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 G**Transport** **of** **vector** **fields**IMan who says it cannot be done should not interrupt m**and**oing it.—Sayings **of** Vattay GáborNTHISAPPENDIX we show that the multidimensional Lyapunov exponents **and**relaxation exponents (dynamo rates) **of** **vector** **fields** can be expressed in terms**of** 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 **vector**ηtransported 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 **vector**ηwe 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 **vector**η(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 st**and**ard 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 **vector**s,is given by rescaling **of** (G.1):exercise G.1The 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.u ′ = R t (x, u)=1Λ t (x, u) Jt (x)u.(G.3)794appendApplic - 20sep2009 **Chaos**Book.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 kki**and** in the neighborhood **of** the e (i) eigen**vector** 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 eigen**vector**s{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 toexp**and** the variation around the eigen**vector** 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 **Chaos**Book.org version13.3, Sep 23 2010appendApplic - 20sep2009 **Chaos**Book.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 Aurell**and** 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, **and**1/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 stabilities**of** opposite signs, the cycle expansion reduces to1/ζ dyn (s)=1. (G.33)ExercisesG.1. Stretching factor. Prove the multiplicative property**of** 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 **of**1/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 zeros**of** 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 evaluation**of** 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 **Chaos**Book.org version13.3, Sep 23 2010refsAppApplic - 20sep2009 **Chaos**Book.org version13.3, Sep 23 2010