fat-tailed statistics leading to domina-tion of a single event and lack of self-averaging of measure-ments. In active systems where fluctuating amplificationcan occur, even rare trajectories can generate large-sizedfluctuations. This is well-known for multiplicative stochas-tic processes [2] and chaotic dynamical systems that dis-play intermittency and multifractality [3].Countless example are present in the physical, biolog-ical and even social sciences. A case of experimental rele-vance is provided by optical media with diffusion and am-plification of light, as it occurs in random lasers whereheavy-tailed distributions of emission intensities, charac-terized by L´evy -stable statistics [4] have been predicted[5] (see also [6, 7]) and confirmed in experiments [8, 9, 10].An experimental system that encompasses propertiesof a dynamical systems on graphs and non trivial statis-tics is the lasing network , recently introduced in [11]. Itconsists of active and passive optical fibers, connected toform a graph structure. The connectivity induces a formof topological disorder and can be viewed as a discreterandom laser, with a controllable complexity. The pres-ence of the optical gain and disorder induce wild emissionfluctuations whose origin is not fully understood [11]. Soa related question is how such fluctuations relate to thenetwork structure and connectivity. The existence of fat tails is intimately related to thethe possibility for a spontaneous fluctuation to grow wellbeyond the average. The indicators to quantify this are thefinite-time and generalized Lyapunov exponents [12]. Formultiplicative noise they been shown that they are usefultools to yield an intuitive derivation of the form of theprobability distribution’s tail in the presence of additivenoise [13].Another possibility to have stationary fat-tailed statis-tics for multiplicative growing processes is to consider re-setting, namely a random process where the variable is setto a given value with some given protocol [14]. The casesof stochastic partial differential equations like the Kardar-Parisi-Zhang equation of fluctuating interfaces has beenals considered [15].In the present paper we study a simple chaotic mapthat couples chaotic diffusion and random amplification.Nonlinear maps are thoroughly investigated as mathemat-ically simple model to analyze the connection betweenmacrolaws and microscopic chaos [16]. It can be regardedas a toy model for coupled active chaotic cavities or thelasing networks mentioned above [11]. The idea is thatlight rays can be treated as particles undergoing chaoticdiffusion and amplification. Indeed, the classical dynamicsof particles on graphs is a chaotic type of diffusive process[17]. Trajectories of a particle on a graph, undergoingscattering at its vertices, are in one-to-one correspondencewith the ones of one-dimensional piecewise chaotic maps[17, 18].The model is simple enough to allow for a very de-tailed analysis, demonstrating power-law distributions ofthe invariant measure. It allows to elucidate in a sim-ple mathematical framework the conditions for observing
Preprint submitted to Chaos, Solitons and Fractals June 22, 2020 ´evy statistical regimes and chaotic intermittency in suchsystems. It also serves as an example to demonstrate theusefulness of generalized Lyapunov exponents to assess thepossibility of power-law fluctuations. Moreover we extendthe concepts to the case of open systems (like chaotic repel-lors), a case that, to our knowledge, has not been studiedin this terms.In Section 2 the map model is presented and its relationwith the physical systems is sketched. The stationary in-variant measure is computed along with an effective masterequation. The connection between power-law and gener-alized Lyapunov exponents is discussed in Section 3. Thisrelation is extended to the case of open maps in Section4. The connection between the model and the calcula-tion of the spectrum of the lasing network is given in theAppendix.
2. Map model
We consider the following map ( x n +1 = f ( x n ) E n +1 = g ( x n ) E n + s (1)where f is chaotic with a positive Lyapunov exponent λ .For definiteness, let us consider x to belong to the unitinterval and E n , g positive and s ≥ x n todescribe couples the position of a ”ray” undergoing chaoticmotion during which it acquires an ”energy” that increasesor decreases according to whether g il larger or smallerthan one. Thus chaotic diffusion and amplification arecoupled since the acquired energy depends on the trajec-tory. In Fig.1 we sketch a physical reference system in-spired from the lasing network experimentally studied in[11, 19].The term s represent some form of energy injectionand is needed to avoid that E n = 0 is not and ”absorbing”point. The case s = 0 leads to a non-stationary distri-bution that for large times is log-normal. This is read-ily understood as the variable log E n = z n performs adiscrete-time biased random walk [20]. The average ve-locity h log g ( x n ) i ≡ λ is the (second) Lyapunov exponentof (1). The situation is drastically different in presenceof a term s >
0, that act as a source term. If λ < z n is attracted towards the source and this yield astationary measure. In the stochastic case this mechanismof repulsion has been shown to generically yield power-law decaying stationary distributions [21, 22, 23]. Similarconsiderations apply for extended stochastic systems likethe non-linear diffusion equation with multiplicative noise[24]. On the other hand, for λ > f ( x ) = p x ≤ x ≤ p/ − p x + − p − p ) p/ < x ≤ / − p x − − p ) / < x ≤ − p/ p x + 1 − /p − p/ < x ≤ f can be derived exactly as a suitablePoincar´e section as described in [17]. The map is ev-erywhere expanding and is invariant for x → − x , butit is straightforward to generalize to an asymmetric caseand/or more complex graphs.Since the invariant measure of the map is constant itsthe Lyapunov exponent λ = − p log p − (1 − p ) log(1 − p ).So λ > p approaching 0and 1 where the map has weakly unstable orbits. Also, letus consider a piece-wise constant gain function gg ( x n ) = ( g < x n ≤ l < x n < g ≥ < l ≤
1. This is merely a choice of sim-plicity and it entails that the sequence of multipliers g ( x n )is in one-to-one correspondence with symbolic dynamics ofthe map f . Even in this simple example, for p = 1 / g ( x n ) is correlated in time and theamplification fluctuations change accordingly. n f ( x n ) gainloss Figure 1: Left: the chaotic map (2) for p = 0 .
25. Right: a sketchof two coupled chaotic cavities as in a double-ring lasing networkconnected by a coupler that transmits with a given probability (noreflections). One cavity contains an active amplifying medium, theother is dissipative. In the graph interpretation, the map f can bederived exactly as a suitable Poincar´e section as done in [17]. Let us focus on the case where the solution does notdiverge namely h log g ( x ) i <
0. Before entering the math-ematical analysis, in Fig.2 we report some representative Thus the ”physical” time t n corresponding to the n th iterationof the map and depends on the length on each bond . For instance,for the graph in Fig.1 with bond lengths L and L , t n +1 = t n + T ( x n ) T ( x n ) = T or T for x n < / x n > / v being the particle velocity, T , = L , /v ). This description can begeneralized to arbitrary graphs associated with Markov dynamics,see [17]. )02000400060008000012345 E n p=0.2p=0.5p=0.6 Figure 2: Time series of E n , g = 1 . l = 0 . s = 10 − for differentvalues of p . The trajectory is highly intermittent with large excursionof short duration. time-series of the the variable E n . The dynamics is highlyintermittent with large-amplitude spikes lasting tenths ofiterates. A finite value of the source term insures that thevariable does not vanish at long times, according to themechanism mentioned above.We are interested in the statistics of the variable E n .The time evolution of the measure P n ( x, E ) can be com-puted as solution of Perron-Frobenius operator P n +1 ( x, E ) = (3)= ( pg P n ( y , E − sg ) + (1 − p ) l P n ( y , E − sl ) 0 ≤ x ≤ − p ) g P n ( y , E − sg ) + pl P n ( y , E − sl ) < x ≤ y = px , y = (1 − p ) x + , y = (1 − p ) x + p − and y = px + 1 − p are the preimages of x .As a general approach, one may consider expanding P n on the basis of the eigen-functions of the Frobenius-Perronoperator of the map f . This would allow to describe thefull evolution of the measure in time, including transientsassociated with possibly slow chaotic diffusion. Since weare mostly interested in steady-state results, let us look forpiecewise-constant in x solutions of the form P n ( x, E ) = P ,n ( E ) for 0 < x < / P n ( x, E ) = P ,n ( E ) for1 / ≤ x < P ,n +1 ( E ) = pg P ,n (cid:18) E − sg (cid:19) + 1 − pl P ,n (cid:18) E − sl (cid:19) P ,n +1 ( E ) = (1 − p ) g P ,n (cid:18) E − sg (cid:19) + pl P ,n (cid:18) E − sl (cid:19) (4) We expect that this equation is valid when chaotic diffu-sion is sufficiently rapid to ensure homogeneization of themeasure on the time scale faster than the typical growth.It has a form of a master equation for probabilities of thevariable E on each side of the interval. Also, a standardKramers-Moyal expansion may be used to show that itcorresponds to a set of coupled Langevin equations withmultiplicative noise for the energies on the two sides ofthe interval. The connection between this equation andthe one used in the calculation of the spectrum of the las-ing network is given in the Appendix.If we look for stationary solutions that decay as a powerlaw for large E ≫ s , P , ( E ) ∝ A , E − (1+ α ) (5)where A , are constants, the dependence on s can be ne-glected and we get the self-consistency condition p ( g α + l α ) − (2 p − g α l α = 1 (6)The latter, along with the the stability condition λ < lg ≤ α < s is essential to yield a sta-tionary distribution, its value does enter in the exponentof asymptotic decay (5). We also checked numerically thatbasically the same statistics if found if s is replaced by arandom positive number (additive noise).To emphasize the importance of the separation of timescales leading to equations (4) and thus to power-laws letus compare with a situation when chaotic diffusion is rel-atively slow. For instance, in Fig.3 we consider the case p is small. In this limit, the map has a weakly unstable pe-riod two orbit and λ ≈ p . Not surprisingly, the invariantmeasure is non uniform and fractal in the direction of thevariable z .
3. Generalized Lyapunov exponents
The generalized Lyapunov exponents L ( q ) define thegrowth of the q th moment of the perturbation [25, 3, 26,27]. In general, for a perturbation δu of a dynamical sys-tem, which evolves according to the linearized equation ofmotion, let R ( τ ) = k δu ( t + τ ) k / k δu ( t ) k be the responsefunction after a time τ to a disturbance at time t . Then,for large times R q ( τ ) ∼ exp( L ( q ) τ ) where the overline de-note a time average. If L ( q ) > q thenthere is a finite probability that a small perturbation growvery large. Moreover, the deviation of L ( q ) from a linearbehavior in q signal an intermittent dynamics [12].The condition for power-law stationary tails can be ob-tained from generalized Lyapunov exponents [13]. In thepresent case we are interested in the generalized exponentsassociated with the E n variable when no source term is3 x n -6-5-4 l og E n Figure 3: Iterates of the map x n , log E n , g = 1 . l = 0 . s = 10 − for p = 0 .
05, yielding λ ≈ . λ = − .
290 giving a Lyapunov(Kaplan-Yorke) dimension 1 . present and unbounded growth or decay at large times ispossible. Actually, using equations (4) with s = 0 we canwrite the evolution map for the moments h E q , i and obtain L as the logarithm of its largest eigenvalue, L ( q ) = log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ( g q + l q ) + p p ( g q + l q ) − p − g q l q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7)In the simplest case p = 1 / L ( q ) = log (cid:16) g q + l q (cid:17) and themoment multiplier is just the arithmetic average of l q and g q . Note that, by construction, the standard Lyapunovexponent λ = L ′ ( q = 0) = log( gl ) / p while the L ( q ) do.The stability condition implies that the Lyapunov ex-ponent is negative. On the other hand, L ( q ) ≈ q log g for q large and positive. So a positive solution for L ( q ∗ ) = 0 ex-ist and coincides with the condition for a power-law decaygiven for to (6) for q ∗ = α . In the Gaussian approximationthis is seen immediately since in this case L ( q ) ≈ −| λ | q + µq where µ is the variance of λ . In this approximation q ∗ ≈| λ | /µ that makes transparent the fact that fluctuationsin the gain have to be of the same order as λ to observelarge fluctuations. This is in agreement with the generalscenario described in [13].In Fig.4 we report the generalized Lyapunov exponentsand the distributions of the z n variables. The estimatedexponents are in very good agreement with the simula-tions. For instance in the case p = 0 . q ∗ = 0 . .. .When the condition of stability is violated, some fur-ther mechanism of saturation is needed to have a steady distribution. In this case the term s can be ignored. Forinstance one may consider a nonlinear term in the form ofa chaotically-driven logistic map ( x n +1 = f ( x n ) E n +1 = g ( x n ) E n − E n . (8)This a particular case of the systems thoroughly studied in[20, 28] and will not thus be considered any further here.In presence of the nonlinear term, λ remains negative andthe invariant measure remains broad but has an exponen-tial cutoff at large E n . Detailed predictions on the natureof the chaotic intermittency close to the transition can begiven [20, 28, 29], including universality of power spectraand Lyapunov exponents. In the next section, we consideran alternative possibility for obtaining steady-state powerlaws even in the unstable case. -2 0 2 4q00,10,20,3 L ( q ) p=0.2p=0.3p=0.4p=0.6p=0.7p=0.8 -5 0 5 10 15z10 -7 -6 -5 -4 -3 -2 P D F p=0.2p=0.5p=0.6 Figure 4: The generalized Lyapunov exponents L ( q ) for differentvalues of the map parameter p and the distributions of log E , g =1 . l = 0 . s = 10 − . An exponential tail of P ( z ) ∼ exp( − αz )correspond to a power-law decay E − − α .
4. Open maps
We now discuss another possibility to have steady fluc-tuations with power-law tails namely an open setup wherethe trajectory are allowed to escape (and be re-injected).The idea is that the distribution of the values of E n is in4his case determined by the combined effect of the fluc-tuations of growth rates (as measured by finite-time Lya-punov exponents) and the statistical distribution of theescape events.Let us consider the growth of a perturbation over a fi-nite time τ , E ∝ exp( λ ( τ ) τ ) where λ ( τ ) is the finite-timeLyapunov exponent [26]. Defining z = log E its distribu-tion Q ( z ) is given by Q ( z ) = Z Z dλdτ δ ( z − λ ( τ ) τ ) P ( τ ) P ( λ, τ ) (9)basically an average of the growth rates on the distributionof escape times P ( τ ). As usual, for large τ we introducethe large-deviation function of the form P ( λ, τ ) ∼ exp( − U ( λ ) τ ) (10)In most cases, the distribution of escape times is Poisso-nian P ( τ ) = r exp( − rτ ) where r is the escape rate. Substi-tuting this expression in equation (9), the resulting integralcan be evaluated using the saddle-point approximation: ifwe denote by λ ∗ the saddle point, one obtains the con-dition λ ∗ U ′ ( λ ∗ ) − U ( λ ∗ ) = r . Then, recalling that thegeneralized exponents are the Legendre transform of thelarge-deviation function L ( q ) = qλ − U ( λ ) with q = U ′ ( λ )one obtains that the asymptotic decay of the distribution Q ( z ) ∼ exp( − q ∗ z ); L ( q ∗ ) = r (11)Changing back to the original variable E one obtains againa power-law tail, E − − q ∗ for large E . This last expressiongeneralizes the one given above and confirms that also inthe open setup the generalized exponents can be used toestimate the power law decay.A consequence of the above is that, in the open case wecan also consider the unstable case λ > q ∗ relatively close to zero. In the Gaussianapproximation L ( q ) = λ q + µq and for small r one has q ∗ ≈ r/λ . This nicely fits with the estimate given in [5]for observing L´evy fluctuations in amplifying diffusive me-dia with absorbing boundaries, upon identifying 1 /r withthe average residence time in the medium and λ with thetypical amplification time.To verify the above argument we consider first the sim-pler case of the map (8) (with s = 0) undergoing a stochas-tic resetting dynamics. With some preassigned small prob-ability r (which represents the escape rate) the variable E n is reset to some arbitrary value (with no modification onthe x n dynamics).The second case is a deterministic type of resetting,where the x n dynamics is given by the map, see Fig.5 f ( x ) = x ≤ x ≤ / a ( x − /
4) + / < x ≤ / a ( x − /
4) + / < x ≤ x − / < x ≤ a > / − / a, / − / a ). Thus for a → + the escaperate from the associated chaotic repellor is r ≈ ( a − / E n to one and x n to a random uniformly-distributed value. x n f ( x n ) gainloss Figure 5: Left: the chaotic map (12) for a >
2. Right: a sketch ofthe physical realization the open system in the case of two chaoticcavities connected by a coupler with leaks that allow escape of rayswith some probability.
In Fig.6 we considered both examples in the unstableand unstable regimes. The distributions are clearly dif-ferent from the one of the closed map given in Fig.4. Asexpected, the deterministic and stochastic case are sim-ilar. The generalized Lyapunov exponents as given byformula (7) are also reported. In both cases, the distribu-tions have double-exponential shape with rates in excellentagreement with the one given by (11), represented graph-ically in the leftmost panels of Fig.6. In the deterministiccase with a not too close from 2, the escape rate has beenestimated numerically.We conclude with a remark on the finer-scale structureof the distribution. A feature of the model is that the z n variable occurs almost in discrete values. Fig. 7 shows thatthe distribution has a finer structure with narrow peaksalmost equally-spaced (see inset of Fig 7). That should becontrasted with the case of the closed map where z n hascontinuous values and a smooth distribution.
5. Conclusions
Motivated by recent experiments of lasing networks[11, 19], we have introduced a toy map model describingthe effect of chaotic diffusion and amplification on a graphstructure. Since the motion is purely classical, it shouldapply when the wavelength is small with respect of thebond lengths. Evidence of large fluctuations and intermit-tency for the lasing network has been indeed been providedboth experimentally and by Monte-Carlo simulation [11].5 L ( q ) -6 -4 -2 0 2 410 -6 -4 -2 stochastic-5 0 5 10 15log E10 -6 -4 -2 -1 0 1 2 300,1 L ( q ) -10 -5 0 5 1010 -8 -6 -4 -2 deterministic0 10 20 30log E10 -8 -6 -4 -2 Figure 6: Leftmost panels: the generalized Lyapunov exponents for p = 0 . l = 0 .
8, the first row for the stable case g = 1 .
2, the secondfor the unstable one g = 1 .
4. Central panels: distributions of the variable z = log E for the map with stochastic resetting with probability r = 0 .
1. Dashed lines correspond to the exponential behaviors predicted by (11). Rightmost panels: distribution of the variable z = log E forthe deterministic map (12), a = 2 . , . , . a = 2 . r ≈ . -2 0 2 4 6 log E P D F Figure 7: Finer structure of the distribution of z = log E for theopen map (12) with a = 2 . l = 0 . g = 1 . gl ). Starting from a ”Lagrangian” description in terms ofchaotic trajectories we derived the corresponding ”Eule-rian” equations for the probabilities. We discussed thesimplest graph, but the generalization to larger graphs ispretty straightforward especially at the level of the mas-ter equation (4). In this case the equation can be easilyformulated in terms of the transition matrix of the under-lying diffusive process and the matrix for stochastic gainor loss terms (see also the Appendix below). Chaotic diffusion and amplification yield multiplica-tive fluctuations and power-law steady distributions. Insome regimes the variance can diverge leading to L´evy-likestatistics. We have confirmed that the Generalized Lya-punov exponents can give a precious hint on the statistics,both in the stable and unstable cases. We have extendedthis concept to open systems through equation (9) thatconnects the Generalized exponents with the escape rate.This result should apply under quit general conditions, asdemonstrated by the case of random resetting dynamics.
Acknowledgements
I acknowledge Stefano Gelli for contributing to the ini-tial stage of this work.
Appendix
In [11] the steady state modes of the lasing networkshave been computed using an approach extending the oneused for quantum graphs [30]. This is accomplished im-posing that a suitable network matrix N = SP has aneigenvalue equal to one. Physically, S is the scatteringmatrix of the optical couplers (splitters) and P is the socalled propagation matrix along the optical fibers and con-tains both the metric information on the bond length thanthe gain coefficients [19].6o clarify the connection with the map model studiedhere, let us first consider expressing equation (4) in the”physical” time t (neglecting the term s ) P ( E, t ) = pg P (cid:18) Eg , t − T (cid:19) + 1 − pl P (cid:18) El , t − T (cid:19) P ( E, t ) = (1 − p ) g P (cid:18) Eg , t − T (cid:19) + pl P (cid:18) El , t − T (cid:19) where T , = L , /v are the travel times (see the footnotein the main text). Taking the Laplace transform in t andintroducing the averages h ( E, z ) = Z ∞ h ( E, t ) e − zt dt, I , ≡ Z ∞ EP , ( E, z ) dE one obtains the condition (cid:18) I I (cid:19) = W G (cid:18) I I (cid:19) , where W ≡ (cid:18) p − p − p p (cid:19) , G ≡ (cid:18) ge zT le zT (cid:19) and W is recognized to be the stochastic matrix for a ran-dom walk on a graph with two states. To have non-trivialsolutions we impose det( W G −
1) = 0 that determines allpossible values of z . This equation is obtained by the con-dition that N has a eigenvalue one, by taking | N | , i.e. thematrix whose elements are the square moduli of it. Thestochastic matrix of the graph is thus the square modu-lus on the scattering matrix S of the coupler W = | S | while G = | P | . Viewed in this way one can recognizethe similarity with the ”quantization” procedure outlinedin [31, 18] where the classical stochastic transition ma-trix is replaced by an unitary one describing a quantummap. The generalization of the above to arbitrary graphsis straightforward. References [1] M. A. Porter, J. P. Gleeson, Dynamical Systems on Networks:A Tutorial, Springer series Frontiers in Applied Dynamical Sys-tems: Reviews and Tutorials, Switzerland, 2016.[2] J. Garc´ıa-Ojalvo, J. Sancho, Noise in spatially extended sys-tems, Springer Verlag, 1999.[3] A. Crisanti, G. Paladin, A. 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