Transport of vector fields - Chaos: Classical and Quantum

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Transport of vector fields - Chaos: Classical and Quantum

Here A(x) is (4.3), the stability matrix (velocity gradients matrix) of the flow. Wewrite the solution asx(t)= f t (x 0 ), η(t)= J t (x 0 )η 0 ,(G.1)Appendix GTransport of vector fieldsIMan who says it cannot be done should not interrupt mandoing it.—Sayings of Vattay GáborNTHISAPPENDIX we show that the multidimensional Lyapunov exponents andrelaxation exponents (dynamo rates) of vector fields can be expressed in termsof leading eigenvalues of appropriate evolution operators.G.1 Evolution operator for Lyapunov exponentsLyapunov exponents were introduced and computed for 1−dimensionalmaps in sect. 17.3.2. For higher-dimensional flows only the Jacobian matrices aremultiplicative, not individual eigenvalues, and the construction of the evolutionoperator for evaluation of the Lyapunov spectra requires the extension of evolutionequations to the flow in the tangent space. We now develop the requisite theory.Here we construct a multiplicative evolution operator (G.4) whose spectraldeterminant (G.8) yields the leading Lyapunov exponent of a d-dimensional flow(and is entire for Axiom A flows).with the tangent space vectorηtransported by the Jacobian matrix J t (x 0 )=∂x(t)/∂x 0(4.6).As explained in sect. 4.1, the growth rate of this vector is multiplicative alongthe trajectory and can be represented asη(t) = |η(t)|/|η(0)| u(t) where u(t) is a“unit” vector in some norm||.||. For asymptotic times and for almost every initial(x 0 ,η(0)), this factor converges to the leading eigenvalue of the linearized stabilitymatrix of the flow.We implement this multiplicative evaluation of Floquet multipliers by adjoiningthe d-dimensional transverse tangent spaceη∈TM x ;η(x)·v(x)=0 to the (d+1)-dimensional dynamical evolution space x∈M⊂R d+1 . In order to determine thelength of the vectorηwe introduce a homogeneous differentiable scalar functiong(η)=||η||. It has the property g(Λη)=|Λ| g(η) for anyΛ. An example is theprojection of a vector to its dth component⎛η 1ηg 2⎜⎝···⎞⎟⎠ =|η d|.η dAny vectorη(0)∈TM x can now be represented by the productη=Λu, whereu is a “unit” vector in the sense that its norm is||u||=1, and the factorΛ t (x 0 , u 0 )=g(η(t))=g(J t (x 0 )u 0 )is the multiplicative “stretching” factor.(G.2)Unlike the leading eigenvalue of the Jacobian the stretching factor is multiplicativealong the trajectory:The key idea is to extend the dynamical system by the tangent space of theflow, suggested by the standard numerical methods for evaluation of Lyapunovexponents: start at x 0 with an initial infinitesimal tangent space vector in the d-dimensional tangent spaceη(0)∈TM x , and let the flow transport it along thetrajectory x(t)= f t (x 0 ).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 space of unit transverse tangent vectors,is given by rescaling of (G.1):exercise G.1The dynamics in the tangent bundle (x,δx)∈TM is governed by the systemof equations of variations (4.2):ẋ=v(x), ˙η= A(x)η.u ′ = R t (x, u)=1Λ t (x, u) Jt (x)u.(G.3)794appendApplic - 20sep2009 ChaosBook.org version13.3, Sep 23 2010


Eqs. (G.1), (G.2) and (G.3) enable us to define a multiplicative evolution operatoron the extended space U× ET g,xTheδu i component does not contribute to this sum since g(e (i) + du i e (i) )=1+du iimplies∂g(e (i) )/∂u i = 1. Indeed, infinitesimal variationsδu must satisfyL t (x ′ , u ′ ; x, u)=δ ( x ′ − f t (x) ) δ ( u ′ − R t (x, u) )|Λ t (x, u)| β−1 , (G.4)g(u+δu)=g(u)=1 =⇒d∑l=1δu l∂g(u)∂u l= 0,whereβis a variable.To evaluate the expectation value of log|Λ t (x, u)| which is the Lyapunov exponentwe again have to take the proper derivative of the leading eigenvalue of (G.4).In order to derive the trace formula for the operator (G.4) we need to evaluateTrL t = ∫ dxduL t (u, x; u, x). The ∫ dx integral yields a weighted sum over primeperiodic orbits p and their repetitions r:TrL t =∆ p,r =∑∫pg∞∑T pr=1δ ( t−rT p)| det (1− M r p)| ∆ p,r,du δ( u−R T pr (x p , u) )|Λ T pr (x p , u)| β−1 , (G.5)where M p is the prime cycle p transverse stability matrix. As we shall see below,∆ p,r is intrinsic to cycle p, and independent of any particular periodic point x p .We note next that if the trajectory f t (x) is periodic with period T, the tangentspace contains d periodic solutionse (i) (x(T+ t))=e (i) (x(t)), i=1,..., d,so the allowed variations are of form∑()δu= e (k) − e (i)∂g(e(i) )c k , |c k |≪1,∂u kkiand in the neighborhood of the e (i) eigenvector the ∫ du integral can be expressedas∫g∫ ∏du= dc k .kiInserting these variations into the ∫ du integral we obtain∫du δ ( e (i) +δu−R T (e (i) )−δR T (e (i) )+... )g=∫ ∏dc k δ((1−Λ k /Λ i )c k +...)ki∏ 1=|1−Λ k /Λ i | ,kicorresponding to the d unit eigenvectors{e (1) , e (2) ,···,e (d) } of the transverse stabilitymatrix, with “stretching” factors (G.2) given by its eigenvaluesM p (x)e (i) (x)=Λ p,i e (i) (x), i=1,..., d. (no summation on i)The ∫ du integral in (G.5) picks up contributions from these periodic solutions. Inorder to compute the stability of the ith eigen-direction solution, it is convenient toexpand the variation around the eigenvector e (i) in the stability matrix eigenbasisδu= ∑ δu l e (l) . The variation of the map (G.3) at a complete period t=T is thengiven byδR T (e (i) ) ==(Mδug(Me (i) ) − Me(i) ∂g(e (i) ))g(Me (i) ) 2 Mδu∂u∑ ( )Λ p,ke (k) − e (i)∂g(e(i) )δu k .Λ p,i ∂u kki(G.6)and the ∫ du trace (G.5) becomes∆ p,r =d∑ 1 ∏ 1|Λ r p,i |β−1 | 1−Λ r (G.7)p,k /Λr p,i|.i=1kiThe corresponding spectral determinant is obtained by observing that the Laplacetransform of the trace (18.23) is a logarithmic derivative TrL(s)=− d dslog F(s)of the spectral determinant:⎛ ⎞∑F(β, s)=exp⎜⎝ − esT prr|det (1− M r p)| ∆ p,r(β) ⎟⎠ .p,r(G.8)This determinant is the central result of this section. Its zeros correspond to theeigenvalues of the evolution operator (G.4), and can be evaluated by the cycleexpansion methods.appendApplic - 20sep2009 ChaosBook.org version13.3, Sep 23 2010appendApplic - 20sep2009 ChaosBook.org version13.3, Sep 23 2010


In the case of a flow governed by a map, we can adapt the formulas (G.27)and (G.28) for the dynamo determinants by simply making the substitutionz np = e sT p , (G.30)CommentaryRemark G.1 Lyapunov exponents.Sect. G.1 is based on ref. [G.1].where n p is the integer order of the cycle. Then we find the spectral determinantF e (z) given by equation (G.27) but withRemark G.2 Dynamo zeta. The dynamo zeta (G.32) has been introduced by Aurelland Gilbert [G.3] and reviewed in ref. [G.4]. Our exposition follows ref. [13.21].t p = znp|Λ p |(G.31)for the weights, and1/ζ dyn (z)=Π p(1−σp z np )for the zeta-function(G.32)For maps with finite Markov partition the inverse zeta function (G.32) reducesto a polynomial for z since curvature terms in the cycle expansion vanish. Forexample, for maps with complete binary partition, and with the fixed point stabilitiesof opposite signs, the cycle expansion reduces to1/ζ dyn (s)=1. (G.33)ExercisesG.1. Stretching factor. Prove the multiplicative propertyof the stretching factor (G.2). Why should we extend thephase space with the tangent space?G.2. Dynamo rate. Suppose that the fluid dynamics ishighly dissipative and can be well approximated by thepiecewise linear map{1+ax if x0,(G.35)on an appropriate surface of section (a, b>2). Supposealso that the return time is constant T a for x0. Show that the dynamo zeta is1/ζ dyn (s)=1−e sTa + e sTb . (G.36)Show also that the escape rate is the leading zero of1/ζ 0 (s)=1−e sTa /a−e sTb /b. (G.37)Calculate the dynamo and the escape rates analyticallyif b=a 2 and T b = 2T a . Do the calculation for the casewhen you reverse the signs of the slopes of the map.What is the difference?For such maps the dynamo spectral determinant is simply the square of the scalaradvection spectral determinant, and therefore all its zeros are double. In otherwords, for flows governed by such discrete maps, the fast dynamo rate equals thescalar advection rate.In contrast, for 3-dimensional flows, the dynamo effect is distinct from thescalar advection. For example, for flows with finite symbolic dynamical grammars,(G.29) implies that the dynamo zeta function is a ratio of two entire determinants:1/ζ dyn (s)=F dyn (s)/F0 2 (s). (G.34)This relation implies that for flows the zeta function has double poles at the zerosof the scalar advection spectral determinant, with zeros of the dynamo spectraldeterminant no longer coinciding with the zeros of the scalar advection spectraldeterminant; Usually the leading zero of the dynamo spectral determinant is larger exercise G.2than the scalar advection rate, and the rate of decay of the magnetic field is nolonger governed by the scalar advection.References[G.1] P. Cvitanović and G. Vattay, Entire Fredholm determinants for evaluationof semi-classical and thermodynamical spectra, Phys. Rev. Lett. 71, 4138(1993); arXiv:chao-dyn/9307012.[G.2] Ya.B. Pesin, Uspekhi Mat. Nauk 32, 55 (1977), [Russian Math. Surveys 32,55 (1977)].[G.3] E. Aurell and A. Gilbert, Geophys.& Astrophys. Fluid Dynamics (1993).[G.4] S. Childress and A.D. Gilbert Stretch, Twist, Fold: The Fast Dynamo,Lecture Notes in Physics 37 (Springer Verlag, Berlin 1995).[G.5] J. Balatoni and A. Renyi, Publi. Math. Inst. Hung. Acad. Sci. 1, 9 (1956);english translation 1, 588 (Akademia Budapest, 1976).[G.6] R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, J. Phys. A17, 3521 (1984).appendApplic - 20sep2009 ChaosBook.org version13.3, Sep 23 2010refsAppApplic - 20sep2009 ChaosBook.org version13.3, Sep 23 2010

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