Ulam method and fractal Weyl law for Perron--Frobenius operators
aa r X i v : . [ n li n . C D ] F e b EPJ manuscript No. (will be inserted by the editor)
Ulam method and fractal Weyl law for Perron–Frobeniusoperators
L.Ermann and D.L.Shepelyansky Laboratoire de Physique Th´eorique (IRSAMC), Universit´e de Toulouse, UPS, F-31062 Toulouse, France LPT (IRSAMC), CNRS, F-31062 Toulouse, France Abstract.
We use the Ulam method to study spectral properties of the Perron-Frobenius operators ofdynamical maps in a chaotic regime. For maps with absorption we show that the spectrum is characterizedby the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaoticscattering with the Weyl exponent ν = d −
1, where d is the fractal dimension of corresponding strange setof trajectories nonescaping in future times. In contrast, for dissipative maps we find the Weyl exponent ν = d/ d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed viathe relation ν = d / d is the fractal dimension of the invariant sets. We also discuss the propertiesof eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law. The Weyl law gives a fundamental relation between anumber of quantum states in a given classical phase spacevolume and an effective Planck constant ~ for Hermitianoperators [1]. Recently, this relation has been extendedto nonunitary quantum operators which describe complexspectrum of open systems or poles of scattering problem.In this case the fractal Weyl law determines a number ofGamow eigenstates in a complex plane of eigenvalues withfinite decay rates γ via a fractal dimension d of a classicalfractal set of nonescaping orbits. The Gamow eigenstatesfind applications in various types of physical problems in-cluding decay of radioactive nuclei [2], quantum chemistryreactions [3], chaotic scattering [4] and chaotic microlasers[5]. It is interesting that the fractal Weyl law was first in-troduced by mathematicians via rigorous mathematicalbounds [6]. Later, numerical simulations for systems withquantum chaotic scattering and open quantum maps con-firmed the mathematical bounds and determined a num-ber of interesting properties of such nonunitary quantumoperators [7,8,9,10,11]. Open quantum maps with absorp-tion, e.g. the Chirikov standard map [12], are very con-venient for numerical studies that allowed to establish anumber of intriguing properties of decay rates and quan-tum fractal eigenstates in the limit of large matrix sizeand small scale quantum resolution [13,10].The fractal Weyl law gives the following scaling for thenumber of Gamow states N γ with the decay rate in a finiteband width 0 ≤ γ ≤ γ b : N γ ∝ N ν , N = V / ~ , ν = d − , (1) where N is a matrix size given by a number of quantumstates in a volume V and the exponent ν is determined bya fractal dimension d of classical set formed by classicaltrajectories nonescaping in future times (see Fig.1).In view of the result (1) it is natural to assume thatthe fractal Weyl law should also work for other type ofnonunitary matrix operators. An important type of suchmatrices is generated by the Ulam method [14] appliedto the Perron-Frobenius operators of dynamical systems[15]. The method is based on discretization of the phasespace and construction of a Markov chain based on prob-ability transitions between such discrete cells given by thedynamics. It is proven that for hyperbolic maps in oneand higher dimensions the Ulam method converges to thespectrum of continuous system [16]. While the spectrumof such Ulam matrix approximant of continuous operatorhas been studied numerically for various dynamical maps(see e.g [17] and Refs. therein) the validity of the fractalWeyl law has not been investigated. Mathematical resultsfor the Selberg zeta function [19] indicate that the law(1) should remain valid but, as we show here, for certaindynamical systems the exponent ν starts to depend onfractal dimension d in a different way.It is known that in certain cases the Ulam method givessignificant modifications of the spectrum compared to thecase of the continuous Perron-Frobenius operators [16]. Infact discretization by phase-space cells effectively intro-duces small noise added to dynamical equations of mo-tion. For Hamiltonian systems with divided phase spacethis noise destroys the invariant curves and drasticallychanges the eigenstate of the Perron-Frobenius operator(see e.g. discussion in [18]). However, for homogeneouslychaotic systems the effect of this noise is rather weak com- L.Ermann, D.L.Shepelyansky: Ulam method and fractal Weyl law for Perron–Frobenius operators
Fig. 1. (Color online) Phase space representation of eiges-tates ψ i of the Ulam matrix approximant S of the Perron-Frobenius operator for models 1 and 2 at N = 110 × | ψ i | with red/gray for maximumand blue/black for zero). Left column shows eigenstatesfor the model 1 at K = 7 , a = 2 for maximum λ = 0 . λ = − .
01 + ı .
513 (bottom panel),the space region is ( − aK/ ≤ y ≤ aK/ , ≤ x ≤ π )and the fractal dimension of the strange repeller is d =1 . K = 7 , η = 0 . λ = 1 (top panel) and λ = − .
258 + ı .
445 (bottom panel), the space region is( − π ≤ y ≤ π, ≤ x ≤ π ) and the fractal dimension ofthe strange attractor is d = 1 . To study the validity of the fractal Weyl law we use theChirikov standard map [12]. We consider two models: themap with absorption that corresponds to the classicallimit of the quantum model studied in [13,10] (model 1)and the map with dissipation (model 2) also known asthe Zaslavsky map [20]. In the first model the dynamics is described by the map (cid:26) ¯ y = y + K sin( x + y/ x = x + ( y + ¯ y ) / π ) (2)where bar notes the new values of dynamical variablesand K is the chaos parameter. The map is written in itssymmetric form and all orbits going out of the interval − aK/ ≤ y ≤ aK/ K = 7 and varythe classical escape time by changing a in the interval0 . ≤ a ≤ η < (cid:26) ¯ y = ηy + K sin x ¯ x = x + ¯ y (mod2 π ) (3)with periodic boundary conditions in y ∈ [ − π, π ). Dueto dissipation and chaos the dynamics converges to a strangeattractor (see e.g. [21]).To construct the Ulam matrix approximant for a con-tinuous Perron-Frobenius operator in the two-dimensionalphase space we divide the space of dynamical variables( x, y ) on N = N x × N y cells with N x = N y . Then N c tra-jectories are propagated on one map iteration from a cell j , and the elements S ij are taken to be equal to a relativenumber N i of trajectories arrived at a cell i ( S ij = N i /N c and P i S ij = 1). Thus the matrix S gives a coarse-grainedapproximation of the Perron-Frobenius operator for thedynamical map. The map gives about K links for eachcell. We use N c values from 10 to 10 where the resultsare independent of N c . The fractal dimension d of thestrange repeller and attractor depends on system param-eters and is computed as a box counting dimension usingstandard methods [21].We also used another method to construct the Ulammatrix based on a one trajectory for the dynamics witha strange attractor in the model 2. In the one trajectoryUlam method we iterate one trajectory up to time t =100; after that we continue iterations of the trajectory upto time t = 10 and determine the matrix elements S ij as the ration between the number of transitions from cell j to cell i divided by the total number of transitions N c from cell j to all other cells (in this way P i S ij = 1). Thisapproach has certain advantages since it gives the Ulammatrix restricted to a dynamics only on the attractor. Fora given cell size this method gives a significantly smallermatrix size N a ≪ N since the number of cells N a locatedon the attractor is much smaller than the total number ofcells N . When speaking about the results based on the onetrajectory Ulam method we always directly specify this. The eigenvalues λ i and right eigenvectors ψ i of the matrix S ( S ψ i = λ i ψ i ) are obtained by a direct diagonalization.Examples of the eigenstates with maximal absolute valuesof λ i are shown in Fig.1. The fractal structure of eigen-states is evident. For the model 1 the measure is decreasing .Ermann, D.L.Shepelyansky: Ulam method and fractal Weyl law for Perron–Frobenius operators 3 Fig. 2. (Color online) Distribution of eigenvalues λ in thecomplex plane for the Ulam matrix approximant S forthe parameters of Fig.1 for the models 1 (top panel) and2 (center panel). Bottom panel shows the spectrum forthe model 2, with the same parameters as for the centralpanel, obtained via the one trajectory Ulam approximant(see text). Color/grayness of small squares is determinedby the value of overlap measure µ defined in the text andshown in the palette.due to absorption and λ <
1, the state with λ representsa set of strange repeller formed by orbits nonescaping infuture. For the model 2 all measure drops on the strangeattractor and in agreement with the Perron-Frobenius the-orem we have λ = 1 [15]. Other eigenstates with smallervalues of | λ | are located on the same fractal set as thestates with maximal λ but have another density distri-bution on it.The spectrum of matrix S in the complex plane isshown in Fig.2. It has a maximal real value λ isolated by a gap from a cloud of eigenvalues more or less homoge-neously distributed in a circle of radius r λ . For the model2 the dense part of the spectrum has r λ ≈ η (at least atsmall values of η ) that physically corresponds to the factthat η gives the relaxation rate to the limiting set of thestrange attractor. The gap between λ and other eigen-values in the model 1 is probably related to a dynamicson the strange repeller. According to [10] the decay rateof total probability in (2) is exponential in time with therate γ c = 0 .
270 (for parameters of Figs.1,2). This agreeswell with the numerical value λ = 0 . ≈ exp( − γ c ). Thedata of Fig.1 indicate that the states with i > λ ( i = 1). In aquantitative way this overlap can be characterized by anoverlap measure defined as µ i = P l ψ ( l ) | ψ i ( l ) | where thesum runs over all N cells. For µ close to unity an eigen-state ψ i has a strong overlap with the steady state ψ andsuch states can be viewed as higher mode excitations onthis domain. For µ ≪ ψ . The data of Fig.2 show thatstates with small values of | λ | have small µ . ln N l n N γ a=1a=2 η =0.3 η =0.6 Fig. 3. (Color online) Dependence of the integrated num-ber of states N γ with decay rates γ ≤ γ b = 16 on thesize N of the Ulam matrix S for the models 1 and 2 at K = 7. The fits of numerical data, shown by dashedstraight lines, give ν = 0 . , d = 1 .
643 (at a = 1); ν = 0 . , d = 1 .
769 (at a = 2); ν = 0 . , d = 1 . η = 0 . ν = 0 . , d = 1 .
723 (at η = 0 . | λ | →
0. The numberof states within a finite band with 0 ≤ γ ≤ γ b , where | λ | = exp( − γ/ N with theexponent ν < N . Typicalexamples of such a dependence are shown in Fig.3 for bothmodels.The spectrum for the one trajectory Ulam approxi-mant is shown in the bottom panel of Fig.2 (here N a =2308 while N = 12100). The spectrum with one trajectoryhas a structure similar to the spectrum of the usual Ulammethod. This shows that the spectrum is mainly deter-mined by the diffusive type excitations on the attractor. L.Ermann, D.L.Shepelyansky: Ulam method and fractal Weyl law for Perron–Frobenius operators d W / d γ γ Fig. 4. (Color online) Dependence of density of states dW/dγ on the decay rate γ for the Ulam matrix S forthe model 1 (top panel), and model 2 (center panel) atdifferent sizes N = N x × N y given in the inset. Bottompanel shows the spectral density for the one trajectoryUlam method for parameters of the central panel. Data areshown for parameters of Fig.1, the density is normalizedby the condition R dW/dγdγ = 1.Our matrix sizes N are sufficiently large and allow toreach asymptotic behavior in the limit of large N . This isconfirmed by the fact that the density of states dW/dγ in γ becomes independent of N as it is show in Fig. 4. Thisdirectly demonstrates that the Ulam method is stable forour models and that it converges to the continuous limitof the Perron-Frobenius operator. The density of statesfor the Ulam matrix obtained with one trajectory has thedensity of states very close to the one obtained by theusual Ulam method. This shows that the spectrum withfinite values of γ is determined by the dynamics on theattractor.The density is mainly determined by the cloud of statesin the radius r λ , the contribution of the isolated eigenvalue λ is only weakly visible at minimal γ . The density has abroad maximum around γ ≈
3, for the model 2 this valueis compatible with the value − η which determines theglobal relaxation rate to the strange attractor. It is in-teresting to note that for the model 1 the spectral den-sity of the Perron-Frobenius operator (Fig.4, top panel)is rather different from the spectral density in the cor-responding quantum problem (Fig.4 in [10]). Indeed, thedensities dW/dγ for the classical and quantum systemsare very different: the classical model 1 has one isolatedeigenvalue λ and a broad maximum around γ ≈
3. Thequantum model of [10] has a peaked distribution around γ c = − λ corresponding to the classical state at λ and a monotonically decreasing density at larger values of γ . At the same time the eigenstates with minimal γ arelocated on the strange set of trajectories nonescaping infuture times, both in the classical and quantum cases (see Fig.1 here and in [10]). Thus the semiclassical correspon-dence between classical and quantum cases of model 1 stillrequires a better understanding. d ν Fig. 5. (Color online) Fractal Weyl law for three differentmodels: models 1 (black points at K = 7) and 2 (red/graysquares at K = 12, blue/black triangles at K = 7) andH´enon map (green/gray diamonds at a = 1 .
2; 1 . b = 0 . ν is shown asa function of fractal dimension d of the strange forwardtrapped set in model 1 and strange attractor in model 2and Henon map. The straight dashed lines show the laws(4) (upper line) and (1) (bottom line). We used a ∈ [0 . , η ∈ [0 . ,
1] for model 2.We determine the exponent ν as it is shown in Fig. 3 forboth models at different values of parameters. At the sametime we compute the fractal dimension d of the strange setof trajectories nonescaping in future using box countingdimension with a box size ǫ . In this way the size of theUlam matrix is N = 1 /ǫ while the number of cells onthe fractal set scales as N f ∝ /ǫ d = N d/ . In this way wedetermine the dependence of ν on d . The data are shown inFig. 5. For the model 1 we find that the usual fractal Weyllaw with ν = d − d . Relatively small deviations can be attributed to afinite accuracy in computation of ν at finite matrix sizes.In contrast to that for the model 2 we find absolutelyanother relation which can be approximately described as N γ ∝ N ν , ν = d/ . (4)This relation works rather well for K = 12 while for K = 7 the deviations are a bit larger. We attribute this tothe fact that at K = 7 there is a small island of stabilityat η = 1 [22] which does not influence the dynamics in thecase of absorption (2) but can produce certain influencefor the dissipative case (3). To check that the law (4) worksfor other systems with strange attractors we computed ν and d for the H´enon map (¯ x = y + 1 − ax , ¯ y = bx , seee.g. [21]) at standard parameter values of a, b . The resultsconfirm the validity of the fractal Weyl law also for theH´enon map (see Fig. 5).The physical origin of the law (4) can be understood ina simple way: the number of states N γ with finite values .Ermann, D.L.Shepelyansky: Ulam method and fractal Weyl law for Perron–Frobenius operators 5 of γ is proportional to the number of cells N f ∝ N d/ onthe fractal set of strange attractor. Indeed, the results forthe overlap measure µ (see Fig. 2) show that these stateshave strong overlap with the steady state while the stateswith λ → N states have eigenvalues λ → N γ ∝ N f ∝ N d/ ≪ N hasfinite values of λ . We also checked that the participationratio ξ of the eigenstate of model 2 at λ = 1, defined as ξ = ( P l | ψ ( l ) | ) / P l | ψ ( l ) | , grows as ξ ∼ N f ∝ N d/ .The fractal Weyl laws (1) and (4) have two differentexponents ν but they correspond to two different situa-tions: for (1) the law describes the systems with absorp-tion when all measure escapes from the system and only asmall fractal set remains inside; for (4) all measure dropson a fractal set inside the system. Due to that reasons theexponents are different. d ν d d Fig. 6. (Color online) Fractal Weyl law for three differentmodels as a function of the dimension of the invariant set d ; the models and their parameters are the same as in Fig.5. The fractal Weyl exponent ν is shown as a function offractal dimension d of the strange repeller in model 1 andstrange attractor in model 2 and Henon map. The straightdashed line show the theoretical dependence ν = d / d of trajectories nonescaping in future and thefractal repeller dimension d for the case of model 1; thedashed straight line shows the theoretical dependence d = d / ν on d in Eqs. (1,4) canbe reduced to one dependence if to express ν via the fractaldimension d of the invariant sets. Indeed, for the model(2) all trajectories drop on the strange attractor which canbe considered as an invariant set with the fractal dimen-sion d = d . For the model 1 we have the set of trajectoriesnonescaping in future with dimension d , there is also thefractal set of trajectories nonescaping in the past whichhas also the dimension d due to symmetry between thefuture and the past present in the model 1 (symmetry toreflection x, y → − x, − y in (2)). Then the invariant set ofa strange repeller corresponds to the intersection of these two sets of trajectories nonescaping neither in the futureneither in the past with the fractal dimension d . As itis known, see e.g. [21], we have 2 = d + d − d so that d = d / d of the invariant set N γ ∝ N ν , ν = d / . (5)This global dependence is confirmed by the data shown inFig. 6.The numerical data for the one trajectory Ulam methodgives always N γ ∝ N a . This satisfies the relation (5) sinceby definition N a ∝ N d / . In summary, our results show that the Ulam method givesvery efficient possibility to study the spectral properties ofthe Perron-Frobenius operators for systems with dynam-ical chaos. Their spectrum is characterized by the fractalWeyl law with the Wyel exponent determined by the frac-tal dimension of dynamical system according to relations(1), valid for systems with absorption or chaotic scatter-ing, or (4), valid for dissipative systems with strange at-tractors.It is interesting to note that for dynamical systems theUlam method naturally generates directed Ulam networks[18] which have certain similarities with the propertiesof the Google matrix of the World Wide Web (WWW).However, for the model 2 and the H´enon map consideredabove, there is a finite gap between λ = 1 and other eigen-values while for the WWW there is no such gap [23,24].In this sense the above models are more close to ran-domized directed networks considered in [25] which havea relatively large gap. We note that the PageRank vec-tor ψ with λ = 1, used by Google for ranking of webpages, corresponds in our case to a strange attractor. Inthis case the probability p l ∼ ψ ( l ) is distributed over allcells N f ∝ N d/ occupied by the strange attractor. Thenumber of such cells grows infinitely with N that corre-sponds to a delocalized phase of the PageRank similar tothe cases discussed in [18]. In contrast to that the WWWis characterized by a localized PageRank with an effectivefinite number of populated sites independent of N . In spiteof that it is not excluded that the future evolution of theWWW can enter in a delocalized regime of the PageRank.Therefore, we think that the fractal Weyl law discussedhere can be useful not only for the Perron-Frobenius op-erators of dynamical systems but also for various types ofrealistic directed networks. References
1. H. Weyl, Math. Ann. , 441 (1912).2. G. A. Gamow, Z. f¨ur Phys. , 204 (1928).3. N. Moiseyev, Phys. Rep. , 211 (1998). L.Ermann, D.L.Shepelyansky: Ulam method and fractal Weyl law for Perron–Frobenius operators4. P. Gaspard, Chaos, Scattering and Statistical Mechanics ,Cambridge Univ. Press, Cambridge (1998).5. C. Gmachl, F. Capasso, E.E. Narimanov, J.U. N¨ockel,A.D. Stone, J. Faist, D.L. Sivco and A. Y. Cho, Science , 1556 (1998).6. J. Sj¨ostrand, Duke Math. J. , 1 (1990); M. Zworski,Not. Am. Math. Soc. , 319 (1999); J. Sj¨ostrand andM. Zworski, Duke Math. J. , 381 (2007); S. Nonnen-macher and M. Zworski, Commun. Math. Phys. , 311(2007).7. W. T. Lu, S. Sridhar and M. Zworski, Phys. Rev. Lett. ,154101 (2003).8. H. Schomerus and J. Tworzydlo, Phys. Rev. Lett. ,154102 (2004).9. J.P. Keating, M. Novaes, S.D. Prado and M. Sieber,Phys. Rev. Lett. , 150406 (2006); S. Nonnenmacher andM. Rubin, Nonlinearity , 1387 (2007).10. D.L. Shepelyansky, Phys. Rev. E , 015202(R) (2008).11. L. Ermann, G.G. Carlo, and M. Saraceno, Phys. Rev. Lett. , 054102 (2009); J.M. Pedrosa, G.G. Carlo, D.A. Wis-niacki and L. Ermann, Phys. Rev. E , 016215 (2009).12. B. V. Chirikov, Phys. Rep. , 263 (1979); B.Chirikov andD.Shepelyansky, Scholarpedia, (3): 3550 (2008).13. F. Borgonovi, I. Guarneri and D.L. Shepelyansky, Phys.Rev. A , 4517 (1991); G. Casati, G. Maspero andD.L. Shepelyansky, Physica D , 311 (1999).14. S.M. Ulam, A Collection of mathematical problems , Vol.8 of Interscience tracs in pure and applied mathematics,Interscience, New York, p. 73 (1960).15. M. Brin and G. Stuck,
Introduction to dynamical systems ,Cambridge Univ. Press, Cambridge, UK (2002).16. T.-Y. Li, J. Approx. Theory , 177 (1976); M. Blank,G. Keller, and C. Liverani, Nonlinearity , 1905 (2002);D. Terhesiu and G. Froyland, Nonlinearity , 1953 (2008).17. Z. Kov´acs and T. T´el, Phys. Rev. A , 4641 (1989);G. Froyland, R. Murray and D. Terhesiu, Phys. Rev. E , 036702 (2007).18. D.L. Shepelyansky and O.V. Zhirov,arXiv:0905.4162v2[cs.IR] (2009).19. L. Guillop´e, K.K. Lin, and M. Zworski, Commun. Math.Phys. , 149 (2004).20. G.M. Zaslavsky, Phys. Lett. A , 145 (1978); Scholarpe-dia, (5): 2662 (2007).21. E. Ott, Chaos in Dynamical Systems , Cambridge Univ.Press, Cambridge (1993).22. B.V. Chirikov, arXiv:nlin/0006013 (2000).23. S. Brin and L. Page, Computer Networks and ISDN Sys-tems , 107 (1998).24. A. M. Langville and C. D. Meyer, Google’s PageRank andBeyond: The Science of Search Engine Rankings , Prince-ton University Press (Princeton, 2006).25. O. Giraud, B. Georgeot and D. L. Shepelyansky, Phys.Rev. E80