Numerical study of rotation numbers and its variationals for parametric families of circle diffeomorphisms

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–7) Numerical study of rotation numbers and its variationals for parametric families of circle diffeomorphisms A. Luque 1, J. Villanueva1 1Dpt. Matem`atica Aplicada I, Universitat Polit`ecnica de Catalunya, Av. Diagonal, 647, Aptdo. 08028, Barcelona. E-mails: [email protected], [email protected]. Palabras clave: Rotation number, circle diffeomorphisms, Arnold Tongues Resumen We present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphisms with high accuracy. Our methodology is an extension of an existing approach to compute rotation numbers, that it is based on suitable averages of the iterates of the map and Richardson extrapolation. In order to justify the method, we require the family of maps to be differentiable with respect to the parameters and the rotation number to be Diophantine. The method is used to compute the Taylor expansions of Arnold tongues with high precision. 1. Introduction The rotation number is a very important topological invariant in the study of the dynamics associated to circle maps and, by extension, invariant curves for maps or two dimensional invariant tori for vector fields. For this reason, during the last years, several numerical methods for approximating rotation numbers have been developed. We refer to the works [1, 2, 3, 4] as examples of methods of different nature and complexity. Recently, a new method for computing rotation numbers of circle diffeomorphisms with high precision at low computational cost has been introduced in [6]. Concretely, this method is built assuming that the circle map is conjugate to a rigid rotation in a sufficiently smooth way and, basically, it consist in averaging the iterates of this circle map together with Richardson extrapolation. From the practical point of view, it is specially suited if we are able to compute the iterates of the map with high precision, for example if we can work with a computer arithmetic having a large number of decimal digits. The goal of this paper is to extend the method of [6] in order to compute derivatives of the rotation number with respect to parameters in parametric families of circle diffeomorphisms. Our idea is based in the same averaging-extrapolation process applied to 1 A. Luque, J. Villanueva the derivatives of the iterates of the map. To this end, we only require the family to be differentiable with respect to the parameters and the rotation number to be Diophantine. Hence, we are able to obtain accurate variational information at the same time that we approximate the rotation number. Consequently, the method allows to study parametric families of circle maps from a point of view that is not given by the previously mentioned methods. 2. Review of the method of [6] to compute rotation numbers Let T=R/Zbe the real circle and Diff r +(T), r∈[0,+∞)∪{∞, ω}, the group of orientation-preserving homeomorphisms of Tof class Cr. Given f∈Diff r +(T), we identify fwith its lift to Rby fixing the normalization condition f(0) ∈[0,1). Definition 2.1. Let fbe the lift of an orientation-preserving homeomorphism of the circle such that f(0) ∈[0,1). Then the rotation number of fis defined as the limit ρ(f) := l´ım |n|→∞ fn(x0)−x0 n, that exists for all x0∈R, is independent of x0and satisfies ρ(f)∈[0,1). Given f∈Diff 2 +(T) with ρ(f)∈R\Q, Denjoy’s theorem ensures that fis topologically conjugate to the rigid rotation Rρ(f), i.e., there exist η∈Diff 0 +(T) such that f◦η=η◦Rρ(f), where Rθ(x) = x+θ. In addition, if we require η(0) = x0, for fixed x0∈[0,∞), then ηis unique. The theoretical support of the method is provided by the regularity of η, that follows from the next result: Theorem 2.2. Assume that f∈Diff r +(T)has Diophantine rotation number θ=ρ(f)of (C, τ)-type, i.e., there exist constants C > 0and τ≥1such that |1−e2πikθ|−1≤C|k|τ,∀k∈Z∗.(1) Then, if τ+1 < r,fis conjugated to Rρ(f)by means of a conjugacy η∈Diff r−τ−ε +(T), for any ε > 0. Note that Diff ω +(T) = Diff ω−τ−ε +(T)while the domain of analyticity is reduced. From now on we focus in the analytic case, although finite differentiability is enough. Let us consider f∈Diffω +(T) with rotation number θ=ρ(f)∈ D. Notice that we can write η(x) = x+ξ(x), being ξa 1-periodic function normalized in such a way that ξ(0) = x0, for a fixed x0∈[0,1). Now, we can write the following expression for the iterates under the lift: fn(x0) = fn(η(0)) = η(nθ) = nθ +X k∈Z ˆ ξke2πiknθ,∀n∈Z,(2) where the sequence {ˆ ξk}k∈Zdenotes the Fourier coefficients of ξ. We introduce the following recursive sums for p∈N S0 N(f) := fN(x0)−x0, Sp N(f) := N X j=1 Sp−1 j(f).(3) 2 Numerical study of rotation numbers and its variationals Then, the result presented in [6] says that, under the previous hypotheses, the following sums (3) satisfy the expression (basically, the idea is to use (2) and the fact that the Fourier coefficients decay very fast due to the analyticity of η) µN+p p+ 1 ¶−1 Sp N(f) = θ+ p X l=1 Ap l Nl+Ep(N),(4) where the coefficients Ap ldepend on fand pbut are independent of N. Furthermore, the remainder Ep(N) is uniformly bounded by an expression of order O(1/Np+1). Then, if we select a maximum number N= 2qof iterates, we obtain θ= Θq,p(f) + O(2−(p+1)q),Θq,p(f) := p X j=0 cp jµN+p p+ 1 ¶−1 Sp N(f),(5) for some universal coefficients cp jwhich are given by cp l= (−1)p−l2l(l+1)/2 δ(l)δ(p−l),(6) where we define δ(n) := (2n−1)(2n−1−1) ···(21−1) for n≥1 and δ(0) := 1. The operator Θq,p corresponds to the Richardson extrapolation of equation (4) of order p. As far as the behavior of the error is concerned, by comparing (5) with 2q−1and 2q iterates, we obtain the following (heuristic) expression |θ−Θq,p(f)| ≤ ν 2p+1 |Θq,p(f)−Θq−1,p(f)|,(7) where νis a “safety parameter” whose role is to prevent the oscillations in the error as a function of qdue to the quasi-periodic part. We take ν= 10. 3. Derivatives of the rotation number with respect to parameters Now we adapt the previous ideas in order to compute derivatives of the rotation number with respect to parameters (assuming that they exist). For a sake of simplicity, we introduce the method for one-parameter families of circle diffeomorphisms, albeit the construction can be adapted to deal with multiple parameters. Thus, consider µ∈I⊂R7→ fµ∈Diff ω +(T) depending on µin a regular way. The rotation numbers of the family {fµ}µ∈Iinduce a function θ:I→[0,1) given by θ(µ) = ρ(fµ). Our goal is to generalize formula (5) to approximate numerically the derivative Dd µθfor any d, when it exists. For a full discussion of the construction we refer to [5]. We assume that the family µ7→ fµ∈Diff ω +(T) depends Cd-smoothly with respect to the parameter. As before, we define the recursive sums Dd µS0 N(fµ) := Dd µ(fn µ(x0)−x0), Dd µSp N(fµ) := N X j=0 Dd µSp−1 j(fµ). Then, if θ(µ0)∈ D and Dd µθ(µ0)6= 0, we obtain (omitting the point µ0) 3 A. Luque, J. Villanueva µN+p p+ 1 ¶−1 Dd µSp N(fµ) = Dd µθ+ p−d X l=1 Dd µˆ Ap l Nl+O(1/Np−d+1),(8) where the remainder Dd µEp(N) is of order O(1/Np−d+1). Therefore, according to formula (8), we can extrapolate the d-th derivative of the rotation number as Dd µθ= Θd q,p,p−d(f) + O(2−(p−d+1)q),Θd q,p,m(f) := m X j=0 cm jµN+p p+ 1 ¶−1 Dd µSp N(fµ), and we observe that now the extrapolation order is p−dinstead of p. Finally, we obtain the following heuristic formula for the extrapolation error |Dd µθ−Θd q,p,p−d(f)| ≤ ν 2p−d+1 |Θd q,p,p−d(f)−Θd q−1,p,p−d(f)|.(9) Remark 3.1. Up to this point we have assumed that Dd µθ6= 0 at the computed point. However, if we know a priori that Dr µθ= 0 for r= 1, . . . , d, then the remainder Dd µEp(N) is now of order O(1/Np+1). This allows to approximate Dd µθwith the same extrapolation order as the averaging order p. Indeed, we obtain 0 = Dd µθ= Θd q,p,p(f) + O(2−(p+1)q). 4. Application to the Arnold family As an example, we consider the Arnold family of circle maps, given by fα,ε :S−→ S x7−→ x+ 2πα +εsin(x),(10) where (α, ε)∈[0,1) ×[0,1) are parameters and S=R/(2πZ). Notice that, for this family of maps, it is convenient to take the angles modulo 2πjust for avoiding the lost of significant digits due to the factors (2π)d−1that would appear in the d-derivative of the map. 4.1. Stepping up to a Devil’ staircase Let us fix the value of ε∈[0,1) and consider the one-parameter family {fα}α∈[0,1) given by equation (10), i.e. fα:= fα,ε. The map α7→ ρ(fα) gives rise to a “staircase” with a dense number of stairs, that is usually called a Devil’ staircase. To illustrate the behavior of the method we have computed the above staircase for ε= 0,75. The computations have been performed by taking 104points of α∈[0,1), using 32-digit arithmetics, and a fixed averaging order p= 8. In addition, we estimate the error in the approximation of ρ(fα) and Dαρ(fα) using formulas (7) and (9), respectively. Then, we stop the computations for a tolerance of 10−26 and 10−24, using at most 222 = 4194304 iterates. Let us discuss the obtained results. First, we point out that only 11,4 % of the selected points have not reached the previous tolerances for 222 iterates. Moreover, we observe that 4 Numerical study of rotation numbers and its variationals 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 1.16 1.18 1.2 1.22 1.24 0.282 0.283 0.284 0.285 0.286 0.287 0.288 0.289 0.29 0.291 0.292 Figura 1: Devil’ staircase (top-left) and its derivative (top-right) for the Arnold family with ε= 0,75. The plots in the bottom correspond to some magnifications of the derivatives top-right one. the rotation number for 98,8 % of the points has been obtained with an error less that 10−20, while the estimated error in the derivatives is less than 10−18 for 97,7 % on the points. In figure 1 we show α7→ ρ(fα) and its derivative α7→ Dαρ(fα) for those points that satisfy that the estimated error is less than 10−18 and 10−16, respectively. We recall that the rational values of the rotation number correspond to the constant intervals in the top-left plot, and note that looking at the derivative (top-right plot) we can visualize the density of the stairs better than looking at the staircase itself. We remark that both these rational rotation numbers and their vanishing derivatives have been computed as well as the Diophantine cases. Moreover, at the bottom of the same figure, we plot some magnifications of the derivative to illustrate the non-smoothness of a Devil’ staircase. These magnifications have been computed by taking 105and 106points, respectively. We refer to [5] for more details on these computations and further discussion of the results. 4.2. Computation of the Taylor expansion of Arnold tongues All through the rest of the section, we assume that θ∈ D and consider the Arnold Tongue Tθ={(ε, α(ε)) : ε∈[0,1)}. Then, we can expand αat the origin as α(ε) = θ+α0(0) 1! ε+α00(0) 2! ε2+···+α(d)(0) d!εd+O(εd+1),(11) 5 A. Luque, J. Villanueva and our goal is to compute the derivatives α(r)(0), r≤d, that appear in this power series. d2πα(d)(0) e1 0 3.883222077450933154693731259925391915269339787692096599014776434 - 1 5.289596087298835974306750728481413682115174017433159533705768026 ·10−54 2·10−50 2 -1.944003667801032197325141712953470682792841985057545477738933600 ·10−17·10−50 3 6.353866339253870417285870622952031667026712174414003758743809499 ·10−52 3·10−48 4 9.865443989835495993231949890783720243438883460505483297079900562 ·10−12·10−47 5 4.733853534850495777271526084574485398105534790325269345544052633 ·10−49 2·10−45 6 -1.451874181864020963416053802229271731186248529989217665545212404 ·1016·10−45 7 -1.986768674642925514096249083525472601734104441662711304098209993 ·10−47 7·10−44 8 1.673363822376717001078781931538386967523434046199355922539083323 ·1018·10−42 9 -5.559060362825539878039137008326038842079877436013501651866007318 ·10−44 2·10−40 10 1.974679484744669888248485084754876332689468886829840384314732615 ·1042·10−39 11 4.019718902900154426125206309959051888079502318143227318836414835 ·10−42 1·10−38 12 3.594891944526889578314748272295019294147597687816868847742850594 ·1056·10−37 13 -4.123166034989923032518732576715313341946051550138603536248010821 ·10−39 2·10−35 14 2.198602821435568153883567054383394767567371744732559263055644337 ·1063·10−33 15 1.307318024754974551233761145122558811543944190022138837513637182 ·10−35 6·10−32 16 -4.009257214040427899940043656551946700300230713255210114705187412 ·1010 4·10−31 17 -6.641638995605492204184114438636683272452899190211080822408603857 ·10−33 4·10−29 18 -2.582559893723659427522610275977697024396910000154382754643273110 ·1012 1·10−27 19 -4.366235264281358239242428788236090577328510850575386329987344515 ·10−30 2·10−26 Tabla 1: Derivatives of 2πα(ε) at the origin for θ= (√5−1)/2. The column e1corresponds to the estimated error using (7). Our idea is to use the fact that the rotation number is constant on the tongue combined with remark 3.1. To this end, we consider the one-parameter family {fα(ε),ε}ε∈[0,1) of circle diffeomorphisms, where the graph of αparametrizes the tongue Tθ. For this family, we have ρ(fα(ε),ε) = θfor any ε∈[0,1), and hence, from remark 3.1 we read the expression 0 = Θd q,p,p(fα(ε),ε) + O(2−(p+1)q).(12) Then, let us assume that the values α0(0), . . . , α(d−1)(0) are known, and isolate the derivative α(d)(0) from Dd ε(fn(x0))|ε=0. Proceeding by induction with respect to nwe obtain the following formula Dd ε(fn(x0))|ε=0 = 2πnα(d)(0) + gd n,(13) where gd:= {gd n}n=1,...,N is a sequence that only requires the known derivatives α(r)(0), for r < d. Concretely, gd n=Dd−1 ε(∂εg(fn−1(x0))|ε=0 + d−1 X r=1 µd−1 r¶Dr ε(∂xg(fn−1(x0)))Dd−r ε(fn−1(x0))|ε=0 +gd n−1. Hence, by introducing an extrapolation operator for the previous sequence we obtain Θd q,p,p(f)|ε=0 = 2πα(d)(0) + Θq,p,p(gd) = O(2−(p+1)q). 6 Numerical study of rotation numbers and its variationals Therefore, the Taylor expansion (11) follows from the sequential computation of α(d)(0) by means of the expression 2πα(d)(0) = −Θq,p,p(gd) with an error of order O(2−(p+1)q). In particular, in table 1 we show the computations of 2πα(d)(0), for 0 ≤d≤19, that correspond to the Arnold Tongue associated to θ1,1= (√5−1)/2. The computations are performed using 64-digit arithmetics. The implementation parameters are selected as p= 11 and q= 23. Referencias [1] H. Broer y C. Simo. Hill’s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomera. Bol. Soc. Brasil. Mat. (S.N.) 29 (2) 253-293, 1998. [2] R. de la Llave y N. Petrov. Regularity of conjugacies between critical circle maps: an experimental study. Exp. Math. 11 (2), 219-241, 2002. [3] G. G´omez, J.M. Mondelo y C. Sim´o. Refined Fourier analysis: procedures, error estimates and applications. http://www.maia.ub.es/dsg/2001/index.html, 2001 (preprint). [4] J. Laskar, C. Froeschl´e y A. Celletti. The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping. Physica D, 56 (2-3), 253-269, 1992. [5] A. Luque y J. Villanueva. Computation of variationals of the rotation number for parametric families of circle diffeomorphisms. In progress. [6] T. M-Seara y J. Villanueva. On the numerical computation of Diophantine rotation numbers of analytic circle maps. Physica D, 107-120, 2006. 7