Eigenfunctions of the Perron-Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolic area-preserving maps
Kensuke Yoshida, Hajime Yoshino, Akira Shudo, Domenico Lippolis
EEigenfunctions of the Perron-Frobenius operatorand the finite-time Lyapunov exponents inuniformly hyperbolic area-preserving maps
Kensuke Yoshida , , Hajime Yoshino , Akira Shudo andDomenico Lippolis Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji,Tokyo 192-0397, Japan Institute for Applied Systems Analysis, Jiangsu University, 212013 Zhenjiang, ChinaE-mail: [email protected]
Abstract.
The subleading eigenvalues and associated eigenfunctions of thePerron-Frobenius operator for 2-dimensional area-preserving maps are numericallyinvestigated. We closely examine the validity of the so-called Ulam method, anumerical scheme believed to provide eigenvalues and eigenfunctions of the Perron-Frobenius operator, both for linear and nonlinear maps on the torus. For the nonlinearcase, the second-largest eigenvalues and the associated eigenfunctions of the Perron-Frobenius operator are investigated by calculating the Fokker-Planck operator withsufficiently small diffusivity. On the basis of numerical schemes thus established, wefind that eigenfunctions for the subleading eigenvalues exhibit spatially inhomogeneouspatterns, especially showing localization around the region where unstable manifoldsare sparsely running. Finally, such spatial patterns of the eigenfunction are shown tobe very close to the distribution of the maximal finite-time Lyapunov exponents.
Submitted to:
J. Phys. A: Math. Theor. a r X i v : . [ n li n . C D ] J a n igenfunctions of Perron-Frobenius operator
1. Introduction
Uniformly hyperbolic systems, or, more loosely, strongly chaotic systems, exhibitunstable dynamics everywhere and the orbits asymptotically explore the whole phasespace. Locally, or for short times, chaotic dynamical systems are characterized bylocal instability everywhere in the phase space, but, after a sufficiently long time,local information is averaged out, and a uniquely determined equilibrium distributionis achieved eventually.This does not necessarily mean that no characteristic structures remain andeverything proceeds homogeneously both in space and time, rather, there appear richand varied structures worth examining even in a uniformly hyperbolic system. Recentdecades have actually witnessed significant progress in characterizing spatial patternsand temporal behaviors of strongly chaotic systems. This includes developing a newconcept such as the almost-invariant set [1, 2], which is proposed to capture regionsin which the orbits tend to stay for a relatively long time compared to other regions.Another important finding as for the spatial structure hidden in the uniformly hyperbolicsystem is the position dependence of the escape rate [3, 4, 5, 6]. The escape rate measureshow fast the orbits initially launched in the phase space enter a hole punched at a certainplace. For uniformly hyperbolic systems, the escape rate is supposed to be proportionalto the hole size, and this is actually the case in the leading order. However, it wasfound that there is a substantial and actually observable correction originating from theperiodic orbits of the system. Further analyses unveiled that the position dependence ofthe escape rate can be observed not only for mathematically well controlled situations[3, 4, 5], but also for more generic settings [7, 6, 8].The strange eigenmode is an additional important topic, which should be mentionedin this context. In the field of the chaotic advection, it was reported that the distributionof the tracer particles stirred by external forces shows a fractal-like inhomogeneouspattern for a long time [9]. Namely, the residence time of the tracer particles dependson the position in the state space, and the pattern is called the strange eigenmode [9].Note that the existence of the strange eigenmode for the advection-diffusion equationhas been rigorously proved in the systems satisfying a suitable condition [10]. However,it is yet unclarified what geometric structures of the underlying deterministic systemthe strange eigenmode reflects [11].In this paper, we explore spatial and temporal properties of strongly chaoticsystems, especially 2-dimensional area-preserving maps, by investigating the eigenvaluesand eigenfunctions of the Perron-Frobenius operator. The Perron-Frobenius operatordescribes the time-evolution of distribution functions defined on the phase space. Inparticular, we focus on the leading sub-unit eigenvalue of the Perron-Frobenius operatorand associated eigenfunction, which are referred to respectively as the second-largesteigenvalue and the second eigenfunction in our subsequent descriptions. As long as thesecond-largest eigenvalue is nonzero and is isolated from other eigenvalues, it controlsthe decay of correlations of observables, as well as the relaxation to the steady state [12]. igenfunctions of Perron-Frobenius operator C : T (cid:55)→ T , known as theArnold cat map: C : ( x, y ) (cid:55)→ ( x + y, x + 2 y ) mod 1 , (1)and another one is the so-called perturbed cat map given as C (cid:15),ν : ( x, y ) (cid:55)→ C ◦ F (cid:15),ν ( x, y ) mod 1 , (2)where F (cid:15),ν : ( x, y ) (cid:55)→ (cid:16) x − (cid:15)ν sin(2 νπy ) , y (cid:17) . (3)Here the perturbation strength (cid:15) is assumed to be real and positive, and the perturbationfrequency ν is a positive integer.There are several works in the physics literature, where eigenvalues andeigenfunctions of the Perron-Frobenius operator have been calculated numerically[16, 17, 18, 19, 20, 21]. Apparently, numerical calculations can easily be implemented,for instance using the Ulam method introduced below. However, because of the highlypathological nature of eigenfunctions of the Perron-Frobenius operator in chaos, possiblycoming from the fact that they are hyperfunctions [22], some straightforward numericalschemes can be unstable and thus unreliable. This is particularly so for the cat map C .Hence one of our main aims in this paper is to make clear to what extent one can accesseigenfunctions of the Perron-Frobenius operator with numerical calculations. This pointof view has not been seriously examined so far, although rigorous mathematical worksstrongly suggest that eigenfunctions, and eigenvalues as well, are not so easily handled.The answer depends on the functional spaces on which the Perron-Frobenius operatoracts, and rigorous arguments cannot be developed without specifying such settings.After establishing a reliable numerical scheme, we focus on the spatial signature ofthe second eigenfunctions. We pay particular attention to fractal-like- and at the same igenfunctions of Perron-Frobenius operator §
2, we introduce the Perron-Frobenius operator. In § §
4, we provide numerical results for second-largesteigenvalues and associated eigenfunctions of the Perron-Frobenius operator obtainedfirst by using the Ulam method, and then by applying the Fokker-Planck to approximatethe Perron-Frobenius operator. In §
5, we present numerical observations showingthat eigenfunctions of the leading sub-unit eigenvalue exhibit spatially inhomogeneouspatterns, and strong localization appears around the region where unstable manifoldsare sparsely running. In §
6, we introduce the maximum finite-time Lyapunov exponent(mFT Lyapunov exponent, for short) to characterize the spatial pattern of the secondeigenfunctions of the Perron-Frobenius operator. In §
7, we draw conclusions and discussour results.
2. The Perron-Frobenius operator
The Perron-Frobenius operator L for a map f = C or C (cid:15),ν is introduced as L ρ ( x ) = (cid:90) T δ ( x − f (¯ x )) ρ (¯ x )d¯ x , (4)where x = ( x, y ) ∈ T and the function ρ ( x ) is defined on the phase space T .In the case of an area-preserving map, the Perron-Frobenius operator becomesunitary when it acts on the L space [23]. As a result, the Perron-Frobenius operatorfor an area-preserving map in L does not have any non-zero sub-unit eigenvalues, wheninstead we are interested in the decay of correlations, as mentioned in the introduction.For that reason, we shall consider the Perron-Frobenius operator acting on otherfunctional spaces. A typical setting in the mathematical treatment is to introduce aproper functional space for which the Perron-Frobenius operator is quasi-compact. Aquasi-compact operator has a positive essential spectral radius, defined as the minimumradius of the disk such that all eigenvalues lying outside the disk are isolated and haveat most finite degeneracy [12]. In particular, the maximal eigenvalue is simple andequal to one, if the system is mixing, and its eigenfunction is a uniform distributionon T , for an area-preserving map. The quasi-compactness is a necessary condition forthe Perron-Frobenius operator to have a non-zero second-largest eigenvalue. It is notguaranteed that the desired functional space can be accessed within reach of numericalcomputations.In the subsequent sections, we investigate the validity of numerical schemesapproximating the Perron-Frobenius operator. We denote eigenvalues and thecorresponding eigenfunctions of the Perron-Frobenius operator by σ and ψ respectively,and do not distinguish the ones obtained numerically from exact eigenvalues and igenfunctions of Perron-Frobenius operator σ hereafter.
3. Numerical approximation of Perron-Frobenius operator I: the Ulammethod
The Ulam method is one of the most commonly used numerical schemes forapproximating the Perron-Frobenius operator [24]. Let us consider a map f anda partition of the phase space R = { R , R , . . . } . In general, the partition R = { R i } i =1 , ,...,n of the space X is a family of subsets such that X = (cid:83) ni =1 R i and R i ∩ R j = ∅ ( i (cid:54) = j ). In the Ulam method, one approximates the Perron-Frobeniusoperator for the map f with the Ulam matrix, defined as( L ) ij = m ( R i ∩ f − ( R j )) m ( R i ) ( i, j = 1 , , . . . ) , (5)where f − is the inverse mapping of f and m ( · ) the Lebesgue measure normalized inthe phase space.It was shown for some maps that the eigenfunction of L associated with theeigenvalue 1 weakly converges to a measure which is absolutely continuous in the sense ofLebesgue or Sinai-Ruelle-Bowen, for an infinitely refined partition [25, 26, 27, 28, 29, 30].The Ulam method has also been used to approximate sub-unit eigenvalues and theireigenfunctions [19, 20, 21]. On the other hand, rigorous results on the Ulam methodare rather limited [31, 32, 13], and the results for hyperbolic linear maps are brieflymentioned in our subsequent arguments.In the following, we will apply the Ulam method to the cat map and the perturbedcat map in order especially to explore to what extent the Ulam method works incalculating sub-unit eigenvalues and the corresponding eigenfunctions for the Perron-Frobenius operator.We numerically construct the Ulam matrix in the following way. First put N j initial points randomly chosen in the j -th partition R j ∈ R where N j is taken to beproportional to the area of R j , and then count the points whose backward iterations arecontained in the i -th partition R i . We denote the number of such points by N ij : N ij = { x ( n ) ∈ R j ( n = 1 , , . . . , N j ) | f − ( x ( n ) ) ∈ R i } . (6)The ( i, j )-component of the Ulam matrix is set to ( L ) ij = N ij (cid:80) i N ij . In this subsection, we apply the Ulam method for theunperturbed cat map C by using the Markov partitions to generate the Ulam matrix.Given a partition R = { R , R , . . . , R N } , where each region R i has a boundary ∂R i = ∂ S R i (cid:83) ∂ U R i , and ∂ S R i and ∂ U R i respectively in the stable and unstable manifolds, R isa Markov patition if C ( (cid:83) Ni =1 ∂ S R i ) ⊂ (cid:83) Ni =1 ∂ S R i and C − ( (cid:83) Ni =1 ∂ U R i ) ⊂ (cid:83) Ni =1 ∂ U R i [33]. igenfunctions of Perron-Frobenius operator M A andthe BSTV partition by M B .For a Markov partition M , either M A or M B , the finer partition obtained by M ∨ C ( M ) = { E M ( x ) ∩ E C ( M ) ( x ); x ∈ T } , (7)is also a Markov partition where C is the cat map. Here E M ( x ) denotes the elementsof the partition M containing the point x . An example illustrating the procedure givenin equation (7) is shown in figure 1. The partition A ∨ B represents the finest partitiongenerated from the partitions A = { A , A , A , A } and B = { B , B , B } . Figure 1.
Illustration of the partition A ∨ B , which is generated by the partitions A and B . The finer partition generated iteratively up to the t max -th image of the partition C t max ( M ) is denoted by M t max = M ∨ C ( M ) ∨ C ( M ) . . . ∨ C t max ( M ) . (8)Note that M t max is also a Markov partition. We call M t max A (resp .B ) the t max -th AW(resp.BSTV) partition. As shown in figure 2, the AW partition M A is composed of tworectangles, and does not have any spatial symmetry whereas the BSTV partition M B has three rectangles, and is point symmetric in the torus T . igenfunctions of Perron-Frobenius operator (a) (b) (c)(d) (e) (f) Figure 2. (a)-(c) The AW partition M t max A with (a) t max = 0, (b) t max = 3 and (c) t max = 4. (d)-(f) The BSTV partition M t max B with (d) t max = 0, (e) t max = 3 and (f) t max = 4. Let L A (resp .B ) be the Ulam matrix generated with the AW(resp. BSTV) partition.Brini et al. proved, by extending the results in [34], that the sets of the eigenvaluesof L A and L B are { , ( λ ( C ) ) − , } and { , ( λ ( C ) ) − , ( λ ( C ) ) − , } , respectively [31]. Thenumerically obtained eigenvalues of L A (resp .B ) are shown in figure 3. One can find thatthe second-largest eigenvalues are ( λ ( C ) ) − (resp . ( λ ( C ) ) − ), independent of the cardinalityof the partitions. The complex eigenvalues placed around the origin (0 ,
0) are spurious,that appear as a result of numerical artifacts. All spurious eigenvalues approach theorigin of the complex plane with the increase of the initial points N j [cf. equation (6)].Here λ ( C ) = (3 + √ / ∂C i /∂x j . We call λ ( C ) the stability multiplier of the cat map. The multiplicity of the non-zero eigenvaluesof L A (resp .B ) is 1 [31]. To our knowledge, it has not been clarified what functional spacethe Perron-Frobenius operator approximated by the Ulam matrix acts on, when thecardinality of the partition is taken to be infinity [35]. On the other hand, the Perron-Frobenius operator for the area-preserving map becomes unitary if it is defined as theoperator acting on the Hilbert space L ( T ), and in such a case the second-largesteigenvalue turns out to be 0 [23]. Even if one restricts the functional space on which thePerron-Frobenius operator acts to smooth functions, sub-leading eigenvalues, which arecalled the Ruelle-Pollicott resonances , are shown to be all zero [18, 36]. These resultsimply that the resulting eigenvalues depend on the method we use and the functionalspace we consider. igenfunctions of Perron-Frobenius operator (a) (b) Figure 3.
Eigenvalues of the Ulam matrix generated with the Markov partitions (a)AW partition and (b) BSTV partition.
The second eigenfunctions of L A or L B obtained by performing numericalcalculations are presented in figure 4. The second eigenfunction of L A appears to bea binary valued function, and the second eigenfunction of L B seems a ternary valuedfunction within the scale of the color bar of the figure, strongly suggesting that they aretruly binary or ternary, although no rigorous proofs exist, to our knowledge. We showthe eigenfunction associated with the third-largest eigenvalue of L B in figure 4(e)-(f).Although the third-largest eigenvalue of L B is identical to the second-largest eigenvalueof L A , the eigenfunction associated with the third-largest eigenvalue of L B looks aternary valued function within the scale of the color bar of the figure. Therefore, even ifthe eigenvalues take the same value, the eigenfunctions of the Ulam matrices can exhibitdifferent patterns depending on the choice of the Markov partition.On the other hand, one can find some properties common to eigenfunctions obtainedfrom the AW partition and BSTV partition. The pattern of the second eigenfunctionbecomes finer with increase in t max (compare the left and the right panel in figures 4(a)and (b)). The second eigenfunction of the Ulam matrix generated with M t max A (resp .B ) showsthe pattern with a smaller scale of that obtained by M t max − A ( B ) . Because of this property,we can expect that the second eigenfunction tends to a fractal function as t max → ∞ .The second eigenfunction is smooth along unstable manifolds and wildly oscillatingin the direction of stable manifolds. Such a pattern is commonly observed in numericallyobtained eigenfunctions associated with sub-unit eigenvalues of the Ulam matrix for thestandard map [13]. Furthermore, such patterns can be obtained not only by the Ulammethod but also by other methods [16, 17]. In particular, it was reported that in thequadbaker map, that is also uniformly hyperbolic, piecewise linear, and area-preserving,the absolute value of the second eigenfunction generated by a certain partition smoothlychanges along the unstable manifold, and it wildly oscillates along the stable manifold.It was rigorously proved that it converges to the second eigenfunction of the Perron-Frobenius operator on a suitable functional space as the partition tends to an infiniterefinement [13]. Note that in [13] the author used a partition consisting of equally-sizedsquares, whose sides are parallel to the stable and unstable manifolds. igenfunctions of Perron-Frobenius operator (a) (b)(c) (d)(e) (f) Figure 4. (a)-(b) The second eigenfunction generated using the AW partition with(a) t max = 3 and (b) t max = 4. (c)-(d) The second eigenfunction generated using theBSTV partition with (c) t max = 3 and (d) t max = 4. (e)-(f) The third eigenfunctiongenerated using the BSTV partition with (e) t max = 3 and (f) t max = 4. igenfunctions of Perron-Frobenius operator We here investigate how the calculation based on theUlam method with a uniform discretization of the phase space works for the unperturbedcat map C . The equally-spaced partition will hereafter be referred to as the uniformpartition . Previously, there has been an attempt in the same direction [20]. As willbe illustrated below, however, the Ulam method with uniform partitions is not stableenough to obtain eigenvalues and the corresponding eigenfunctions, and thus we haveto say that it does not necessarily lead to conclusive results.We here construct the uniform partition R M by dividing the phase space into the M × M equally spaced squared boxes and prepare the Ulam matrix associated withthe partition R M . Note that for the unperturbed cat map C and any M , each matrixelement takes the value of either 0 or 1 / L M . Notice that thereappear several second-largest eigenvalues for every given M , taking the same absolutevalue within numerical errors, and the number of second-largest eigenvalues depends on M . Note that for M = 97, numerous second-largest eigenvalues appear so as to form acircle in the complex plane. Additionally, as seen in figure 5(b), the absolute value ofthe second-largest eigenvalue largely fluctuates even with increase in M , and does notseem to converge to an asymptotic value, up to M = 100. Such a fluctuating behaviorof the eigenvalues has been already reported in [32].It was proved that the Ulam matrix generated with the uniform partition has aself-similar structure, meaning that L kM is composed of the combination of L M where k is a positive integer [32]. Although L M has such a clear structure, it is only proved thatall of the eigenvalues of the L M are contained in the set of the eigenvalues of the L kM for any given positive integer k [32]. We cannot conclude whether the second-largesteigenvalue of L M converges to a certain value in the limit of M → ∞ or not. igenfunctions of Perron-Frobenius operator (a) (b) Figure 5. (a) Second-largest eigenvalues of the Ulam matrix generated with theuniform partition. The outer solid circle is the unit circle. For M = 97, numeroussecond-largest eigenvalues are overlapped and form a circle. (b) The absolute valueof the second-largest eigenvalue on the real axis generated with the uniform partitionplotted as a function of the partition size M . Since the entries of the Ulam matrix generated with the uniform partition are allreal but the matrix is non-symmetric, the eigenfunctions become complex-valued, ingeneral. We hereafter plot their magnitudes. In figure 6, we first present a pair ofsecond eigenfunctions, whose eigenvalues have the same absolute value but differentphases. It was proved that eigenfunctions for L M show repeating patterns [32], howeverit is not clear whether the mathematical statement presented in [32] justifies periodicpatterns observed in figure 6, and also the results reported in [20]. (a) (b) Figure 6.
Second eigenfunctions (absolute value) for the cat map obtained using theUlam matrix generated with the uniform partition. (a) and (b) are associated withdifferent second-largest eigenvalues. M = 100 was taken. Since, as seen above, the numerical evaluation of the eigenvalues is rather unstable,it would be reasonable to expect that the same is true for eigenfunctions. In order to igenfunctions of Perron-Frobenius operator M and M (cid:48) as∆ ψ ( M,M (cid:48) ) i,j := 1¯ M × ¯ M ¯ M × ¯ M (cid:88) k =1 | ψ ( M ) i ( k ) − ψ ( M (cid:48) ) j ( k ) | . (9)Here ψ ( M ) i ( k ) denotes the value of a second eigenfunction supported on a finer uniformgrid of ¯ M × ¯ M squares at the k -th interval (1 ≤ k ≤ ¯ M × ¯ M ). The index i runs in theset of the second eigenfunctions, and ¯ M stands for the least common multiple of M and M (cid:48) . Obviously, ∆ ψ ( M,M (cid:48) ) i,j is non-negative and equal to 0 if M = M (cid:48) . If min i,j ∆ ψ ( M,M (cid:48) ) i,j is close to zero, then L M and L M (cid:48) have second eigenfunctions with a similar spatialpattern. (a) (b) Figure 7. (a) The plot of min i,j ∆ ψ ( M,M (cid:48) ) i,j . (b) The normalized distribution ofmin i,j ∆ ψ ( M,M (cid:48) ) i,j for M > M (cid:48) . The result of figure 7(a) tells us that the values of ∆ ψ ( M,M (cid:48) ) i,j are non-negligiblysmall, except on the trivial line ( M = M (cid:48) ). As shown in figure 7(b), the distribution ofmin i,j ∆ ψ ( M,M (cid:48) ) i,j exhibits a bell-shaped profile, and its mean value is around 0.4. In figure8(a), we present a pair of second eigenfunctions, whose difference ∆ ψ ( M,M (cid:48) ) i,j is relativelysmall compared with the mean value of min i,j ∆ ψ ( M,M (cid:48) ) i,j . These spatial patterns happento be similar to each other, but such a pair is rarely found. As shown in figure 8(b) and(c), the typical pairs of second eigenfunctions, whose differences ∆ ψ ( M,M (cid:48) ) i,j are close tothe mean value of min i,j ∆ ψ ( M,M (cid:48) ) i,j , show totally different spatial patterns. From thesecalculations, we must conclude the Ulam method with uniform partitions is not suitableto calculate both eigenvalues and eigenfunctions in the case of the unperturbed cat map C . igenfunctions of Perron-Frobenius operator (a)(b)(c) Figure 8.
Pairs of second eigenfunctions (absolute values) whose differences arerelatively small. (a) (
M, M (cid:48) ) = (92 ,
94) with ∆ ψ ( M,M (cid:48) ) i,j (cid:39) . M, M (cid:48) ) =(98 , ψ ( M,M (cid:48) ) i,j (cid:39) . M, M (cid:48) ) = (99 , ψ ( M,M (cid:48) ) i,j (cid:39) . igenfunctions of Perron-Frobenius operator Next we apply the Ulam method to the perturbed cat map C (cid:15),ν . It was shown thatthe perturbed cat map is topologically conjugate to the cat map if the condition0 < (cid:15) < ( √ − / √ π ≈ .
069 is satisfied [37, 38, 39]. Furthermore, we show thatthe eigenvalues of the Jacobian for the perturbed cat map are both real and positivefor 0 ≤ (cid:15) < / π ≈ .
159 (see § ν = 1 but the following numericalresults are essentially the same in the ν > R M introduced in the previous subsection.Figures 9(a)-(c) display the second-largest eigenvalues of the Ulam matrix. In eachcase, there appear multiple second-largest eigenvalues having the same modulus butdifferent phases. As the perturbation strength (cid:15) increases we find a tendency of themultiple second-largest eigenvalues to come close to each other.As shown in figure 9(d), the convergence of the second-largest eigenvalues as afunction of the size M × M of the Ulam matrix improves compared to the case of theunperturbed cat map C , although fluctuations are still noticeable. A similar scenario isfound in the patterns of the corresponding eigenfunctions. As seen in figure 10, spatialpatterns depend on M . We notice, however, that the difference of spatial patternsbetween the case with M = 99 and M = 100 is even smaller as the perturbation strength (cid:15) is increased. In addition, it should be noted that localized regions become more clearlyrecognizable, and computational stability improves as (cid:15) increases. This localized patternreminds us of the scarring phenomenon in the quantum chaotic system [40]. As will beclosely discussed in the next section, localized regions appear along unstable manifoldsand so this will become an important signature characterizing the pattern of secondeigenfunctions. igenfunctions of Perron-Frobenius operator (a) (b)(c) (d) Figure 9.
Second-largest eigenvalues of the Ulam matrices for the perturbed catmap with (a) (cid:15) = 0 .
05, (b) (cid:15) = 0 . (cid:15) = 0 . ν = 1 is taken. The broken line isthe circle whose radius equals | σ | for M = 100. The outer solid curve represents theunit circle. (d) The absolute value of a second-largest eigenvalue of the Ulam matrixas a function of M . ν = 1 is taken. igenfunctions of Perron-Frobenius operator (a) (b)(c) (d)(e) (f) Figure 10.
Magnitudes of the second eigenfunctions of the Ulam matrices for theperturbed cat map with (a) (cid:15) = 0 . M = 99, (b) (cid:15) = 0 . M = 100, (c) (cid:15) = 0 . M = 99, (d) (cid:15) = 0 . M = 100, (e) (cid:15) = 0 . M = 99, and (f) (cid:15) = 0 . M = 100. ν = 1 is taken all through. In closing this section, we briefly summarize the results obtained so far. The Ulam igenfunctions of Perron-Frobenius operator
4. Numerical approximation of Perron-Frobenius operator II:Fokker-Planck operator method
The Ulam method is unstable for calculating eigenvalues and eigenfunctions of thePerron-Frobenius operator for the unperturbed cat map C , whereas it works reasonablywell for the perturbed cat map C (cid:15),ν . In order to confirm that the characteristiclocalization patterns observed in the second eigenfunctions shown in figure 10 areinherent of the perturbed cat map, we approximate the evolution operator using aFourier basis. One reason for this choice is, as shown below, that it yields betterconvergent results, and can be used to examine the Perron-Frobenius operator by takingthe noiseless limit of the Fokker-Planck operator. Another reason is that it is worthinvestigating physically in its own right since it naturally introduces a cutoff of spatialresolution, which is inevitable when the system is subject to external noise or coupledwith a heat bath, etc. Introducing noise, we replace the Perron-Frobenius operator withthe Fokker-Planck operator.The Fokker-Planck operator for the map f = C or C (cid:15),ν is defined as follows[16, 41, 36]. L ∆ ρ ( x ) = (cid:90) T (cid:88) k ∈ Z e π i k · ( f ( x ) − ¯ x ) − k ∆ ρ (¯ x )d¯ x , (10)where k denotes the wavenumber vector and ∆ a positive real parameter controlling thevariance of the Gaussian integral kernel of the operator, which represents the amplitudeof the noise, and makes the eigenvalues and eigenfunctions converge faster by suppressingthe high-frequency Fourier modes. We call ∆ diffusivity hereafter.If one restricts the functions on which the Fokker-Planck operator L ∆ acts to theHilbert space L ( T ), the Fokker-Planck operator L ∆ becomes a matrix of entries equalto the transition probabilities between Fourier modes k and q [41, 36]( ˜ L ∆ ) kq = e − ∆ q (cid:90) T e π i( q · x − k · f ( x )) d x . (11) igenfunctions of Perron-Frobenius operator L for area-preserving maps acting on L isunitary. This implies that the sub-unit eigenvalues of the Perron-Frobenius operatorare all zero since the spectrum lies entirely on the unit circle of the complex plane. Onthe other hand, Faure and Roy proved that, for a class of nonlinear uniformly hyperbolic2-dimensional maps, which includes C and C (cid:15),ν with small enough (cid:15) , the eigenvalues ofthe Fokker-Planck operator ˜ L ∆ tend to the Ruelle-Pollicott resonances of the Perron-Frobenius operator as ∆ → C (cid:15),ν with a larger perturbation, we may identify the numericallyobtained spectra of ˜ L ∆ with the Ruelle-Pollicott resonances, in the noiseless limit.Since we are not aware of any proof of that, we refrain from using the Ruelle-Pollicottterminology in what follows. For the cat map C , we can easily show the sub-unit eigenvalues of the Fokker-Planckoperator are all zero for any ∆. Since the matrix elements of ˜ L ∆ can be explicitly givenas [36], ( ˜ L ∆ ) kq = e − ∆ q δ k · ¯ C, q , (12)and ˜ L ∆ only has one nonzero diagonal element ( ˜ L ∆ ) , while all other diagonal entriesare equal to 0. Here ¯ C = (cid:32) (cid:33) and = (0 , For the perturbed cat map C (cid:15),ν , the integral (11) is evaluated in terms of the Besselfunction of the first kind I ω , as follows [41]:( ˜ L ∆ ) kq = (cid:40) e − ∆ q δ , − Q x ( − Qyν I − Qyν (cid:0) ( k x + k y )2 π (cid:15)ν (cid:1) Q y ν ∈ Z , . (13)where ( Q x , Q y ) = k · C − q . For numerical calculations, we truncate the wavenumber, sothat k , q ∈ [ − K, K ] × [ − K, K ] , K >
0, and the Fokker-Planck operator ˜ L ∆ is expressedas a (2 K + 1) × (2 K + 1) -dimensional matrix.In the following calculation, we take ν = 1 because the second-largest eigenvalueis real-valued, isolated, and non-degenerate in magnitude, as explained below. On theother hand, the number of second-largest eigenvalues increases for ν ≥ igenfunctions of Perron-Frobenius operator We first provide numerical evidencethat the Fokker-Planck operator for the perturbed cat map has non-zero second-largesteigenvalues. In figure 11, we give the location of the second-largest eigenvalues for (cid:15) = 0 .
1. It is clearly seen that with decrease in diffusivity ∆ the location of the second-largest eigenvalue gets stabilized and tends to be fixed. In this calculation, the maximumwavenumber is taken to be K = 100. As shown in table 1, the second-largest eigenvaluegradually decreases with decrease in the perturbation (cid:15) . This behavior implies that allsub-unit eigenvalues tend to 0 as (cid:15) →
0, which is consistent with the mathematicalresult for the cat map [36].Perturbation (cid:15)
Second-largest eigenvalue σ .
12 0 . . . . . . . . . .
08 0 . . . . .
06 0 . . . . .
04 0 . . . . Table 1.
The second-largest eigenvalue of the Fokker-Planck operator with differentperturbation (cid:15) . We take K = 100 and ∆ = 10 − .(a) (b) Figure 11.
The second-largest eigenvalue of the Fokker-Planck operator for theperturbed cat map C (cid:15),ν . We take (cid:15) = 0 . , ν = 1 and K = 100. The circle representsthe unit circle. (b) is the magnification of (a). We can more clearly show that the second-largest eigenvalues actually converge tocertain non-zero constant values, by plotting the position of eigenvalues as a function of∆. Figure 12 illustrates these results for several values of the maximum wavenumber K .For (cid:15) = 0 .
1, the curves relative to different K are indistinguishable, as they all collapseto a single curve, and they all tend to the same value for ∆ →
0. It is noted that theconvergence is stable compared to the result obtained using the Ulam method. igenfunctions of Perron-Frobenius operator (cid:15) = 0 .
15, the cutoff K = 10 is not enough to achievewell-convergent result, but for larger K values, we find that | σ | seems to stabilize on alimit value for decreasing diffusivity, implying that the second-largest eigenvalue takesa non-zero value in this case as well in the noiseless limit. We have verified that allthese results do not change even if we replace the numerical precision from double toquadruple.We can further confirm that the second-largest eigenvalue thus obtained is isolatedfrom the third-largest and other eigenvalues as noticed in figure 13. Here we plot thedistance δσ , := | σ | − | σ | , where σ and σ denote the second and third-largesteigenvalue for the Fokker-Planck operator, respectively. The third-largest eigenvalueis also real-valued. The distance δσ , again tends to a fixed value, meaning that thereexists a finite gap between the second-largest and other eigenvalues and so the second-largest one is isolated.These numerical calculations altogether strongly suggest that the second-largesteigenvalue of the Fokker-Planck operator is real valued, strictly positive, and isolated,which is in contrast with what seen for the linear cat map C . When developing arigorous argument, one must treat the order of the limit ∆ → K → ∞ carefully. (a) (b) Figure 12.
Absolute value of the second-largest eigenvalue of the Fokker-Planckoperator for the perturbed cat map vs the diffusivity ∆. (a) (cid:15) = 0 . (cid:15) = 0 . igenfunctions of Perron-Frobenius operator Figure 13.
The difference between the absolute values of the second-largest eigenvalueand third-largest eigenvalue as a function of K . We take ∆ = 10 − . On the basis of the well convergentnature of eigenvalues confirmed above, we might expect that the associated secondeigenfunctions can also be obtained via the Fokker-Planck operator. As is shown infigure 14, it is indeed the case. We explain this by focusing on the K -dependence with∆ being fixed. As is seen at the first column of figure 14, the pattern becomes stabilizedas the maximum wavenumber increases. Since we may expect that the smaller ∆ is, thefiner spatial structure is developed, there should exist a minimum cutoff K such thatthe spatial pattern produced by the Fokker-Planck operator with diffusivity ∆ could beresolved. The convergence of the observed patterns could be achieved because such adesired condition is satisfied in this case.In order to estimate the maximum wavenumber necessary to provide well-convergent eigenfunctions, we present eigenfunctions in the wavenumber representation.As shown in figures 14 (b),(d) and (f), the distribution in the wave number space exceedsthe wave range if K = 10 is taken. However, the distribution is reasonably confined inthe range with K = 50, and well confined in the case of K = 100. These observationsare consistent with the phase space representations shown in the left panels in figure 14the spatial patterns for the cases with K = 50 and 100 are almost the same, suggestingthat these results are convergent.As shown in figure 15, the localized region of the second eigenfunction becomesnarrower with decrease of the diffusivity ∆. From this observation, we would expectthat the spatial pattern tends to a fractal set as ∆ goes to zero, while maintaining thespatial inhomogeneity that surrounds the fixed point.From this convergent nature in the second-largest eigenvalues and the correspondingeigenfunctions for the Fokker-Planck operator, we might expect that the second-largesteigenvalue for the Perron-Frobenius operator takes a finite value, and observed spatialinhomogeneity survives even in the limit of ∆ →
0. The small diffusivity limitis highly nontrivial and subtle, in particular for the eigenfunctions since they mustbecome a class of hyperfunctions. The eigenfunction for the quadbaker map, which was igenfunctions of Perron-Frobenius operator igenfunctions of Perron-Frobenius operator (a) (b)(c) (d)(e) (f) Figure 14.
Magnitude of the second eigenfunctions of the Fokker-Planck operatorfor the perturbed cat map (a)-(b) K = 10, (c)-(d) K = 50 and (e)-(f) K = 100.Left panels: phase space representation. Right panels: wavenumber representation. (cid:15) = 0 . ν = 1 and ∆ = 5 × − are taken. igenfunctions of Perron-Frobenius operator (a) (b) Figure 15.
The second eigenfunctions of the Fokker-Planck operator for theperturbed cat map (a) ∆ = 9 × − and (b) ∆ = 10 − . We take (cid:15) = 0 . ν = 1and K = 100. Our numerical calculations strongly suggest that the second-largest eigenvalue ofthe Fokker-Planck operator of the perturbed cat map tends to a finite positive value as∆ →
0, and the second eigenfunction exhibits localization similarly observed in the oneobtained using the Ulam method (cf. figure 10). In the subsequent sections, we willexplore the origin of localization and what quantity characterizes localization patterns.
5. Inhomogeneity of unstable manifolds
It is worth noting that, as was observed in spatial patterns obtained via the Ulammethod, strongly localized regions appear in the eigenfunctions of the evolution operator.This tells us that the second eigenfunction, which provides information on the decayof correlations in the dynamics, is spatially inhomogeneous. Such a signature remindsus of the position dependence of the escape rate, a subject that has recently attractedmuch attention [3, 4, 5, 6].Here we just point out that the localized region in the eigenfunction patterns alsofeatures a sparse unstable manifold. As displayed in figure 16(a), the density of anunstable manifold emanating from a fixed point is not uniform but inhomogeneous.Sparse regions seem to coincide with localized regions of the second eigenfunction ofthe Fokker-Planck operator (cf. figure 14) and the Ulam matrix (cf. figure 10). Spatialprofiles of the second eigenfunction of the Fokker-Placnk operator for smaller diffusivitycase is close to those obtained based on the Ulam method.In what follows, we show that unstable manifolds are sparser in proximity of shortperiodic orbits with small stability exponents. Recall that the larger eigenvalue of theJacobian matrix of the map is here referred to as the stability multiplier. For the igenfunctions of Perron-Frobenius operator C (cid:15),ν it is given as λ ( P ) (cid:15),ν ( y ) = − (2 π(cid:15) cos(2 νπy ) −
3) + (cid:112) (2 π(cid:15) cos(2 νπy ) − − , (14)which takes the minimum value on the line νy ∈ Z provided that 0 < (cid:15) < / π is satisfied. One can verify that the set of periodic orbits { ( x, y ) = ( p/ν − (cid:98) p/ν +1 / (cid:99) , q/ν − (cid:98) q/ν + 1 / (cid:99) ); p, q = 0 , , . . . , ν − } has minimum stability multiplier.As demonstrated in figure 16, the region where the unstable manifold is sparselyrunning is populated with relatively shorter, and less unstable periodic orbits. Herethe stability is measured by the local instability, defined by | λ ( P ) (cid:15),ν | p , where p denotesthe period of the periodic orbit. We here present not only the ν = 1 case but alsothe ν ≥ igenfunctions of Perron-Frobenius operator (a) (b)(c) (d) Figure 16.
Solid line: unstable manifold of the perturbed cat map for (a) ν = 1, (b) ν = 2, (c) ν = 3, and (d) ν = 4. (cid:15) = 0 .
6. Finite-time Lyapunov exponent and second eigenfunction ofPerron-Frobenius operator
In this section we explore what signatures of dynamics are imprinted in the spatialpatterns of the second eigenfunction. As shown in the previous section, spatial patternsof the second eigenfunction well reflect the density or sparseness of the unstable manifold.In particular, it was observed that the second eigenfunction is localized around therelatively lesser unstable periodic orbits.We here present an example demonstrating that, unlike in the argument developedin quantum scarring, the local instability of the periodic orbits does not necessarilycontrol the localization observed in the second eigenfunction of the Perron-Frobeniusoperator. Figure 17 portrays a second eigenfunction in the case of the perturbed catmap with ν = 2. As noticed, this second eigenfunction is not localized around theperiodic orbit with the smallest instability, that is the fixed point at the origin, but igenfunctions of Perron-Frobenius operator Figure 17.
The absolute value of the second eigenfunction of the Fokker-Planckoperator associated with the perturbed cat map for (cid:15) = 0 . ν = 2. Note thatthe corresponding eigenvalue takes a complex value (see below). We take K = 50 and∆ = 10 − . The fixed point (+) and period-3 orbit ( × ) are marked in red. Here we consider the maximal finite-time (mFT) Lyapunov exponent as a possiblecandidate to understand localization. The mFT Lyapunov exponent is a characteristicquantity defined at each point in phase space, but it carries information about a longertime scale than that of linearized dynamics.
We first introduce the mFT Lyapunov exponent. To this end, define the matrix A as, A ( x , t ) = t − (cid:89) k =0 J ( x k ) , J ij ( x ) = ∂f i ∂x j , (15)where x k = f ( x k − ) and J ( x ) is the Jacobi matrix of f at x . Let a ( x , t ) be the largereigenvalue of A ( x , t ) in magnitude. The mFT Lyapunov exponent at time t is expressedas, Λ( x , t ) = 1 t ln | a ( x , t ) | . (16)In figure 18, we show the distribution of the mFT Lyapunov exponent for the perturbedcat map as a function of x t = ( x t , y t ). We take the perturbation strength (cid:15) = 0 . ν = 1. For t = 1, the mFT Lyapunov exponent is just thelogarithm of the larger eigenvalue of the Jacobian matrix. As shown in figure 18 (a), thedistribution does not reflect structures of the unstable manifold, but only the linearizeddynamics, therefore what is imprinted in the localized pattern of eigenfunctions is notinformation associated with local stability multipliers. For t >
1, as shown in figure18 (b) and (c), the distribution of the mFT Lyapunov exponents is smooth along the igenfunctions of Perron-Frobenius operator igenfunctions of Perron-Frobenius operator (a) (b)(c) (d) Figure 18. (a)-(c) Distribution of the mFT Lyapunov exponent of the perturbed catmap with (cid:15) = 0 . ν = 1. The time interval to evaluate the mFT Lyapunov exponentis (a) t = 1, (b) t = 5 and (c) t = 15. 10 initial points randomly distributed inthe entire phase space are taken for calculations. (d) The second eigenfunction of theFokker-Planck operator of the perturbed cat map for (cid:15) = 0 . ν = 1. We take K = 100 and ∆ = 5 × − . We here consider the relation between the second eigenfunction of the Perron-Frobeniusoperator and the distribution of the mFT Lyapunov exponents by assuming that thePerron-Frobenius operator for a 2-dimensional uniformly hyperbolic area-preservingmap f : M → M has a real isolated second-largest eigenvalue σ of eigenfunction ψ . For this purpose, let us consider the time evolution of a distribution function ρ ( x ),which is initially smooth in M and takes positive values. In the long time limit, ρ ( x )tends to a uniform distribution ( cf. § L t ρ ( x ) ∼ cσ t ψ ( x ) , ( t → ∞ ) (17) igenfunctions of Perron-Frobenius operator c ∈ R is some constant. Note that the convergence to the equilibrium state occursnot in a point-wise manner but in the sense of weak convergence at most [43]. Fromequation (17), we infer that a proper time for evaluating the mFT Lyapunov exponent is t ∼ − / log | σ | . The distributions of figure 18, and the eigenvalues of the Fokker-Planckoperator reported in figure 19 enable us to verify this estimate.We introduce a coordinate ξ ∈ ( −∞ , ∞ ) along the unstable manifold in M , andconsider the dynamics f u restricted on the unstable manifold. The Perron-Frobeniusoperator associated with f u is expressed as L u ρ u ( ξ ) = (cid:90) M δ ( ξ − f u ( ¯ ξ )) ρ u ( ¯ ξ )d ¯ ξ = (cid:90) M δ ( ξ − η ) ρ u ( f − u ( η )) | f (cid:48) u ( η ) | − d η = | f (cid:48) u ( ξ ) | − ρ u ( f − u ( ξ )) , so the t -time iteration of the Perron-Frobenius operator is L tu ρ u ( f t ( ξ )) = t − (cid:89) k =0 (cid:12)(cid:12) f (cid:48) u ( f ku ( ξ )) (cid:12)(cid:12) − ρ u ( ξ ) , (18)where f (cid:48) u ( ξ ) denotes the derivative of f u at ξ . Since the expanding rate of f u is thestability multiplier of f , the following holds: t − (cid:89) k =0 (cid:12)(cid:12) f (cid:48) u ( f ku ( ξ )) (cid:12)(cid:12) − = e − Λ ( x ,t ) t , (19)where Λ ( x , t ) is the mFT Lyapunov exponent of f . Since the map f u defined on( −∞ , ∞ ) is monotonically expanding, |L tu ρ u ( f t ( ξ )) | → t → ∞ ,consistently with the decay of ρ ( x ) to the equilibrium distribution ρ ∞ = 1.We hereafter consider the time evolution of the distribution function ρ ( x ), whichis assumed to be smooth and localized around an arbitrary point x = x ∈ M andproperly normalized in M . Since the unstable manifold is dense in M , for any closeneighborhood of x , there exists a point x on the unstable manifold. Then we introducethe function ρ u ( ξ ), which is defined by restricting the function ρ ( x ) onto the unstablemanifold on which x is located. Here we denote the coordinate of x along the unstablemanifold by ξ .As mentioned above, any smooth function tends to the equilibrium distribution as t → ∞ , so for any x the difference |L t ρ ( f t ( x )) − | should also tend to zero as t → ∞ .Therefore if the relaxation towards equilibrium can be described as |L t ρ ( f t ( x )) − | ∼ |L t ρ ( f t ( x )) − | ∼ |L tu ρ u ( f tu ( ξ )) | , (20)we obtain | cσ t ψ ( f t ( x )) | ∼ e − Λ ( x ,t ) t ρ u ( ξ ) . (21)The constants c, σ do not depend on t , and ρ u ( ξ ) is a value of the initial distribution ρ u ( ξ ) at ξ = ξ . Hence the following equivalence relation holds for any pair of points igenfunctions of Perron-Frobenius operator x (1)0 , x (2)0 ∈ M ,Λ ( x (1)0 , t ) ≤ Λ ( x (2)0 , t ) ⇔ | ψ ( f t ( x (1)0 )) | ≥ | ψ ( f t ( x (2)0 )) | . (22)This explains a good agreement between the second eigenfunction for the Perron-Frobenius operator and the spatial distribution of the mFT Lyapunov exponent, asshown in figure 18 [Recall that, in our notation from section 6.1, x t = f t ( x )].We now provide further evidence in support of the previous argument, by examiningother examples of perturbed cat maps. In the case of ν = 1, the second-largest eigenvalueis unique, so the above argument can be applied. On the other hand, for ν = 2, thereappear multiple second-largest eigenvalues having the same absolute value, hereafterdenoted respectively by σ , real , σ , comp and σ ∗ , comp , and shown in figure 19(a).For such a degeneracy, neither the localization pattern of the eigenfunction of theeigenvalue σ , real [figure 20(a)], nor that of the eigenvalue σ , comp [figure 21(a)] single-handedly matches the spatial distribution pattern of the mFT Lyapunov exponent[figure 22(a)]. However, when superposing the eigenfunctions of σ , real and σ , comp , theresulting localization pattern agrees with that of the distribution of the mFT Lyapunovexponent. We notice that the latter might reflect a superposed state of two degeneratedeigenfunctions, although degeneracy here occurs only in the sense of the modulus. Weclarify this picture by introducing an additional perturbation to the perturbed cat map C (cid:15), , in order to lift the degeneracy of the second eigenvalues: C (cid:15), ¯ (cid:15) = C (cid:15), ◦ F ¯ (cid:15) , (23) F ¯ (cid:15) : ( x, y ) (cid:55)→ (cid:16) x − ¯ (cid:15) πy ) cos(2 πy ) , y (cid:17) . (24)In the range 0 < (cid:15) < / π , where the original perturbed cat map C (cid:15), is also hyperbolic,the larger eigenvalue of the Jacobian matrix of C (cid:15), ¯ (cid:15) is real and greater than unityeverywhere in the phase space, as long as the condition 0 < ¯ (cid:15) < / π − (cid:15) is satisfied.We will fix (cid:15) = 0 . C (cid:15), ¯ (cid:15) . Unlike the originalperturbed cat map C (cid:15), , the Fokker-Planck operator for the map C (cid:15), ¯ (cid:15) does not havean analytic expression, so we need to carry out numerical integration in order tocompute the matrix elements (11). We approximate the integral (11) using the followingsummation: (cid:90) T e π i( q · x − k · C ¯ (cid:15) ( x )) d x ∼ M − (cid:88) i =0 M − (cid:88) j =0 e π i( q · x ij − k · C ¯ (cid:15) ( x ij )) M , (25)where M denotes the lattice number in the discretization of the phase space, and x ij = (cid:0) iM , jM (cid:1) ( i, j = 0 , . . . , M −
1) represents the ( i, j )-th lattice point.As illustrated in figure 19, the three-fold degeneracy in magnitude of the second-largest eigenvalues is lifted with the perturbation ¯ (cid:15) . Note that the real eigenvalue σ , real gradually increases while staying real-valued, whereas the magnitude of σ , comp , σ ∗ , comp decreases, and the magnitude of the third-largest one σ (¯ (cid:15) ) exceeds that of σ , comp , σ ∗ , comp around ¯ (cid:15) ∼ . σ (¯ (cid:15) ) increases as a function of ¯ (cid:15) as well, nevertheless σ , real and σ (¯ (cid:15) ) igenfunctions of Perron-Frobenius operator (cid:15) > (a) (b)(c) (d) Figure 19. (a)-(c) Eigenvalues σ , real (¯ (cid:15) ) , σ , comp (¯ (cid:15) ) , σ ∗ , comp (¯ (cid:15) ) and σ (¯ (cid:15) ) of theFokker-Planck operator for the map C (cid:15), ¯ (cid:15) with (cid:15) = 0 .
1. We take K = 30 and ∆ = 10 − .(b) A magnification of (a) around σ , real and σ . (c) A magnification of (a) around σ . comp . (d) Absolute values | σ , real (¯ (cid:15) ) | , | σ , comp (¯ (cid:15) ) | and | σ (¯ (cid:15) ) | as a function of ¯ (cid:15) . When there exists a unique, isolated second-largest eigenvalue, we can expect theeigenfunction of the Perron-Frobenius operator to reflect the spatial distribution of themFT Lyapunov exponent, as is confirmed in the case of ν = 1. Indeed, as shownin figure 20(b), the eigenfunction of the Fokker-Planck operator associated with σ , real gives a similar spatial dependence of the mFT Lyapunov exponent distribution, whichis presented in figure 22(b). Both the eigenfunction and the distribution of the mFTLyapunov exponents are localized in the region along the unstable manifold emanatingfrom the fixed point (0 , . , .
5) [the unstable manifold is split into six branches by themodulo operation, cf. figure 20(b)]. Since | σ , comp | is close to | σ , real | , the amplitudearound the period 3 orbit is still higher than that in other regions. On the otherhand, as shown in figure 21, the eigenfunction associated with σ , comp shows a spatialpattern, which is different from that of the mFT Lyapunov exponent distribution. This igenfunctions of Perron-Frobenius operator (a) (b) Figure 20.
Eigenfunction of the Fokker-Planck operator for the map C (cid:15), ¯ (cid:15) associatedwith σ , real (absolute value). (a) (cid:15) = 0 . (cid:15) = 0. (b) (cid:15) = 0 . (cid:15) = 0 . K = 30 and ∆ = 10 − .(a) (b) Figure 21.
Eigenfunction of the Fokker-Planck operator for the map C (cid:15), ¯ (cid:15) associatedwith σ , comp (absolute value). (a) (cid:15) = 0 . (cid:15) = 0. (b) (cid:15) = 0 . (cid:15) = 0 . K = 30 and ∆ = 10 − . igenfunctions of Perron-Frobenius operator (a) (b)(c) (d) Figure 22.
Distribution of the mFT Lyapunov exponent of the map C (cid:15), ¯ (cid:15) . (a), (c) (cid:15) = 0 . (cid:15) = 0. (b), (d) (cid:15) = 0 . (cid:15) = 0 . t = 5 and (c), (d) t = 15. The calculationis performed by sampling the phase space with 10 randomly-distributed initial points. In the perturbed situation analyzed above, the second eigenvalue is real-valued,and the corresponding eigenfunction localized around the fixed point, which has theminimal instability. However, we can realize the situation in which the complex-valued eigenfunctions become the second-eigenfunction after resolving the degeneracyby flipping the sign of the perturbation ¯ (cid:15) from positive to negative. In this case, thesecond eigenfunction looks almost the same as the one presented in figure 17. Thismeans that the second eigenfunction is not necessarily localized around the periodicorbit (fixed point) that has the smallest local instability, as already observed in section5. igenfunctions of Perron-Frobenius operator
7. Conclusion and Discussion
We have numerically investigated eigenfunctions associated with the leading sub-uniteigenvalues of the Perron-Frobenius operator in 2-dimensional uniformly hyperbolicarea-preserving maps. We call them the second eigenfunctions and the second-largesteigenvalues, respectively. In the present paper, we have considered the cat map and theperturbed cat map as model systems, since these are known to be uniformly hyperbolicwithin some parameter range.The eigenfunctions of the Perron-Frobenius operator for uniformly hyperbolicsystems are expected to belong to a class of nontrivial hyperfunctions and functionalspaces: one needs special care when calculating them numerically. The Ulam method is awell known and simple numerical technique to obtain eigenvalues and the correspondingeigenfunctions, and thus it has been used as a numerical scheme in the physics literature[19, 20, 21].As demonstrated in section 3, however, numerically calculated second eigenfunc-tions for the unperturbed cat map provide inconsistent spatial patterns, depending onthe choice of partitions constructing the transfer matrix. One might think it more rea-sonable to take Markov partitions in the computation of the transfer matrix of the Ulammethod because the Markov partition is compatible with the dynamics, but it has beenshown that the second-largest eigenvalue is sensitive to the type of Markov partition[31]. This implies that associated eigenfunctions also depend on the choice of the par-tition, and this is actually confirmed in section 3, although the spatial patterns seemto stabilize as the partition is made finer. The same computation is generally unstablewhen the uniform partition is used to evaluate the transfer matrix. The second-largesteigenvalue does not converge, and spatial patterns of the second eigenfunction vary sen-sibly as a function of the grid resolution, if we adopt the uniform partition. From theobservations presented here, we must conclude that the Ulam method is not reliable forthe unperturbed cat map.The Ulam method is shown to be relatively efficient and seems to work reasonablywell when it is applied to the perturbed cat map. In particular, for a large enoughnonlinear perturbation, we obtain better convergence for the second-largest eigenvalue.What is interesting is that the spatial distribution of the second eigenfunction is highlyinhomogeneous, and localized in some regions of the phase space. Such localizationreminds us of quantum scarring, which is found in quantum eigenfunctions of chaoticsystems, thus it might be called classical scarring although it is not clear to what extentthey share a common origin [15].Next, we have investigated the second-largest eigenvalues and the associatedeigenfunctions of the Perron-Frobenius operator by calculating the Fokker-Planckoperator with sufficiently small diffusivity. As shown in section 4, this approach workswell and more stable compared to the Ulam method: as the maximum wavenumberincluded in constructing the Fokker-Planck kernel is increased, well convergent second-largest eigenvalues and second eigenfunctions are obtained for each value of diffusivity. igenfunctions of Perron-Frobenius operator igenfunctions of Perron-Frobenius operator Acknowledgements
The authors are very grateful for Masato Tsujii, Fr´ed´eric Faure and Yuzuru Sato fortheir critical comments and helpful discussions. This work has been supported by JSPSKAKENHI Grants No. 15H03701 and No. 17K05583. DL is partially supported bythe National Science Foundation of China, Young International Scientists (Grant No.11750110416-1601190090).
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