In this appendix we show that the multidimensional Lyapunov exponents and

Transcription

1 Aendix A6 Lyaunov exonents Man who says it cannot be done should not interrut man doing it. Sayings of Vattay Gábor In this aendix we show that the multidimensional Lyaunov exonents and relaxation exonents (dynamo rates of vector fields can be exressed in terms of leading eigenvalues of aroriate evolution oerators. A6.1 Evolution oerator for Lyaunov exonents Lyaunov exonents were introduced and comuted for 1-dimensional mas in sect For higher-dimensional flows only the Jacobian matrices are multilicative, not individual eigenvalues, and the construction of the evolution oerator for evaluation of the Lyaunov sectra requires the extension of evolution equations to the flow in the tangent sace. We now develo the requisite theory. Here we construct a multilicative evolution oerator (A6.4 whose sectral determinant (A6.8 yields the leading Lyaunov exonent of a d-dimensional flow (and is entire for Axiom A flows. The key idea is to extend the dynamical system by the tangent sace of the flow, suggested by the standard numerical methods for evaluation of Lyaunov exonents: start at x 0 with an initial infinitesimal tangent sace vector in the d- dimensional tangent sace η(0 TM x, and let the flow transort it along the trajectory x(t = f t (x 0. The dynamics in the tangent bundle (x, δx TM is governed by the system of equations of variations (4.2: ẋ = v(x, η = A(x η. 835

2 APPENDIX A6. LYAPUNOV EXPONENTS 836 Here A(x is (4.3, the stability matrix (velocity gradients matrix of the flow. We write the solution as x(t = f t (x 0, η(t = J t (x 0 η 0, (A6.1 with the tangent sace vector η transorted by the Jacobian matrix J t (x 0 = x(t/ x 0 (4.5. As exlained in sect. 4.1, the growth rate of this vector is multilicative along the trajectory and can be reresented as η(t = η(t / η(0 u(t where u(t is a unit" vector in some norm.. For asymtotic times and for almost every initial (x 0, η(0, this factor converges to the leading eigenvalue of the linearized stability matrix of the flow. We imlement this multilicative evaluation of Floquet multiliers by adjoining the d-dimensional transverse tangent sace η TM x ; η(x v(x = 0 to the (d+1-dimensional dynamical evolution sace x M R d+1. In order to determine the length of the vector η we introduce a homogeneous differentiable scalar function g(η = η. It has the roerty g(λη = Λ g(η for any Λ. An examle is the rojection of a vector to its dth comonent g η 1 η 2 η d = η d. Any vector η(0 TM x can now be reresented by the roduct η = Λu, where u is a unit" vector in the sense that its norm is u = 1, and the factor Λ t (x 0, u 0 = g(η(t = g(j t (x 0 u 0 (A6.2 is the multilicative stretching factor. Unlike the leading eigenvalue of the Jacobian the stretching factor is multilicative along the trajectory: Λ t +t (x 0, u 0 = Λ t (x(t, u(t Λ t (x 0, u 0. The u evolution constrained to ET g,x, the sace of unit transverse tangent vectors, is given by rescaling of (A6.1: exercise A6.1 u = R t (x, u = 1 Λ t (x, u Jt (xu. (A6.3 Eqs. (A6.1, (A6.2 and (A6.3 enable us to define a multilicative evolution oerator on the extended sace U ET g,x L t (x, u ; x, u = δ ( x f t (x δ ( u R t (x, u Λ t (x, u β 1, (A6.4 where β is a variable. aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

3 APPENDIX A6. LYAPUNOV EXPONENTS 837 To evaluate the exectation value of log Λ t (x, u which is the Lyaunov exonent we again have to take the roer derivative of the leading eigenvalue of (A6.4. In order to derive the trace formula for the oerator (A6.4 we need to evaluate tr L t = dxdu L t (u, x; u, x. The dx integral yields a weighted sum over rime eriodic orbits and their reetitions r: tr L t =,r = g T r=1 δ ( t rt det (1 M r,r, du δ( u R T r (x, u Λ T r (x, u β 1, (A6.5 where M is the rime cycle transverse stability matrix. As we shall see below,,r is intrinsic to cycle, and indeendent of any articular eriodic oint x. We note next that if the trajectory f t (x is eriodic with eriod T, the tangent sace contains d eriodic solutions e (i (x(t + t = e (i (x(t, i = 1,..., d, corresonding to the d unit eigenvectors {e (1, e (2,, e (d } of the transverse stability matrix, with stretching" factors (A6.2 given by its eigenvalues M (xe (i (x = Λ,i e (i (x, i = 1,..., d. (no summation on i The du integral in (A6.5 icks u contributions from these eriodic solutions. In order to comute the stability of the ith eigen-direction solution, it is convenient to exand the variation around the eigenvector e (i in the stability matrix eigenbasis δu = δu l e (l. The variation of the ma (A6.3 at a comlete eriod t = T is then given by δr T (e (i = = ( Mδu g(me (i Me(i g(e (i g(me (i 2 u ( Λ,k e (k e (i g(e(i Λ,i u k δu k. Mδu (A6.6 The δu i comonent does not contribute to this sum since g(e (i + du i e (i = 1 + du i imlies g(e (i / u i = 1. Indeed, infinitesimal variations δu must satisfy g(u + δu = g(u = 1 = d l=1 δu l g(u u l = 0, so the allowed variations are of form ( δu = e (k e (i g(e(i c k, c k 1, u k aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

4 APPENDIX A6. LYAPUNOV EXPONENTS 838 and in the neighborhood of the e (i eigenvector the du integral can be exressed as du = dc k. g Inserting these variations into the du integral we obtain g du δ ( e (i + δu R T (e (i δr T (e (i +... = dc k δ((1 Λ k /Λ i c k = 1 Λ k /Λ i, and the du trace (A6.5 becomes,r = d i=1 1 1 Λ r,i β 1 1 Λ r,k /Λr,i. (A6.7 The corresonding sectral determinant is obtained by observing that the Lalace transform of the trace (21.19 is a logarithmic derivative tr L(s = d ds log F(s of the sectral determinant: F(β, s = ex e st r r det (1 M r,r(β. (A6.8,r This determinant is the central result of this section. Its zeros corresond to the eigenvalues of the evolution oerator (A6.4, and can be evaluated by the cycle exansion methods. The leading zero of (A6.8 is called ressure" (or free energy P(β = s 0 (β. (A6.9 The average Lyaunov exonent is then given by the first derivative of the ressure at β = 1: λ = P (1. (A6.10 The simlest alication of (A6.8 is to 2-dimensional hyerbolic Hamiltonian mas. The Floquet multiliers are related by Λ 1 = 1/Λ 2 = Λ, and the sectral determinant is given by F(β, z = ex z rn 1 r Λ r (1 1/Λ r 2,r(β,r (β =,r Λ r 1 β 1 1/Λ 2r + Λr β 3 1 1/Λ 2r. (A6.11 aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

5 APPENDIX A6. LYAPUNOV EXPONENTS 839 The dynamics (A6.3 can be restricted to a u unit eigenvector neighborhood corresonding to the largest eigenvalue of the Jacobi matrix. On this neighborhood the largest eigenvalue of the Jacobi matrix is the only fixed oint, and the sectral determinant obtained by keeing only the largest term the,r sum in (A6.7 is also entire. In case of mas it is ractical to introduce the logarithm of the leading zero and to call it ressure" P(β = log z 0 (β. The average of the Lyaunov exonent of the ma is then given by the first derivative of the ressure at β = 1: λ = P (1. By factorizing the determinant (A6.11 into roducts of zeta functions we can conclude that the leading zero of the (A6.4 can also be recovered from the leading zeta function 1/ζ 0 (β, z = ex z rn r Λ r β. (A6.12,r This zeta function lays a key role in thermodynamic alications, see aendix A36. A6.2 Advection of vector fields by chaotic flows Fluid motions can move embedded vector fields around. An examle is the magnetic field of the Sun which is frozen" in the fluid motion. A assively evolving vector field V is governed by an equation of the form t V + u V V u = 0, (A6.13 where u(x, t reresents the velocity field of the fluid. The strength of the vector field can grow or decay during its time evolution. The amlification of the vector field in such a rocess is called the "dynamo effect. In a strongly chaotic fluid motion we can characterize the asymtotic behavior of the field with an exonent V(x, t V(xe νt, (A6.14 where ν is called the fast dynamo rate. The goal of this section is to show that eriodic orbit theory can be develoed for such a highly non-trivial system as well. We can write the solution of (A6.13 formally, as shown by Cauchy. Let x(t, a be the osition of the fluid article that was at the oint a at t = 0. Then the field evolves according to V(x, t = J(a, tv(a, 0, (A6.15 aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

6 APPENDIX A6. LYAPUNOV EXPONENTS 840 where J(a, t = (x/ (a is the Jacobian matrix of the transformation that moves the fluid into itself x = x(a, t. We write x = f t (a, where f t is the flow that mas the initial ositions of the fluid articles into their ositions at time t. Its inverse, a = f t (x, mas articles at time t and osition x back to their initial ositions. Then we can write (A6.15 V i (x, t = d 3 a L t i j (x, av j(a, 0, (A6.16 with j L t i j (x, a = δ(a f t (x x i a j. (A6.17 For large times, the effect of L t is dominated by its leading eigenvalue, e ν 0t with Re(ν 0 > Re(ν i, i = 1, 2, 3,... In this way the transfer oerator furnishes the fast dynamo rate, ν := ν 0. The trace of the transfer oerator is the sum over all eriodic orbit contributions, with each cycle weighted by its intrinsic stability TrL t = T r=1 tr M r det ( δ(t rt. (A M r We can construct the corresonding sectral determinant as usual F(s = ex 1 tr M r r r=1 det ( e srt. (A M r Note that in this formuli we have omitted a term arising from the Jacobian transformation along the orbit which would give 1 + tr M r in the numerator rather than just the trace of M r. Since the extra term corresonds to advection along the orbit, and this does not evolve the magnetic field, we have chosen to ignore it. It is also interesting to note that the negative owers of the Jacobian occur in the denominator, since we have f t in (A6.17. In order to simlify F(s, we factor the denominator cycle stability determinants into roducts of exanding and contracting eigenvalues. For a 3-dimensional fluid flow with cycles ossessing one exanding eigenvalue Λ (with Λ > 1, and one contracting eigenvalue λ (with λ < 1 the determinant may be exanded as follows: det ( 1 M r 1 = (1 Λ r (1 λ r 1 = λ r j=0 k=0 Λ jr λ kr With this decomosition we can rewrite the exonent in (A6.19 as r=1 1 r (λ r + Λ r e srt det ( = 1 M r j,k=0 r=1. (A ( λ Λ j λ k r e st r (λ r +Λ r, (A6.21 aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

7 APPENDIX A6. LYAPUNOV EXPONENTS 841 which has the form of the exansion of a logarithm: [ ( log 1 e st λ Λ 1 j ( λ k + log 1 e st λ Λ j j,k λ 1+k ]. (A6.22 The sectral determinant is therefore of the form, F(s = F e (sf c (s, (A6.23 where F e (s = j,k=0 ( 1 t ( jk Λ, (A6.24 F c (s = j,k=0 ( 1 t ( jk λ, (A6.25 with t ( jk = e st λ λk Λ j. (A6.26 The two factors resent in F(s corresond to the exanding and contracting exonents. (Had we not neglected a term in (A6.19, there would be a third factor corresonding to the translation. For 2-dimensional Hamiltonian volume reserving systems, λ = 1/Λ and (A6.24 reduces to F e (s = 1 k=0 t Λ k 1 k+1, t = est Λ. (A6.27 With σ = Λ / Λ, the Hamiltonian zeta function (the j = k = 0 art of the roduct (A6.25 is given by ( 1/ζ dyn (s = 1 σ e st. (A6.28 This is a curious formula the zeta function deends only on the return times, not on the eigenvalues of the cycles. Furthermore, the identity, Λ + 1/Λ (1 Λ(1 1/Λ = σ + 2 (1 Λ(1 1/Λ, when substituted into (A6.23, leads to a relation between the vector and scalar advection sectral determinants: F dyn (s = F 2 0 (s/ζ dyn(s. (A6.29 The sectral determinants in this equation are entire for hyerbolic (axiom A systems, since both of them corresond to multilicative oerators. aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

8 APPENDIX A6. LYAPUNOV EXPONENTS 842 In the case of a flow governed by a ma, we can adat the formulas (A6.27 and (A6.28 for the dynamo determinants by simly making the substitution z n = e st, (A6.30 where n is the integer order of the cycle. Then we find the sectral determinant F e (z given by equation (A6.27 but with t = zn Λ for the weights, and ( 1/ζ dyn (z = Π 1 σ z n (A6.31 (A6.32 for the zeta-function For mas with finite Markov artition the inverse zeta function (A6.32 reduces to a olynomial for z since curvature terms in the cycle exansion vanish. For examle, for mas with comlete binary artition, and with the fixed oint stabilities of oosite signs, the cycle exansion reduces to 1/ζ dyn (s = 1. (A6.33 For such mas the dynamo sectral determinant is simly the square of the scalar advection sectral determinant, and therefore all its zeros are double. In other words, for flows governed by such discrete mas, the fast dynamo rate equals the scalar advection rate. In contrast, for 3-dimensional flows, the dynamo effect is distinct from the scalar advection. For examle, for flows with finite symbolic dynamical grammars, (A6.29 imlies that the dynamo zeta function is a ratio of two entire determinants: 1/ζ dyn (s = F dyn (s/f0 2 (s. (A6.34 This relation imlies that for flows the zeta function has double oles at the zeros of the scalar advection sectral determinant, with zeros of the dynamo sectral determinant no longer coinciding with the zeros of the scalar advection sectral determinant; Usually the leading zero of the dynamo sectral determinant is larger than the scalar advection rate, and the rate of decay of the magnetic field is no longer governed by the scalar advection. exercise A6.2 Commentary Remark A6.1. Lyaunov exonents. Sect. A6.1 is based on ref. [3]. Remark A6.2. Dynamo zeta. The dynamo zeta (A6.32 has been introduced by Aurell and Gilbert [1] and reviewed in ref. [4]. Our exosition follows ref. [2]. aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

9 APPENDIX A6. LYAPUNOV EXPONENTS 843 References [1] E. Aurell and A. D. Gilbert, Fast dynamos and determinants of singular integral oerators, Geohys. Astrohys. Fluid Dynam. 73, 5 32 (1993. [2] N. J. Balmforth, P. Cvitanović, G. R. Ierley, E. A. Siegel, and G. Vattay, Advection of vector fields by chaotic flows, Ann. New York Acad. Sci. 706, (1993. [3] P. Cvitanović and G. Vattay, Entire Fredholm determinants for evaluation of semi-classical and thermodynamical sectra, Phys. Rev. Lett. 71, (1993. [4] A. D. Gilbert and S. Childress, Stretch, Twist, Fold: the Fast Dynamo (Sringer, Berlin, aendlyaunov - 20se2009 ChaosBook.org edition16.0, Jan

10 EXERCISES 844 Exercises A6.1. Stretching factor. Prove the multilicative roerty of the stretching factor (A6.2. Why should we extend the hase sace with the tangent sace? A6.2. Dynamo rate. Suose that the fluid dynamics is highly dissiative and can be well aroximated by the iecewise linear ma { 1 + ax if x < 0, f (x = 1 bx if x > 0, (A6.35 on an aroriate surface of section (a, b > 2. Suose also that the return time is constant T a for x < 0 and T b for x > 0. Show that the dynamo zeta is 1/ζ dyn (s = 1 e st a + e st b. (A6.36 Show also that the escae rate is the leading zero of 1/ζ 0 (s = 1 e st a /a e st b /b. (A6.37 Calculate the dynamo and the escae rates analytically if b = a 2 and T b = 2T a. Do the calculation for the case when you reverse the signs of the sloes of the ma. What is the difference? exeraalic - 7jul2000 ChaosBook.org edition16.0, Jan