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Another Proof of the Averaging Principle for Fully Coupled Dynamical Systems with Hyperbolic Fast Motions

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Another Proof of the Averaging Principle for Fully Coupled Dynamical Systems with Hyperbolic Fast Motions DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 13, Number 5, December 2005 pp. 1187{1201 ANOTHER PROOF OF THE AVERAGING PRINCIPLE FOR FULLY COUPLED DYNAMICAL SYSTEMS WITH HYPERBOLIC FAST MOTIONS Yuri Kifer Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel Abstract. In the study of systems which combine slow and fast motions which depend on each other (fully coupled setup) whenever the averaging principle can be justi¯ed this usually can be done only in the sense of L1-convergence on the space of initial conditions. When fast motions are hyperbolic (Axiom A) flows or di®eomorphisms (as well as expanding endomorphisms) for each freezed slow variable this form of the averaging principle was derived in [19] and [20] relying on some large deviations arguments which can be applied only in the Axiom A or uniformly expanding case. Here we give another proof which seems to work in a more general framework, in particular, when fast motions are some partially hyperbolic or some nonuniformly hyperbolic dynamical systems or nonuniformly expanding endomorphisms. 1. Introduction. Many real systems can be viewed as perturbations of an idealized system with integrals of motion which start changing slowly after a perturbation producing a combination of slow and fast motions which can only be described by complicated double scale equations whose analysis is not easy. Already in the 19th century in applications to celestial mechanics it was well understood (though without rigorous justi¯cation) that a good approximation of the slow motion can often be obtained by averaging its parameters in fast variables. Later, averaging methods were applied in other mechanical problems, in signal processing both in deterministic and stochastic setups and, rather recently, to model climate{weather interactions (see [13] and [18]). The classical setup of averaging justi¯ed rigorously in [6] presumes that the fast motion does not depend on the slow one and most of the work on averaging treats this case only which is, essentially, completely understood. On the other hand, in real systems both slow and fast motions depend on each other which leads to the more di±cult fully coupled case which we study in this paper. This set up emerges, in particular, in typical perturbations of Hamiltonian systems where motions on constant energy manifolds are fast and across them are slow. " " " Namely, we consider a system of di®erential equations for X = Xx;y and Y = " Yx;y; dX"(t) dY "(t) = "B(X"(t);Y "(t)); = b(X"(t);Y "(t)) (1.1) dt dt with initial conditions X"(0) = x; Y "(0) = y on the product Rd £ M where M is a compact n¡dimensional Riemannian manifold and B(x; y); b(x; y) are bounded and smooth in x; y vector ¯elds on Rd and M; respectively. The solutions of 2000 Mathematics Subject Classi¯cation. Primary: 37D20 Secondary: 34C29, 60F10. Key words and phrases. averaging principle, hyperbolic attractors. The author was partially supported by US-Israel BSF. 1187 1188 YURI KIFER t d t (1.1) determine the flow of di®eomorphisms ©" on R £ M acting by ©"(x; y) = " " t t t (Xx;y(t);Yx;y(t)). Taking " = 0 we arrive at the flow © = ©0 acting by ©0(x; y) = t t t (x; Fxy) where Fx is another family of flows given by Fxy = Yx;y(t) with Y = Yx;y = 0 Yx;y being the solution of dY (t) = b(x; Y (t));Y (0) = y: (1.2) dt It is natural to view the flow ©t as describing an idealized physical system where t parameters x = (x1; :::; xd) are assumed to be constant while the perturbed flow ©" is regarded as describing a real system where evolution of these parameters is also taken into consideration. t Let ¹x be an ergodic invariant measure of the flow Fx. Then the limit Z T ¹ ¹ ¡1 t B(x) = By(x) = lim T B(x; Fxy)dt (1.3) T !1 0 exists for ¹x¡almost all y and it is equal to Z ¹ ¹ B(x) = B¹x (x) = B(x; y)d¹x(y): (1.4) t t If b(x; y) does not, in fact, depend on x then Fx = F and if ¹x = ¹ is also independent of x we arrive at the classical setup. In this case Lipschitz continuity of B implies already that B¹(x) is also Lipschitz continuous in x, and so there exists a unique solution X¹ = X¹x of the averaged equation dX¹ "(t) = "B¹(X¹ "(t)); X¹ "(0) = x: (1.5) dt Then the classical averaging principle says (see [27]) that " ¹ " lim sup jXx;y(t) ¡ Xx(t)j = 0 (1.6) "!0 0·t·T=" for all x and for those y for which (1.3) holds true. In the general case (1.1) the averaged vector ¯eld B¹(x) in (1.3){(1.4) may even not be continuous in x, let alone Lipschitz, and so (1.5) may have many solutions or not at all. Moreover, there d may exist no natural well dependent on x 2 R family of invariant measures ¹x t since dynamical systems Fx may have rather di®erent properties for di®erent x's. Even when all measures ¹x are the same the averaging principle often does not hold true in the form (1.6), for instance, in the presence of resonances (see [22]). Thus even basic results on approximation of the slow motion by the averaged one in the fully coupled case should be formulated usually in a di®erent way and they require stronger and more speci¯c assumptions. If convergence in (1.3) is uniform in x and y then (see, for instance, [21]) any limit ¹ ¹ " " point Z(t) = Zx(t) as " ! 0 of Zx;y(t) = Xx;y(t=") is a solution of the averaged equation dZ¹(t) = B¹(Z¹(t)); Z¹(0) = x: (1.7) dt t It is known that the limit in (1.3) is uniform in y if and only if the flow Fx on M is uniquely ergodic, i.e. it possesses a unique invariant measure, which occurs rather rarely. If, in addition, the uniform convergence in x is required, as well, we arrive at very few speci¯c models of uniquely ergodic families. Thus, the uniform convergence in (1.3) assumption is too restrictive and excludes many interesting cases. Probably, the ¯rst relatively general result on fully coupled averaging is due AVERAGING PRINCIPLE 1189 to Anosov [1] (see also [22] and [21]). Relying on the Liouville theorem he showed t 1 that if each flow Fx preserves a probability measure ¹x on M having a C dependent on x density with respect to the Riemannian volume m on M then for any ± > 0; " ¹ " mesf(x; y) : sup jXx;y(t) ¡ Xx(t)j > ±g ! 0 as " ! 0; (1.8) 0·t·T=" where mes is the product of m and the Lebesgue measure in a relatively compact domain X ½ Rd. An improvement of (1.8) to (1.6) is not possible in a general fully coupled situation (see Appendix in [21]). Observe that Anosov's theorem does not cover other situations where the aver- aging principle is known to work and [20] give necessary and su±cient conditions d which ensure that for given probability measures º on R and ¹x on M, Z Z " ¹ " sup jXx;y(t) ¡ Xx(t)jd¹x(y)dº(x) ! 0 as " ! 0: (1.9) X M 0·t·T=" It was shown in [20] that these conditions hold true when fast motions are hyper- bolic flows. More speci¯cally, [20] considers the situation where b(x; y) is C2 in x and y and for each x in a closure of a relatively compact domain X the flow t Fx is Anosov or, more generally, Axiom A in a neighborhood of an attractor ¤x: SRB t Let ¹x R be the Sinai-Ruelle-Bowen (SRB) invariant measure of Fx on ¤ and set ¹ SRB ¹ B(x) = B(x; y)d¹x (y): It is known (see [9]) that the vector ¯eld B(x) is Lip- schitz continuous in x; and so the averaged equations (1.5) and (1.7) (with right hand side independent of y) have unique solutions X¹ "(t) and Z¹(t) = X¹ "(t="): Still, SRB in general, the measures ¹x are singular with respect to the Riemannian volume on M, and so the method of [1] cannot be applied here. The general conditions from [20] were veri¯ed there for this situation relying on some large deviations argu- SRB ments which yielded (1.9) both with ¹x = ¹x and with ¹x being the Riemannian volume on M. Moreover,it turned out that in this situation the convergence to zero in (1.8) is exponentially fast in 1=". The appropriate large deviations machinery works only in the uniformly hyperbolic situation and in the present paper we give another simpler proof of the averaging principle in the form (1.9) for the above situation which should work also for some partially hyperbolic and nonuniformly hyperbolic fast motions. This simpler proof does not give however the exponentially fast convergence in (1.8). The corresponding problems for the averaging in the discrete time case with di®erence equations in place of (1.1) were studied in ([19]). Namely, consider se- " " " " quences of functions X (n) = Xx;y(n) and Y (n) = Yx;y(n), n = 0; 1; 2; :::, given by recurrence relations X"(n + 1) ¡ X"(n) = "ª(X"(n);Y "(n));X"(0) = x; (1.10) Y "(n + 1) = F (X"(n);Y "(n));Y "(0) = y; where ª is a smooth vector function and Fx = F (x; ¢): M ! M is a smooth map (a di®eomorphism or an endomorphism).
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