The Koopmanoperator action on a state space function a(x) is to replace it by

Transcription

1 Appendix A9 Implementing evolution A9. Koopmania The Koopmanoperator action on a state space function a(x) is to replace it by its downstream value time t later, a(x) a(x(t)), evaluated at the trajectory point x(t): [ K t a ] (x) = a( f t (x))= dyk t (x, y) a(y) K t (x, y) = δ ( y f t (x) ). (A9.) Given an initial density of representative points ρ(x), the average value of a(x) evolves as a (t) = = ρ ρ dx a( f t (x))ρ(x)= dx [ K t a ] (x)ρ(x) ρ dx dy a(y)δ ( y f t (x) ) ρ(x). The propagator δ ( y f t (x) ) can be interpreted as belonging to the Perron-Frobenius operator (9.0), so the two operators are adjoint to each other, dx [ K t a ] (x)ρ(x)= dy a(y) [ L t ρ ] (y). (A9.2) This suggests an alternative point of view, which is to push dynamical effects into the density. In contrast to the Koopman operator which advances the trajectory by time t, the Perron-Frobenius operator depends on the trajectory point time t in the past 93

2 APPENDIX A9. IPLEENTING EVOLUTION 94 The Perron-Frobenius operators are non-normal, not self-adjoint operators, so their left and right eigenvectors differ. The right eigenvectors of a Perron- Frobenius operator are the left eigenvectors of the Koopman, and vice versa. While one might think of a Koopman operator as an inverse of the Perron- Frobenius operator, the notion of adjoint is the right one, especially in settings where flow is not time-reversible, as is the case for dissipative PDEs (infinite dimensional flows contracting forward in time) and for stochastic flows. The family of Koopman s operators { K t} t R + forms a semigroup parameterized by time (a) K 0 = (b) K t K t =K t+t t, t 0 (semigroup property), with the generator of the semigroup, the generator of infinitesimal time translations defined by ( A= lim t0 + K t ). t (If the flow is finite-dimensional and invertible,a is a generator of a group). The explicit form of A follows from expanding dynamical evolution up to first order, as in (2.6): ( A a(x)= lim t0 + a( f t (x)) a(x) ) = v i (x) i a(x). (A9.3) t Of course, that is nothing but the definition of the time derivative, so the equation of motion for a(x) is ( ) d dt A a(x)=0. (A9.4) The finite time Koopman operator (A9.) can be formally expressed by expo- appendix A9.2 nentiating the time-evolution generator A as K t = e ta. (A9.5) The generator A looks very much like the generator of translations. Indeed, exercise A9. for a constant velocity field dynamical evolution is nothing but a translation by time velocity: exercise 9.0 e tv x a(x)=a(x+tv). (A9.6) As we will not need to implement a computational formula for general e ta in what follows, we relegate making sense of such operators to appendix A9.2. Here appendix A9.2 we limit ourselves to a brief remark about the notion of spectrum of a linear operator. The Koopman operator K acts multiplicatively in time, so it is reasonable to suppose that there exist constants >0,β 0 such that K t e tβ for all appendeasure - 8may202 ChaosBook.org version5.9, Jun

3 APPENDIX A9. IPLEENTING EVOLUTION 95 t 0. What does that mean? The operator norm is define in the same spirit in which we defined the matrix norms in sect. A45.2: We are assuming that no value ofk t ρ(x) grows faster than exponentially for any choice of functionρ(x), so that the fastest possible growth can be bounded by e tβ, a reasonable expectation in the light of the simplest example studied so far, the exact escape rate (20.4). If that is so, multiplyingk t by e tβ we construct a new operator e tβ K t = e t(a β) which decays exponentially for large t, e t(a β). We say that e tβ K t is an element of a bounded semigroup with generatora β. Given this bound, it follows by the Laplace transform 0 dt e st K t =, Re s>β, s A (A9.7) that the resolvent operator (s A) is bounded ( resolvent =able to cause section A45.2 separation into constituents) s A dt e st e tβ = s β. 0 If one is interested in the spectrum ofk, as we will be, the resolvent operator is a natural object to study. The main lesson of this brief aside is that for the continuous time flows the Laplace transform is the tool that brings down the generator in (9.26) into the resolvent form (20.32) and enables us to study its spectrum. A9.2 Implementing evolution (R. Artuso and P. Cvitanović) We now come back to the semigroup of operatorsk t. We have introduced the generator of the semigroup (9.24) as A= d dt K t t=0. If we now take the derivative at arbitrary times we get ( ) d dt K t ψ (x) ψ( f t+η (x)) ψ( f t (x)) = lim η0 η = v i ( f t (x)) ψ( x) x i = ( K t Aψ ) (x) x= f t (x) which can be formally integrated like an ordinary differential equation yielding exercise A9. K t = e ta. (A9.8) This guarantees that the Laplace transform manipulations in sect. 9.5 are correct. Though the formal expression of the semigroup (A9.8) is quite simple one has appendeasure - 8may202 ChaosBook.org version5.9, Jun

4 APPENDIX A9. IPLEENTING EVOLUTION 96 to take care in implementing its action. If we express the exponential through the power series K t = k=0 t k k! Ak, (A9.9) we encounter the problem that the infinitesimal generator (9.24) contains noncommuting pieces, i.e., there are i, j combinations for which the commutator does not satisfy [ ], v j (x) = 0. x i To derive a more useful representation, we follow the strategy used for finitedimensional matrix operators in sects. 4.3 and 4.4 and use the semigroup property to write K t = t/δτ m= K δτ as the starting point for a discretized approximation to the continuous time dynamics, with time stepδτ. Omitting terms from the second order onwards in the expansion ofk δτ yields an error of order O(δτ 2 ). This might be acceptable if the time stepδτ is sufficiently small. In practice we write the Euler product K t = t/δτ m= ( +δτa(m) ) + O(δτ 2 ) (A9.0) where ( A(m) ψ ) (x)=v i ( f mδτ (x)) ψ x i x= f mδτ (x) As far as the x dependence is concerned, e δτa i acts as e δτa i x x i x d x. (A9.) x i +δτv i (x) x d We see that the product form (A9.0) of the operator is nothing else but a pre- exercise 2.6 scription for finite time step integration of the equations of motion - in this case the simplest Euler type integrator which advances the trajectory byδτ velocity at each time step. A9.2. A symplectic integrator The procedure we described above is only a starting point for more sophisticated approximations. As an example on how to get a sharper bound on the appendeasure - 8may202 ChaosBook.org version5.9, Jun

5 APPENDIX A9. IPLEENTING EVOLUTION 97 error term consider the Hamiltonian flowa=b+c,b= p i q i,c= i V(q) p i. Clearly the potential and the kinetic parts do not commute. We make sense of the exercise A9.2 formal solution (A9.0) by splitting it into infinitesimal steps and keeping terms up toδτ 2 in where K δτ = ˆK δτ + 24 δτ3 [B+2C, [B,C]]+, (A9.2) ˆK δτ = e 2 δτb e δτc e 2 δτb. (A9.3) The approximate infinitesimal Liouville operator ˆK δτ is of the form that now generates evolution as a sequence of mappings induced by (9.27), a free flight by δτb, scattering 2 byδτ V(q ), followed again by 2δτB free flight: { } e 2 q δτb p { } q e δτc p { } e 2 q δτb p { } { q q δτ p = 2 p } p { } { q q } p = p +δτ V(q ) { } { q q δτ } = 2 p p p (A9.4) Collecting the terms we obtain an integration rule for this type of symplectic flow which is better than the straight Euler integration (A9.) as it is accurate up to orderδτ 2 : q n+ = q n δτ p n (δτ)2 2 V (q n δτp n /2) p n+ = p n +δτ V (q n δτp n /2) (A9.5) The Jacobian matrix of one integration step is given by [ ][ ][ ] δτ/2 0 δτ/2 = 0 δτ V(q. (A9.6) ) 0 Note that the billiard flow (9.) is an example of such symplectic integrator. In that case the free flight is interrupted by instantaneous wall reflections, and can be integrated out. Commentary Remark A9. Koopman operators. The Heisenberg picture in dynamical systems theory has been introduced by Koopman and Von Neumann [A.3, A.4], see also ref. [9.2]. Inspired by the contemporary advances in quantum mechanics, Koopman [A.3] observed in 93 thatk t is unitary on L 2 (µ) Hilbert spaces. The Koopman operator is the appendeasure - 8may202 ChaosBook.org version5.9, Jun

6 EXERCISES 98 classical analogue of the quantum evolution operator exp ( iĥt/ ) the kernel ofl t (y, x) introduced in (9.3) (see also sect. 20.3) is the analogue of the Green s function discussed here in chapter 35. The relation between the spectrum of the Koopman operator and classical ergodicity was formalized by von Neumann [A.4]. We shall not use Hilbert spaces here and the operators that we shall study will not be unitary. For a discussion of the relation between the Perron-Frobenius operators and the Koopman operators for finite dimensional deterministic invertible flows, infinite dimensional contracting flows, and stochastic flows, see Lasota-ackey [9.2] and Gaspard [A.65]. Remark A9.2 Symplectic integration. The reviews [A9.0] and [A9.] offer a good starting point for exploring the symplectic integrators literature. For a higher order integrators of type (A9.3), check ref. [A9.6]. Exercises A9.. Exponential form of semigroup elements. Check that the Koopman operator and the evolution generator commute,k t A =AK t A9.3. Symplectic leapfrog integrator. Implement (A9.5), by considering the action of for 2-dimensional Hamiltonian flows; compare it with both operators on an arbitrary state space function a(x). Runge-Kutta integrator by integrating trajectories in A9.2. Non-commutativity. Check that the commutators in some (chaotic) Hamiltonian flow. (A9.2) are not vanishing by showing that ( [B,C]= p V ) p V. q References [A9.]. Axenides and E. Floratos, Strange Attractors in Dissipative Nambu echanics: Classical and Quantum Aspects, arxiv: [A9.2] G. Levine and. Tabor, Integrating the Nonintegrable: Analytic Structure of the Lorenz odel Revisited, Physica 33, 89 (988). [A9.3]. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction (Wiley, New York, 989). [A9.4] H. P. F. Swinnerton-Dyer, The Invariant Algebraic Surfaces of the Lorenz System, ath. Proc. Cambridge Philos. Soc. 32, 385 (2002). [A9.5]. Kuš, Integrals of otion for the Lorenz System, J. Phys. A 6, L689 (983). refsappeasure - 7jan207 ChaosBook.org version5.9, Jun

7 References 99 [A9.6] B. A. Shadwick, J. C. Bowman, and P. J. orrison, Exactly Conservative Integrators, arxiv:chao-dyn/950702, Submitted to SIA J. Sci. Comput. [A9.7] D. J. D. Earn, Symplectic integration without roundoff error, astro-ph/ [A9.8] P. E. Zadunaiski, On the estimation of errors propagated in the numerical integration of ordinary differential equations, Numer. ath. 27, 2 (976). [A9.9] K. Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. ath. 4, 279 (986). [A9.0] P. J. Channell and C. Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity 3, 23 (990). [A9.] J.. Sanz-Serna and. P. Calvo, Numerical Hamiltonian problems (Chapman and Hall, London, 994). [A9.2] J.. Sanz-Serna, Geometric integration, pp. 2-43, in The State of the Art in Numerical Analysis, I. S. Duff and G. A. Watson, eds., (Clarendon Press, Oxford, 997). [A9.3] K. W. orton, Book Review: Simulating Hamiltonian Dynamics, SIA Review 48, 62 (2006). [A9.4] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics (Cambridge Univ. Press, Cambridge 2005). [A9.5] E. Hairer, Ch. Lubich, and G. Wanner, Geometric Numerical Integration (Springer-Verlag, Berlin 2002). [A9.6]. Suzuki, General theory of fractal path integrals with applications to many-body theories and statistical physics, J. ath. Phys. 32, 400 (99). refsappeasure - 7jan207 ChaosBook.org version5.9, Jun