Drift of slow variables in slow-fast Hamiltonian systems
aa r X i v : . [ m a t h . D S ] O c t Drift of slow variables in slow-fast Hamiltoniansystems
N. Br¨annstr¨om and V. Gelfreich ∗ Mathematics Institute, University of WarwickCoventry, CV4 7AL, United KingdomE-mail:
[email protected]@warwick.ac.uk
October 15, 2007
Abstract
We study the drift of slow variables in a slow-fast Hamiltonian systemwith several fast and slow degrees of freedom. For any periodic trajectoryof the fast subsystem with the frozen slow variables we define an action.For a family of periodic orbits, the action is a scalar function of the slowvariables and can be considered as a Hamiltonian function which generatessome slow dynamics. These dynamics depend on the family of periodicorbits.Assuming the fast system with the frozen slow variables has a pair ofhyperbolic periodic orbits connected by two transversal heteroclinic tra-jectories, we prove that for any path composed of a finite sequence of slowtrajectories generated by action Hamiltonians, there is a trajectory of thefull system whose slow component shadows the path.
We consider a slow-fast Hamiltonian system described by a smooth Hamiltonianfunction H ( p, q, v, u ; ε ) . This system is slow-fast due to a small parameter in thesymplectic form Ω = dp ∧ dq + 1 ε dv ∧ du . ∗ The authors thank Prof. D. Turaev for useful discussions and helpful suggestions. q = ∂H∂p , ˙ p = − ∂H∂q , ˙ u = ε ∂H∂v , ˙ v = − ε ∂H∂u . (1)Equations of this form often arise after rescaling a part of the variables in aHamiltonian system with the standard symplectic form.The variable ( p, q ) are called fast and ( v, u ) are slow. We assume that thesystem has m + d degrees of freedom, where m is the number of fast degrees offreedom and d is the number of slow ones.After substituting ε = 0 into equation (1) we see that the values of ( v, u )remain constant in time and the system can be interpreted as a family of Hamil-tonian systems with m degrees of freedom which depends on 2 d parameters. Wecall it a frozen system : ˙ q = ∂H∂p , ˙ p = − ∂H∂q , ˙ u = 0 , ˙ v = 0 . (2)The case when the fast system has one degree of freedom is relatively wellunderstood. Indeed, in this case the frozen system typically represents a fastoscillator. The averaging method can be used to eliminate the dependence on thefast oscillations from the slow system. Therefore trajectories of the slow systemare close to trajectories of an autonomous system with 2 d degrees of freedom oververy long time intervals (see e.g. [1, 2]).Points of equilibria of the frozen system form surfaces called slow manifolds.Normally hyperbolic slow manifolds persists and normally elliptic slow manifoldsdo not in general. In both cases the dynamics in a neighbourhood of a slowmanifold can be described using normal forms (for a discussion see e.g. [5]).The case when the fast system has more than one degree of freedom is notablymore difficult. The effect of the fast system on the slow variables strongly dependson the dynamics of the fast system. If the frozen fast system oscillates with aconstant vector of frequencies, generalisations of the averaging method can beused [8, 9]. The averaging method can be also used if the frozen system isuniformly hyperbolic [4] or, more generally, if the frozen system is ergodic andtime averages converge sufficiently fast to space averages [7]. In all these casesthe dynamics of the slow variables is described, in the leading order, by the vectorfield obtained by taking an average of the slow component of (1) over the spaceof fast variables ˙ u = ε (cid:28) ∂H∂v (cid:29) , ˙ v = − ε (cid:28) ∂H∂u (cid:29) . (3)This approximation strongly relies on the fact that in an ergodic system the timeaverage over a trajectory equals the space average for almost all trajectories.2he approximation error strongly depends on the rate of convergence for timeaverages. If the number of fast degrees of freedom is larger than one, there isno reason to expect that the time average over a periodic orbit converges to theaverage over the space. Therefore we should expect that the slow component of atrajectory whose fast component stays near a periodic orbit of the frozen systemmay strongly deviate from the averaged dynamics described by (3). Moreover,we note that periodic orbits are dense in the case of an Anosov system.In this paper we assume that the frozen system has a compact invariant setbearing chaotic dynamics of horseshoe type created by transversal heteroclinicsbetween two saddle periodic orbits. This situation typically arises when a periodicorbit has a transversal homoclinic.In this invariant set hyperbolic periodic orbits are dense and every two periodicorbits are connected by a heteroclinic orbit. We select a finite subset of periodicorbits with relatively short periods. We construct trajectories of the full systemwhich switch between neighbourhoods of the periodic orbits in a prescribed way.We show that the slow component of such trajectories drifts in a way quite similarto trajectories of a random Hamiltonian dynamical system with 2 d degrees offreedom.The trajectories constructed in this paper strongly deviate from the averageddynamics. We think this mechanism is responsible for the largest possible ratesof deviation.A similar construction is used in [6] for studying drift of the enrgy in a Hamil-tonian system which depends on time explicitly and slowly. In particular, itwas shown in [6] that switching between fast periodic orbits indeed provides thefastest rate of energy growth in several situations.The rest of the paper has the following structure. In Section 2 we state ourmain theorem and discuss its application to systems with one slow degree offreedom. In Section 3 we describe slow dynamics of the full system (1) neara family of periodic orbits of the frozen system. The description is based onan action associated with the frozen periodic orbits and can be of independentinterest. The central ingredient of the proof of the main theorem is preservation ofnormally hyperbolic manifolds formed by families of uniformly hyperbolic orbitsof the frozen system which is explained in Section 4. In this section we explainhow symbolic dynamics can be used to describe the dynamics of the full systemrestricted to an invariant subset close to the hyperbolic invariant set of the frozensystem. The discussion is based on ideas of [6]. Section 5 analyses the long timebehaviour of the slow component of the full dynamics. The last section of thepaper finishes the proof of the main theorem.3 Accessibility and drift of slow variables
The total energy is preserved, so we study the dynamics on a single energy level.Without any loss in generality we may consider the dynamics in the zero energylevel M ε = { H ( p, q, v, u ; ε ) = 0 } . First we state our assumptions on the dynamics of the frozen system. Let D ⊂ R d be in a bounded domain. We assume[A1] the frozen system has two smooth families of hyperbolic periodic orbits L c ( v, u ) ⊂ M defined for all ( v, u ) ∈ D , c ∈ { a, b } .[A2] the frozen system has two smooth families of transversal heteroclinic orbits:Γ ab ( v, u ) ⊂ W u ( L a ( v, u )) ∩ W s ( L b ( v, u )) , Γ ba ( v, u ) ⊂ W u ( L b ( v, u )) ∩ W s ( L a ( v, u )) , ∀ ( v, u ) ∈ D .
We note that under these assumptions the frozen system has a family of uniformlyhyperbolic invariant transitive sets Λ ( v,u ) , also known as Smale horseshoes. Forevery ( v, u ) ∈ D , this set contains a countable number of saddle periodic orbits,which are dense in Λ ( v,u ) . Moreover, every two periodic orbits in Λ ( v,u ) are con-nected by a transversal heteroclinic orbit, which also belongs to Λ ( v,u ) . It is wellknown that the dynamics on the Smale horseshoe can be described using thelanguage of Symbolic Dynamics. We defineΛ := [ ( v,u ) ∈ D Λ ( v,u ) . Before stating our main theorem we give a couple of definitions.
Definition 1
The action of a periodic orbit L c is defined by the integral J c ( v, u ) := I L c ( v,u ) p dq . The function J c is independent of the fast variables and can be considered itself asa Hamiltonian function which generates some slow dynamics of ( v, u ) variables: v ′ = − T c ∂J c ∂u , u ′ = 1 T c ∂J c ∂v , (4)where ′ stands for the derivative with respect to the slow time τ = εt , and T c is the period of L c . System (4) is Hamiltonian with the symplectic form ω c = T c ( v, u ) dv ∧ du . Alternatively the equations can be interpreted as a resultof a time scaling in a standard Hamiltonian system.4n the next sections we will show that for properly chosen initial conditions theslow component of the corresponding trajectory of (1) oscillates near a trajectoryof this slow Hamiltonian system.Inside the Smale horseshoe there are infinitely many periodic orbits connectedby transversal heteroclinics. Each periodic orbit has an action associated withit. We select a finite subset of periodic orbits and consider the collection of theiractions. In general we should expect all those actions to be different.In this paper we prove that there are trajectories of the full system suchthat their slow components follow any finite path composed of segments of slowtrajectories generated by actions. Those trajectories of the full system shadowa chain composed of the periodic orbits and heteroclinic trajectories and spendmost of the time near periodic orbits of the frozen system.Let us give a definition of an accessible path and then state the theorem.Consider a finite family of functions J k : D → R , k = 1 , . . . , n . Let Φ τk bethe Hamiltonian flow with Hamiltonian function J k and the symplectic form ω k = T k ( v, u ) dv ∧ du where T k > z = ( v, u ) ∈ D we define σ k ( z ) = sup { τ : Φ τ ′ k ( z ) ∈ D for all τ ′ ∈ (0 , τ ) } , which is the time required to leave the domain D . If the trajectory is definedfor all τ > σ k ( z ) = + ∞ . Obviously, σ k ( z ) > z and k due toopenness of D . Definition 2
We say that
Γ : [0 , T ] → D is an accessible path if Γ is a piece-wise smooth curve composed from a finite number of forward trajectories of theHamiltonian systems generated by J k . More formally, Γ is an accessible path if there are 0 = τ < τ < . . . < τ N = T such that the sequence of points z i := Γ( τ i ) breaks the curve Γ into trajectories,i.e., for every i < N , there is k i , 1 ≤ k i ≤ n , such that for τ ∈ [ τ i , τ i +1 ]Γ( τ ) = Φ τ − τ i k i ( z i ) . Of course, the curve Γ is well defined only if0 < τ i +1 − τ i < σ k i ( z i )which ensures that the trajectories do not leave the domain D . Theorem 1 If D is a bounded domain in R d , the frozen fast system satisfiesassumptions [A1] and [A2] , { J k } nk =1 is a set of actions corresponding to a finiteset of frozen periodic orbits in Λ , and Γ is an accessible path, then there is aconstant C > and ε > such that for every ε < ε there is a trajectory of thefull system (1) such that its slow component z ( t ) satisfies k z ( t ) − Γ( εt ) k < C ε provided ≤ t ≤ ε − T . efinition 3 For any z , z ∈ D , we say that z is accessible from z via thesystem { J k } if there is an accessible path such that Γ(0) = z and Γ( T ) = z . In the case d = 1 the accessibility property has a simple geometrical meaningsince trajectories of the Hamiltonian systems generated by J k are level lines of thefunctions J k . In this case the theorem provides trajectories which follow segmentsof the level lines. The main obstacle for the drift in the slow space is providedby level lines common for all J k . Corollary 1
Consider actions generated by two periodic orbits, a and b . Thoselevel lines of J a,b , which are inside D , are closed curves. The non-singular levellines form rings (or disks), D a and D b . Let V = D a ∩ D b ⊂ D . If J a and J b donot have common level lines, then any point z ∈ V is accessible from any point z ∈ V . Corollary 2
Under the same assumptions. Let us take any finite family of opensets V i ⊂ V , which do not depend on ε . Then for all sufficiently small ε , there isa trajectory which visits all the sets V i . If the energy set M ε is compact the slow dynamics never leaves a boundedset. If at the same time D is a connected set, natural questions arise: Is therea point in D which is not accessible from every other point in D ? Is there atrajectory such that its slow component is dense in D ? Now consider the cylinder formed by periodic orbits of the frozen system: S c, = [ ( v,u ) ∈ D L c ( v, u ) ⊂ M . (5)Let γ ε denote a trajectory of the full system (1) and π s : R m +2 d → R d theprojection on the slow variables.In the next section we will prove that some trajectories stay in a neighbour-hood of S c, for a very long time and provide a detailed description for them. Inthis section we show that in this case the evolution of the slow component π s γ ε approximately follows a trajectory of the slow Hamiltonian flow Φ εtc generated bythe action J c . Lemma 1
Let L c be a family of periodic orbits of the frozen system. If γ ε is afamily of solutions of the full system (1) such that there are z ∈ D and C > such that k π s γ ε (0) − z k < C ε, (6)(ii) there are constants C > and τ < σ c ( z ) such that dist( γ ε ( t ) , S c, ) ≤ C ε ∀ t ∈ [0 , ε − τ ] , (7) then there is C > such that k π s γ ε ( t ) − Φ εtc ( z ) k ≤ C ε (8) for all t ∈ [0 , ε − τ ] .Proof. We write ( p c ( t, v, u ) , q c ( t, v, u )) to denote a periodic solution of the frozensystem and use T c ( v, u ) for the corresponding period: p c ( t + T c ( v, u ) , v, u ) ≡ p c ( t, v, u ) ,q c ( t + T c ( v, u ) , v, u ) ≡ q c ( t, v, u ) . (9)Then the action of the periodic orbit L c is given by the following integral J c ( v, u ) = Z T c p c ∂q c ∂t dt . (10)Since L c belongs to the zero energy level we have a useful identity: H ( p c ( t, v, u ) , q c ( t, v, u ) , v, u ; 0) = 0 (11)for all ( v, u ) ∈ D and all t ∈ R .Let Σ denote a smooth hypersurface in R m +2 d transversal to the flow of thefrozen system such that every periodic orbit of the family L c has exactly oneintersection with Σ. Let M i = γ ε ( t i ) be a sequence of consecutive intersectionsof γ ε with Σ and consider the slow components of those points: ˆ z i := π s M i .We note that inequality (7) and the smooth dependence of p c ( s, z ) , q c ( s, z ) on z imply that there is C > i there is s i such that k M i − ( p c ( s i , ˆ z i ) , q c ( s i , ˆ z i ) , ˆ z i ) k ≤ C ε . Since solutions of differential equations depend smoothly on the initial conditionsand vector field, the segment of γ ε ( t ), t i ≤ t ≤ t i +1 is close to L c (ˆ z i ): γ ε ( t ) = ( p c ( s i + t − t i , ˆ z i ) , q c ( s i + t − t i , ˆ z i ) , ˆ z i ) + O ( ε ) (12)and the time of the first return to the section Σ is close to the period of the frozentrajectory: t i +1 − t i = T c (ˆ z i ) + O ( ε ) . (13)7ow we estimate the displacement ˆ z i +1 − ˆ z i . We write ˆ z i = ( v, u ) and ˆ z i +1 = (¯ v, ¯ u )to shorten the notation. Integrating the slow component of the vector field alongthe exact trajectory and using (1) we conclude¯ u − u = Z t i +1 t i ˙ udt = ε Z T c ( v,u )0 ∂H∂v (cid:12)(cid:12)(cid:12)(cid:12) p c ( t,v,u ) ,q c ( t,v,u ) ,v,u dt + O ( ε ) , ¯ v − v = Z t i +1 t i ˙ vdt = − ε Z T c ( v,u )0 ∂H∂u (cid:12)(cid:12)(cid:12)(cid:12) p c ( t,v,u ) ,q c ( t,v,u ) ,v,u dt + O ( ε ) , (14)where the error terms come from replacing the exact trajectory by the frozen oneand from the difference in the return time, see (12) and (13). The integrals in theright hand side can be expressed in terms of derivatives of the action defined byintegral (10). Indeed, differentiating (10) with respect to u , integrating by partsand taking into account (9), we get ∂J c ∂u = Z T c (cid:18) ∂p c ∂u ∂q c ∂t − ∂q c ∂u ∂p c ∂t (cid:19) dt = Z T c (cid:18) ∂p c ∂u ∂H∂p + ∂q c ∂u ∂H∂q (cid:19) dt . Then differentiating identity (11) we get ∂p c ∂u ∂H∂p + ∂q c ∂u ∂H∂q = − ∂H∂u , where the derivatives are evaluated at ( p c ( t, v, u ) , q c ( t, v, u ) , v, u ). Consequently ∂J c ∂u = − Z T c ( v,u )0 ∂H∂u (cid:12)(cid:12)(cid:12)(cid:12) p c ( t,v,u ) ,q c ( t,v,u ) ,v,u dt . Repeating these arguments with u replaced by v we also get ∂J c ∂v = − Z T c ( v,u )0 ∂H∂v (cid:12)(cid:12)(cid:12)(cid:12) p c ( t,v,u ) ,q c ( t,v,u ) ,v,u dt . Substituting the last two equalities into (14) we arrive to¯ u = u − ε ∂J c ∂v + O ( ε ) , ¯ v = v + ε ∂J c ∂u + O ( ε ) . (15)We see that the displacement between two consecutive intersections of γ ε withsection Σ is approximated by the time- εT c shift along a trajectory of the Hamil-tonian vector field (4) generated by the Hamiltonian function J c :ˆ z i +1 = ˆ z i + Φ εT c c (ˆ z i ) + O ( ε ) . z = z + O ( ε ). Then a rather standard stabilityestimate can be used to showˆ z i = Φ iεT c c ( z ) + O ( ε ) for 0 ≤ i ≤ ε − τ . (16)Finally, we note that ˆ z i = π s γ ε ( t i ) where t i = iT c + O ( εi ) due to (13). Be-tween intersections with Σ the slow component π s γ ε changes by a value of theorder of O ( ε ). Therefore the estimate is extendable to values of t between theintersections and (8) follows immediately. (cid:3) We note that J c is preserved by Φ τc and therefore J c ◦ π s is an adiabaticinvariant for the restriction of the full dynamics on a neighbourhood of S c, .In general, we do not expect the estimates to be valid on time intervals longerthan stated by Lemma 1. For example, note that a trajectory of J c may leavethe domain D in finite time.It is interesting that under additional assumptions ˆ J c ( t ) := J c ( π s γ ε ( t )) maystay near its initial value, ˆ J c (0), for much longer time.Indeed, consider the case of one slow degree of freedom, d = 1, and supposethat level lines of J c are closed curves on the plane of ( v, u ) variables. Then σ c ( z ) = + ∞ for all z ∈ D . In the next section we will show that the full systemhas a locally invariant cylinder S c,ε close to S c, . Then equations (14) describea Poincar´e map on the section defined by intersection of S c,ε and Σ. In thecase of one slow degree of freedom we may suppose that the map (14) satisfiesassumptions of the KAM theorem, then ˆ J c ( t ) will stay close to its initial valueforever. Indeed, under these assumptions the trajectories on S c,ε are trappedbetween two KAM tori.We also note that averaging theory can be used to study the dynamics on S c,ε for d ≥ The following arguments are based on [6]. Let w = ( p, q ) and z = ( v, u ) toshorten notation. Then the frozen system (2) has the form˙ w = G ( w, z ) , (17)where G is expressed in terms of partial derivatives of H for ε = 0. The Hamil-tonian function H ( w, z ) is an integral of system (17).Let system (17) have two smooth families of saddle periodic orbits L a : w = w a ( t, z ) and L b : w = w b ( t, z ) for all z ∈ D . Assume that both families belong tothe zero energy level M . Take a pair of smooth cross-sections, Σ a and Σ b , whichare transverse to the vector field and such that each periodic trajectory L a ( z )9nd L b ( z ) has exactly one point of intersection with the corresponding section.Denote the Poincar´e map on Σ c near L c as Π cc ( c = a, b ). The Poincar´e map issmooth and depends smoothly on z .We assume that for all z ∈ D the frozen system has a pair of transversalheteroclinic orbits: Γ ab ⊆ W u ( L a ) ∩ W s ( L b ) and Γ ba ⊆ W u ( L b ) ∩ W s ( L a ).Let Π ab and Π ba be maps defined on subsets of Σ a and Σ b by following orbitsclose to Γ ab and Γ ba , respectively. Therefore Π ab acts from some open set in Σ a into an open set in Σ b , while Π ba acts from an open set in Σ b into an open set inΣ a . There is a certain freedom in the definition of the maps Π ab and Π ba . Indeed,each of these maps acts from a neighbourhood of one point of a heteroclinic orbitto a neighbourhood of another point of the same orbit, therefore different choicesof the points lead to different maps.Figure 1: Poincar´e maps near two periodic orbitsWhen the maps are fixed, every orbit that lies entirely in a sufficiently smallneighbourhood of the heteroclinic cycle L a ∪ L b ∪ Γ ab ∪ Γ ba corresponds to auniquely defined sequence of points M i ∈ Σ a ∪ Σ b such that M i +1 = Π ξ i ξ i +1 M i where ξ i = c if M i ∈ Σ c ( c ∈ { a, b } ) . In this way the trajectory of the frozen system defines a sequence { ξ i } i =+ ∞ i = −∞ whichis called the code of the orbit .The periodic orbits L a and L b are saddle and the intersections of the stable andunstable manifolds of L a and L b that create the heteroclinic orbits are transversedue to the assumptions [A1] and [A2]. Consequently (cf. [3]), one can choose themaps Π ab and Π ba and define coordinates ( x, y, z ) in Σ a and Σ b in such a waythat the following holds.[H1] Σ c ∩ M is diffeomorphic to the product X c × Y c × D where X c and Y c areballs in R m − of a radius R > c, c ′ ∈ { a, b } the Poincar´e map Π cc ′ can be written in thefollowing “cross-form” [12]: there exist smooth functions f cc ′ , g cc ′ : X c × c ′ → X c ′ × Y c such that a point M ( x, y, z ) ∈ Σ c is mapped to ¯ M (¯ u, ¯ w, z ) ∈ Σ c ′ by the map Π cc ′ if and only if¯ x = f cc ′ ( x, ¯ y, z ) , y = g cc ′ ( x, ¯ y, z ) . (18)[H3] There exists λ < (cid:13)(cid:13)(cid:13)(cid:13) ∂ ( f cc ′ , g cc ′ ) ∂ ( x, ¯ y ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ λ < , (19)where the norm of the Jacobian matrix corresponds to max {k x k , k y k} .Inequality (19) implies that for a fixed z ∈ D the set Λ z of all orbits that lieentirely in a sufficiently small neighbourhood of the heteroclinic cycle L a ∪ L b ∪ Γ ab ∪ Γ ba in the energy level M is hyperbolic, a horseshoe. Moreover, one canshow that the orbits in Λ z are in one-to-one correspondence with the set of allsequences of a ’s and b ’s, i.e. for every sequence { ξ i } i =+ ∞ i = −∞ there exists one andonly one orbit in Λ z which has this sequence as its code.Indeed, take any orbit from Λ z and denote by M i ( x i , y i , z ) the sequence of itsintersections with the cross sections. Equation (18), implies that the orbit has acode { ξ i } i =+ ∞ i = −∞ if and only if the coordinates of M i satisfy the equations x i +1 = f ξ i ξ i +1 ( x i , y i +1 , z ) , y i = g ξ i ξ i +1 ( x i , y i +1 , z ) . Therefore the sequence { ( x i , y i ) } + ∞ i = −∞ is a fixed point of the operator { ( x i , y i ) } + ∞ i = −∞
7→ { ( f ξ i − ξ i ( x i − , y i , z ) , g ξ i ξ i +1 ( x i , y i +1 , z ) } + ∞ i = −∞ . (20)Equation (19) implies this operator is a contraction of the space Q + ∞ i = −∞ X ξ i × Y ξ i ,hence the existence and uniqueness of the orbit with the code { ξ i } i =+ ∞ i = −∞ follow(see e.g. [11]).Moreover, the fixed point of a smooth contracting map depends smoothly onparameters. Consequently the orbit depends smoothly on z and the derivativesof ( x i ( z, ξ ) , y i ( z, ξ )) are bounded uniformly for all i . Lemma 2
If the Poincar´e maps satisfy assumptions [H1]–[H3] , then for any twocode sequences ξ (1) = { ξ (1) i } + ∞ i = −∞ and ξ (2) = { ξ (2) i } + ∞ i = −∞ , which satisfy ξ (1) i = ξ (2) i for | i | ≤ n the corresponding intersections with the cross sections are bounded by max (cid:8) k x i ( z, ξ (1) ) − x i ( z, ξ (2) ) k , k y i ( z, ξ (1) ) − y i ( z, ξ (2) ) k (cid:9) ≤ Rλ n −| i | , (21) where the constants R > and λ ∈ (0 , are defined in [H1] and [H3] respectivelyand do not depend on the sequences ξ (1 , . roof. First we note, thatmax (cid:8) k x i ( z, ξ (1) ) − x i ( z, ξ (2) ) k , k y i +1 ( z, ξ (1) ) − y i +1 ( z, ξ (2) ) k (cid:9) ≤ λ max (cid:8) k x i +1 ( z, ξ (1) ) − x i +1 ( z, ξ (2) ) k , k y i ( z, ξ (1) ) − y i ( z, ξ (2) ) k (cid:9) (22)for − n ≤ i < n . Since none of the normes involved exceeds 2 R we immediatelyconclude thatmax (cid:8) k x i ( z, ξ (1) ) − x i ( z, ξ (2) ) k , k y i +1 ( z, ξ (1) ) − y i +1 ( z, ξ (2) ) k (cid:9) ≤ Rλ . (23)Then the following estimate is true for n ′ = 1max { k x i ( z, ξ (1) ) − x i ( z, ξ (2) ) k , k y i +1 ( z, ξ (1) ) − y i +1 ( z, ξ (2) ) k }≤ R (cid:26) λ n ′ − i for 0 ≤ i < n ′ λ n ′ +1+ i for − n ′ − < i < . (24)We continue inductively in n ′ . Assuming that the estimate (24) holds for n ′ replaced by n ′ − n ′ + 1 different values of i in the order of decreasing of | i | . On each step we use the contraction property(22) and the sharper of upper bounds (24) and (23). In the case n ′ = n ,max { k x n ( z, ξ (1) ) − x n ( z, ξ (2) ) k , k y − n ( z, ξ (1) ) − y − n ( z, ξ (2) ) k } ≤ R is used instead of (23). Then (21) follows directly from the last upper bound and(24) taken with n ′ = n . (cid:3) Now let us consider the full system (1) for a small ε >
0. Since the vectorfield depends smoothly on ε , the Poincar´e maps Π cc ′ : Σ c → Σ c ′ are still definedand can be written in the following form: ( ¯ x = f cc ′ ( x, ¯ y, z, ε ) , y = g cc ′ ( x, ¯ y, z, ε )¯ z = z + εφ cc ′ ( x, ¯ y, z, ε ) , (25)where f, g, φ are bounded along with the first derivatives and f, g satisfy (19).For technical reasons we need to assume that the domain D is invariant underthe Poincar´e map, i.e., φ cc ′ ( x, ¯ y, z, ε ) ≡ z ∈ ∂D . If this is not the case, thenwe modify φ cc ′ in a small neighbourhood of the boundary. We note that thenext lemma contains a statement of uniqueness but the surfaces provided by thelemma may depend on the way the functions φ cc ′ have been modified. Thereforethe lemma implies existence but not uniqueness for the original system.The next lemma is of a general nature and has little to do with the Hamilto-nian structure of the equations. Rather we notice that by fixing any code ξ andvarying z ∈ D we obtain at ε = 0 a sequence of smooth two-dimensional surfaces.The i -th surface is the set run by the point M i ( z ) of the uniquely defined orbitwith the code ξ . This sequence is invariant with respect to the correspondingPoincar´e maps and is uniformly normally-hyperbolic — hence it persists at all ε sufficiently small. 12 emma 3 Given any sequence ξ of a ’s and b ’s, there exists a uniquely definedsequence of smooth surfaces L i ( ξ, ε ) : ( x, y ) = ( x i ( z, ξ, ε ) , y i ( z, ξ, ε )) (26) such that Π ξ i ξ i +1 L i = L i +1 . (27) The functions ( x i , y i ) are defined for all small ε and all z ∈ D , they are uniformlybounded along with their derivatives with respect to z and satisfy (21) . Moreover,there is C > independent from the code ξ such that k ( x i ( z, ξ, ε ) − x i ( z, ξ, , y i ( z, ξ, ε ) − y i ( z, ξ, k ≤ Cε , for all i ∈ Z . A proof of this lemma is essentially identical to the proof of Lemma 1 of [6] andis based on contraction mapping arguments: the functions x i , y i are constructedas a fixed point of an operator similar to (20).We note that if ξ = c ∞ is a code which consists of the symbol c only, then L i , x i and y i are independent from i , and we will denote them by L c , x c and y c respectively. Let ξ be a code. The corresponding trajectory of the full system is described bythe dynamics of its slow component: z i +1 = z i + εφ ξ i ξ i +1 ( x i ( z i , ξ, ε ) , y i ( z i , ξ, ε ) , z i , ε ) . (28)If ξ = c ∞ the functions x i and y i do not depend on i , so we denoted them by x c and y c respectively. Then the equation can be written in the form¯ z i +1 = ¯ z i + εφ cc ( x c (¯ z i , ε ) , y c (¯ z i , ε ) , ¯ z i , ε ) , (29)where the bars over z i and z i +1 are used to distinguish trajectories of (29) and(28).The next lemma estimates the difference between these two slow dynamicsfor all sequences which have a large block of c ’s. Lemma 4
Assume the assumptions of Lemma 3 are satisfied. Then for any K > , t > , there is K > and ε > such that for any | ε | < ε and anycode ξ such that for some index jξ j = ξ j +1 = . . . = ξ j + ⌊ t ε ⌋ = c he inequality k z j − ¯ z k ≤ εK implies the corresponding trajectories of (28) and (29) satisfy the inequality (cid:13)(cid:13) z j + N − ¯ z N (cid:13)(cid:13) ≤ εK for all ≤ N ≤ N ( ε ) ≡ (cid:4) t ε (cid:5) .Proof. Using (28) we get z j + N = z j + ε j + N − X i = j φ ξ i ξ i +1 ( x i ( z i , ξ, ε ) , y i ( z i , ξ, ε ) , z i , ε ) . (30)Using (29), we obtain in a similar way¯ z N = ¯ z + ε N − X i =0 φ cc ( x c (¯ z i , ε ) , y c (¯ z i , ε ) , ¯ z i , ε ) . (31)We have assumed k ¯ z − z j k ≤ K ε . Then taking the difference of the equalities(30) and (31), we obtain k z j + N − ¯ z N k ≤ εK + ε N − X i =0 k φ cc ( x j + i ( z j + i , ξ, ε ) , y j + i ( z j + i , ξ, ε ) , z j + i , ε ) − φ cc ( x c (¯ z i , ε ) , y c (¯ z i , ε ) , ¯ z i , ε ) k (32)Consequently, k z j + N − ¯ z N k ≤ εK + ε (cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂z (cid:13)(cid:13)(cid:13)(cid:13) N − X i =0 k z j + i − ¯ z i k (33)+ ε (cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂ ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) N − X i =0 k ( x j + i ( z j + i , ξ, ε ) − x c (¯ z i , ε ) , y j + i ( z j + i , ξ, ε ) − y c (¯ z i , ε )) k . In order to estimate the last term we note that Lemma 3 includes the estimate (21) N − X i =0 k ( x j + i ( z j + i , ξ, ε ) − x c (¯ z i , ε ) , y j + i ( z j + i , ξ, ε ) − y c (¯ z i , ε )) k≤ N − X i =0 k ( x j + i ( z j + i , ξ, ε ) − x c ( z j + i , ε ) , y j + i ( z j + i , ξ, ε ) − y c ( z j + i , ε )) k + N − X i =0 k ( x c ( z j + i , ε ) − x c (¯ z i , ε ) , y c ( z j + i , ε ) − y c (¯ z i , ε )) k≤ N − X i =0 Rλ min { i,N ( ε ) − i } + N − X i =0 max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) ∂x c ∂z (cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13) ∂y c ∂z (cid:13)(cid:13)(cid:13)(cid:13)(cid:27) k z j + i − ¯ z i k≤ R − λ + max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) ∂x c ∂z (cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13) ∂y c ∂z (cid:13)(cid:13)(cid:13)(cid:13)(cid:27) N − X i =0 k z j + i − ¯ z i k k z j + N − ¯ z N k ≤ ε (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂z (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂ ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) ∂x c ∂z (cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13) ∂y c ∂z (cid:13)(cid:13)(cid:13)(cid:13)(cid:27)(cid:19) N − X i =0 k z j + i − ¯ z i k + ε (cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂ ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) R − λ + εK . Let A = (cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂ ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) R − λ + K .B = (cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂z (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) ∂φ cc ∂ ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) ∂x c ∂z (cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13) ∂y c ∂z (cid:13)(cid:13)(cid:13)(cid:13)(cid:27) . Then k z j + N − ¯ z N k ≤ εA + εB N − X i =0 k z j + i − ¯ z i k and consequently k z j + N − ¯ z N k ≤ εA e εNB . So for N ≤ t ε we have k z j + N − ¯ z N k ≤ K ε (34)where K = A e t B . (cid:3) We note that Lemma 4 is also valid for any two sequences with any largecommon block.
Lemma 5
Assume the assumptions of Lemma 3 are satisfied. Then for any K > , t > , there is K > and ε > such that for any | ε | < ε and anytwo codes ξ (1) and ξ (2) such that for some index jξ (1) j + i = ξ (2) j + i ≤ i ≤ N ( ε ) ≡ (cid:22) t ε (cid:23) the inequality (cid:13)(cid:13)(cid:13) z (1) j − z (2) j (cid:13)(cid:13)(cid:13) ≤ εK implies (cid:13)(cid:13)(cid:13) z (1) j + N − z (2) j + N (cid:13)(cid:13)(cid:13) ≤ εK for all ≤ N ≤ N ( ε ) . The proof of this lemma is almost identical to the previous one so we skip it.15
Proof of Theorem 1
Now we have all ingredients necessary to complete the proof of Theorem 1. Eachperiodic orbit L k ∈ Λ is defined by a periodic code. Let ℓ be the longest periodamong the codes corresponding to the periodic orbits selected in the assumptionsof Theorem 1. Then each of the periodic orbits can be uniquely identified by apiece of code c k ∈ { a, b } ℓ .Given an accessible path Γ we define∆ i = τ i +1 − τ i . It is the time the slow motion follows the flow defined by the Hamiltonian function J k i , 1 ≤ i ≤ N , where N is the number of segments in the path. Let N i ( ε ) = (cid:22) ∆ i εℓ (cid:23) . Let ω i = c N i ( ε ) k i be a finite sequence, which consists of N i ( ε ) copies of the symbol c k i . Let ξ ε be any sequence, which contains ω ω . . . ω N starting from position 0.Obviously, the sequence j i = ℓ i X l =0 N l ( ε )indicates the starting positions of the blocks ω i in the code ξ ε .We note that assumptions [A1] and [A2] imply that there are sections andPoincar´e maps of the frozen system (2) which satisfy [H1]—[H3]. In order to applyLemma 3 we have to modify the slow component of the Poincar´e maps to ensureinvariance of D . Since D is open there is δ > δ -neighbourhood ofΓ is inside D . Then we modify φ cc ′ outside this δ -neighbourhood of Γ to ensurethat φ cc ′ vanishes near ∂D . This modification does not affect trajectories whichdo not leave an O ( ε ) neighbourhood of Γ: i.e. while a trajectory of the modifiedmaps stays in the neighbourhood of Γ it is simultaneously a trajectory of theoriginal Poincar´e maps.Lemma 3 implies that there is a sequence of surfaces which corresponds tothe sequence ξ ≡ ξ ε . Now let z = Γ(0) and consider the sequence of points M i ≡ M ξ i ( x i ( z i , ξ, ε ) , y i ( z i , ξ, ε ) , z i ; ε )on those surfaces. The slow component z i satisfies equation (28) and M i ∈ M ε ∩ Σ ξ i . Lemma 4 (or Lemma 5) implies that k z i − ¯ z i k ≤ K ε i such that 0 ≤ i ≤ N ( ε ), where ¯ z i denote the trajectory of (29). Wecontinue inductively to show using Lemma 4 that there are constant K k suchthat k z i − ¯ z i − j k − k ≤ K k ε (35)for all i such that j k − ≤ i ≤ j k and 1 ≤ k ≤ N , where ¯ z l satisfy (28) withinitial condition ¯ z := z j k − . We note that these ¯ z i − j k − all belong to the invariantsurface L c k , then Lemma 3 impliesdist( M i , S c k , ) = O ( ε ) . We consider the trajectory of the full system (1), which we denote by γ ε , suchthat γ ε (0) = M . Since γ ε goes through the points M i it also stays O ( ε )-close to S c k , between the points thereforedist( γ ε ( t ) , S c k , ) = O ( ε )for t ∈ [ τ k − ε − , τ k ε − ]. Then Lemma 1 implies that the slow component π s γ ε ( t )shadows the accessible path Γ. Finally, we note that similar equations arise in the case of a Hamiltonian systemwith the standard symplectic form, Ω st = dp ∧ dq + dv ∧ du when a Hamilto-nian function looses some degrees of freedom at ε = 0. More precisely, if theHamiltonian function has the form H ( p, q, v, u ; ε ) = H ( p, q ) + εH ( p, q, v, u ) , the corresponding Hamilton equations are given by˙ q = ∂H ∂p + ε ∂H ∂p , ˙ p = − ∂H ∂q − ε ∂H ∂q , ˙ u = ε ∂H ∂v , ˙ v = − ε ∂H ∂u . (36)In this equation, adiabatic invariants can be destructed by resonances [10].These equation are quite similar to (1). We note that the frozen fast systemis independent of the slow variables. The theory developed in this paper canbe applied to the system (36). The most notable difference is related to thedescription of the slow motion near a cylinder formed by periodic orbits of thefrozen system. Indeed the slow motion is described by the averaged perturbationterm ˜ J c ( u, v ) = Z T c H ( p c ( t ) , q c ( t ) , u, v ) dt and not by the actions. 17 eferences [1] Arnold V.I. Mathematical Methods of Classical Mechanics. SpringerVerlag, 1978[2] Bogolyubov N.N., Mitropol’skii Yu.A. Asymptotic Methods in the The-ory of Nonlinear Oscillations Gordon and Breach, 1961.[3] Afraimovich, V.S., Shilnikov, L.P., On critical sets of Morse-Smale sys-tems, Trans. Moscow Math. Soc. 28 (1973) 179–212.[4] Anosov D., Averaging in systems of ODEs with rapidly oscillating so-lutions, Izv. Akad. Nauk. SSSR 24 (1960) 721–742[5] Gelfreich V., Lerman L. Long-periodic orbits and invariant tori in asingularly perturbed Hamiltonian system, Physica D Vol 176 Iss. 3–4,(2003) 125–146[6] Gelfreich V., Turaev D., Unbounded energy growth in Hamiltonian sys-tems with a slowly varying parameter, Math. Physics Preprint Archive(