Singularly perturbed Hamiltonian systems: the ... - Semantic Scholar

Singularly perturbed Hamiltonian systems: the dynamics near slow manifold∗ L. Lerman Institute for Applied Math. & Cybernetics and University of Nizhny Novgorod 10 Ul’yanov St., Nizhny Novgorod, 603005 Russia E-mail: [email protected]

Abstract. When studying a slow-fast (singularly perturbed) Hamiltonian system in two degrees of freedom one can say unexpectedly much about its dynamics near its slow manifold. We demonstrate some details of this picture assuming the phase portrait of the slow system be known and generic. Since this latter system is in one degree of freedom, the problem in question is about the structure near the annulus on the slow manifold filled with periodic orbits, near a center singular point, and near a separatrix loop. In all these case we present the details of the orbit structure.

1

Introduction

One of successful approaches to studying nonlinear systems is the detection, if possible, of different time scales and then the investigation of this multiscaled system using the singular perturbation theory [1, 3, 26]. In the most explicit form it can be described by the so-called slow-fast or singularly perturbed system (see, e.g. [3, 26]) ε

dY = Q(X, Y, ε), dτ

dX = P (X, Y, ε), dτ

(1)

where X ∈ Rn , Y ∈ Rm , and ε > 0 is a small real positive parameter. Variables X are usually called fast variables, and Y slow variables. For ε > 0 it is convenient to introduce the fast time t = τ /ε and rewrite (1) in the following form X
= P (X, Y, ε),

Y
= εQ(X, Y, ε),

(2)

where the prime denotes differentiation in the fast time. In our analysis we will use both these forms of the system. If we formally set ε = 0 in (1), (2), then we obtain two limit systems. It is clear from the very beginning that the limit systems of the equations (1) and (2) are different. In ∗

This paper is an overview of recent results [14-16] obtained in the collaboration with V. Gelfreich.

1

(1) the formal limit ε = 0 leads to an algebraic equation P (X, Y, 0) = 0 and a differential system, where X is considered as a function of Y , X = ϕ(Y ). The function ϕ is not necessarily univalued. The set of points (X, Y ) determined by the equation P (X, Y, 0) = 0 is usually called slow manifold , and the corresponding system Y˙ = Q(ϕ(Y ), Y, 0) is called slow system. In the equations (2) the X-projection of the limit ε = 0 gives a so-called fast system, where Y -coordinates are considered as parameters. X-coordinate of a point on the slow manifold is an equilibrium for the fast system. In the space (X, Y ) the limit system (2) has an invariant foliation with invariant leaves Y = Y0 , and a manifold of equilibria – slow manifold. As is common in the perturbation theory, the main task is to understand the properties of the system (1) or (2) for ε positive, knowing properties of both limit systems. The given geometric interpretation suggests that the form (2) is more flexible for the qualitative study. In order to analyse the slow-fast system for ε > 0, we will take into account the behavior of both systems, slow and fast ones, as well as the type of the function ϕ(y) (univalued, multivalued, etc. [3]). We will consider a singularly perturbed two-degrees-of-freedom Hamiltonian system which loses one degree of freedom at ε = 0,

or, with the fast time

εx˙ = ∂H/∂y,

u˙ = ∂H/∂v,

εy˙ = −∂H/∂x,

v˙ = ∂H/∂u,

x
= ∂H/∂y,

u
= ε∂H/∂v,

y
= −∂H/∂x,

v
= −ε∂H/∂u,

(3)

(4)

where H = H(x, y, u, v, ε) is an analytic function in all its variables. The latter system is Hamiltonian with respect to the symplectic form Ω = dy ∧ dx + ε−1 dv ∧ du, and so is the former one with respect to the form εΩ. Later on we assume that the slow manifold is represented by an univalued function of the slow variables (u, v) in some region of these variables. We will suppose the fast variables (x, y) to be elliptic near slow manifold, that is, all equilibria of the fast system corresponding to the points on slow manifold are centers. Since slow system is in one degree of freedom, its behavior is, in principle, well known, if we assume that this system is generic, in particular, all its equilibria are saddles or centers. If the Hamiltonian of slow system possesses, in addition, the property that all its levels are compact sets, then the phase portrait of slow system in an invariant compact region is the following one. There are a finite number of saddle equilibria, their separatrices divide this region into a finite number of cells filled with periodic orbits which are contracted inside one cell to a center equilibrium. Passing to the structure of the whole system near elliptic slow manifold, we then need to understand what orbit structure will be near an annulus filled with periodic orbits of slow system, near a center equilibrium, and, finally, near the object that we call “ghost” separatrix loop. This is merely a closure of a homoclinic orbit 2

on the slow manifold that is supposed to exist for ε = 0. We call this separatrix “ghost” since it does not really exist as an orbit of the whole system at ε = 0 but it is visible at all orders of power expansions in ε > 0. All this was studied in [14, 15, 16] and here we present an overview of these results. One of the results obtained is that in a neighborhood of the “ghost” separatrix there are one-dimensional stable and unstable semi-separatrices of a singular point of saddle-center type [23, 22] such that their splitting is exponentially small. In a particular situation, arguments in favor of exponentially smallness were given in [10, 18, 20]. Here we follow a different approach (see, [13, 12]) which are independent of the particular form of the Hamiltonian (the Hamiltonian nature of the problem was not observed in [10, 18, 20]). The geometric approach for studying singularly perturbed systems goes back to Fenichel [11] for the case, when slow manifold is normally hyperbolic. Here we go into another direction and consider the case, when this manifold is elliptic. This leads to a quite different picture of the dynamics. In fact, the approach we follow is close to the lines of papers by Neishtadt [28, 29, 30].

1.1

The set-up

We consider a Hamiltonian system of the form (3) with an analytic H in all its variables. We impose the following condition. Assumption 1. For ε = 0 a system of equations ∂H/∂y = 0,

∂H/∂x = 0

(5)

y = g(u, v)

(6)

can be solved with respect to x, y x = f (u, v),

with real analytic functions f, g in a domain D0 of variables (u, v). Moreover, we assume an inequality 

∆(u, v) = det 

2 2 ∂xx H ∂xy H 2 2 H ∂yy H ∂yx

     

>0

(7)

x=f (u,v),y=g(u,v),ε=0

for the Hessian fulfills in a region of R4 , which contains the graph of the function (6). In particular, the solutions of (5) are then locally unique due to the Implicit Function theorem. Recall that the manifold given by (6) is usually called slow manifold and the system of equations for u, ˙ v˙ in (3) is called slow system after plugging in the functions (6). It is evident that the slow system is Hamiltonian with the Hamilton function h(u, v) = H(f (u, v), g(u, v), u, v, 0). Let (u0 , v0 ) be an equilibrium of the slow system, then the point O = (f (u0 , v0 ), g(u0, v0 ), u0 , v0 ) 3

(8)

is an equilibrium of the system (4) for ε = 0, since x˙ = y˙ = 0 on the slow manifold. Though all points in the slow manifold are equilibria for (4) at ε = 0, not all of them can be continued as equilibria for ε > 0. Since ∆ = 0 in (7) for positive ε small enough, a solution (f (x, y, ε), g(x, y, ε)) of (5) also exists and is unique for ε small enough in a neighborhood of the slow manifold. Inserting these functions into the third and fourth equations in (3) one finds the resulting system to be ε-close to the slow system, so it has an equilibrium (u(ε), v(ε)) persisting (u0 , v0 ), if the equilibrium (u0 , v0 ) is nondegenerate (with nonzero eigenvalues). Furthemore, it is easily checked that (f (u(ε), v(ε), ε), g(u(ε), v(ε), ε), u(ε), v(ε)) is an equilibrium for (3), we will denote it O(ε). Remark 1 . It is evident that O(ε) depends on ε analytically, so we can shift this point to the origin. We suppose this is done henceforth. It means, in particular, that H does not contain linear terms in the phase variables near the origin. The type of O(ε) depends both on the eigenvalues of the fast system on the leaf u = u0 , v = v0 and eigenvalues of the equilibrium (u0 , v0 ) for the slow system. Generically, due to hamiltonicity of the fast system, eigenvalues of the fast system can be of two types, either a nonzero real pair or a pure imaginary pair.

1.2

Real fast eigenvalues

Let us suppose, for a moment, that ∆ < 0 in (7). Then all equilibria, in particular O, are saddles for the fast system. ∆ is supposed to be negative for equilibria from a connected component of D, which contains O, and, in particular, in a neighborhood of the ghost separatrix. In this case the slow manifold S0 for the system (4) consists of normally hyperbolic singular points, since ∆ is merely a product of fast eigenvalues for ε = 0. Therefore, there is a smooth normally hyperbolic center manifold Sε for ε = 0, the persistence of which is due to the normal hyperbolicity. This center manifold is symplectic with respect to the singular 2-form, and the restriction of (3) to this manifold gives a Hamiltonian system which is close to the slow system, therefore, it has a saddle equilibrium p(ε), which here coincides with O, and homoclinic orbit Γ(ε) associated to it, i.e. a separatrix loop to the saddle O(ε). This loop is a transverse intersection (within the level containing O) of two-dimensional global stable and unstable manifolds of O, that follows immediately from normal hyperbolicity of Sε . Results of [32, 31] says that there is no nontrivial dynamics in a small enough neighborhood of Γ(ε), only a one-parameter family of periodic orbits accumulation at Γ(ε) from one side in terms of values of the Hamiltonian. All other trajectories leave a small neighborhood of Γ(ε). Remark 2 This result holds even for the case of multidimensional fast system provided that fast eigenvalues for the equilibria on the slow manifold have nonzero real parts (bounded away from zero in D) instead of the condition ∆ < 0 to preserve a true invariant slow submanifold for positive ε, and therefore, a homoclinic orbit on it. If the slow subsystem is also multidimensional, then one should require, in addition, that O is a saddle (no eigenvalues with zero real part) and the homoclinic orbit to O for 4

the slow system is transverse, that is, stable and unstable manifolds of the saddle point O belonging both to the level Hslow = Hslow (O) intersect transversally along the homoclinic orbit (here Hslow is the restriction of the Hamiltonian to the persisting slow manifold).

1.3

Imaginary fast eigenvalues

Another, more delicate, situation arises when ∆ is positive, as in (7), so the fast eigenvalues are pure imaginary. Then the equilibrium O(ε) is of saddle-center type for positive ε [6, 23]. Such a singular point has one-dimensional stable and unstable manifolds, a two-dimensional center manifold filled with Lyapunov saddle periodic orbits, and threedimensional center stable and center unstable manifolds. The latters are formed by the union of the stable manifolds and the unstable manifolds of Lyapunov orbits, respectively. The results of [25, 22, 17] suggest there is no a hope to obtain homoclinic orbits to O for all small ε but one can expect the existence of a countable set of values of ε for which so-called multi-bump (or multi-round) homoclinic orbits exist provided the system is reversible Hamiltonian for any ε with respect to some involution G [25, 17]. In the case of a nonreversible Hamiltonian system one can consider a two parametric family of systems, with parameters ε and, at least, one additional one. Then, it is reasonable to expect the existence of countable sets of parameter values at which there are multi-round homoclinic orbits to O [22, 21].

1.4

Changing stability

There is an important situation, when the fast system changes the type of the equilibria when varying (u0 , v0 ). Then the fast system generically undergoes a bifurcation, when a center and a saddle collide and disappear through a parabolic point (sometimes this bifurcation is called saddle-center bifurcation but we prefer to call it parabolic bifurcation, keeping in mind a quite different equilibrium of a Hamiltonian system of the saddle-center type [23]). Usually, slow manifold near the curve C in (u, v)-plane, where the change of stability occurs, consists of two sheets, and projection of slow manifold onto slow variables has a fold singularity along C. Both these sheets lie on one side from C with respect to the projection onto the slow variables. On one sheet of this manifold the equilibria of the fast system are saddles, for the fixed slow variables (u0 , v0 ) the disk u = u0 , v = v0 contains usually a homoclinic orbit (a loop) of the saddle which enclose the center lying on the second branch of the slow manifold. When moving along an orbit of the slow system, the region inside the loop shrinks at the parabolic point when crossing C, and disappear, when moving further along the orbit. To illustrate the situation here let us present an example. Example. The slow-fast Hamiltonian system is governed by the Hamilton function H = y 2/2 + x3 /3 + xu + v with the equations εx˙ = ∂H/∂y = y,

u˙ = ∂H/∂v = 1,

εy˙ = −∂H/∂x = −x2 − u,

v˙ = ∂H/∂u = −x,

5

(9)

√ Slow manifold is given by y = 0, x = ± −u, which has a fold along the curve C : u = 0. For ε = 0 the motion goes in such a way that u monotonically grows, u = t + u0 . To describe the motion between u ∈ (−d, d), d > 0, we can reduce the system to a one degree of freedom nonautonomous system with a Hamiltonian V = v0 − xu − y 2 /2 − x3 /3 and the ”time” u. Then we get, denoting
the differentiation with respect to the new time εx
= −∂V /∂y = y, εy
= ∂V /∂x = −x2 − u,

(10)

or, introducing the fast time u/ε = τ and passing to the second order equation, we come to x

+ ετ + x2 = 0, (11) that is exactly the first Painlev´e equation. Thus, the model problem of understanding different phenomena in the case under consideration, in particular, passage through separatrices of the fast system, is described by different solutions of this equation. However, it is known that solutions of this equation intimately related with this type of problems [9].

2

Almost invariant elliptic manifold

Recall first a theorem that was proved in [14] (see, also [5]). Due to Assumption 1 functions f and g in (6) are analytic in a complex δ-neighborhood D0 + iδ of the real domain D0 . Consider the slow manifold W0slow = { (x, y, u, v) : x = f (u, v), y = g(u, v), (u, v) ∈ D0 } Denote D1 (δ) = W0slow + iδ. Theorem 3 Let the Hamiltonian H be analytic in D1 (δ) for some δ > 0. Assume also ∆(u, v) > C > 0 (see, (7)) for all (u, v) ∈ D0 and some C. Then there is a canonical ˜ c > 0 such that in change of coordinates Φ : (x, y, u, v) → (X, Y, U, V ) and constants δ, the new coordinates the Hamiltonian H takes the form H ◦ Φ−1 = H0 (I, U, V ; ε) + R(X, Y, U, V ; ε), where I =

1 (X 2 2

2

+ Y ) and



c



R = O e− ε .

(12) (13)

The change of coordinates and the new Hamiltonian are analytic with respect to (X, Y, U, V ) in D1 (δ/2), and C ∞ in ε in [0, ε0 ), ε0 = const > 0. Moreover, Φ(x, y, u, v) = (x − f (u, v), y − g(u, v), u, v) + O(ε). In particular, this theorem asserts that the system with the Hamiltonian (12) is integrable up to an exponentially small error, the additional integral for the system with Hamiltonian H0 is I, this integrable system has true invariant slow manifold X = 0, Y = 0, on which the restriction of the vector field XH0 gives a one-degree-of-freedom system with the Hamiltonian h = H0 (0, U, V, ε) being ε-close to the slow system, since the transformation sending the initial Hamiltonian to that of the form (12) is ε-close to the identity in (u, v) variables. 6

3

Over an annulus filled with periodic orbits

The persistence (or non-persistence) of a true slow elliptic manifold depends on the dynamics of the slow system described by the Hamiltonian Hslow (8). This system is of one degree of freedom, so it is natural to assume that it has an annulus K0 ⊂ D0 filled with periodic orbits. We assume the annulus K0 to be closed. Each orbit belongs to a level Hslow = E and its frequency is a function ω0 (E). We say that K0 is filled with a non-degenerate family of periodic orbits if there are positive constants c1 and c2 such that 1) |ω0 (E)| > c1 ; 2) |ω0
(E)| > c2 . Since ω0 is defined on a connected set K0 , the inequalities mean that each of the functions ω0 (E) and ω0
(E) is either strictly positive or strictly negative. All four combinations of signs are possible. Our main analytical result is formulated in the form of the following theorem. We assume henceforth that 0 < ε < ε0 , where ε0 is a sufficiently small positive constant. 0 defined by (6) and assume Theorem 4 Let the system (3) have a slow manifold Wslow the Hamilton function H satisfies the following properties: 0 [H1] Analyticity: H(x, y, u, v, ε) is analytic in all its variables in a neighborhood of Wslow ; 2 0 H) > c3 > 0 at (x, y, u, v) ∈ Wslow ; [H2] Normal ellipticity: det(∂x,y

[H3] the slow system has an annulus K0 ⊂ D0 filled with a non-degenerate family of periodic orbits. Then there is an ε0 > 0 such that for ε ∈ (0, ε0 ) the system (3) has an invariant manifold Wε filled with periodic orbits; Wε is diffeomorphic Kε = K0 \ Rε , where Rε is a union of nε disjoint annuli; Wε can be represented as a graph x = f (u, v, ε) = f0 (u, v) + O(ε), y = g(u, v, ε) = g0 (u, v) + O(ε),

(u, v) ∈ Kε .

The Lebesgue measure of Rε is exponentially small in ε. nε grows at most linearly in 1/ε. Further, let us discuss the existence of invariant tori filled with quasi-periodic orbits. First we study the region outside of an exponentially thin neighborhood of Wε . We show that results of [2] along with estimates of [30] on the size of that part of the phase space, where invariant tori can be destroyed (see also a review in [4], chapter 5, Sec. 5.4 or [8]) can be applied to prove the existence of invariant tori carrying quasiperiodic orbits in any level H = E. The annulus K0 is filled with a non-degenerate family of periodic orbits of the slow system with the Hamiltonian Hslow (u, v). The (u, v) component of the coordinate change 7

is ε-close to identity. Consequently, one has h(0, u, v, ε) = Hslow (u, v) + O(ε), so the new slow system has an annulus ε-close to K0 which is filled with a nondegenerate family of periodic orbits. We denote this annulus with the same letter. We make a canonical change of coordinates (u, v) → (s, ρ) introducing the action-angle variables for the system with Hamiltonian h(0, u, v, ε). Let us denote the new Hamiltonian h0 (ρ, ε). Due to [H3] this function satisfies the following inequalities: |∂ρ2 h0 (ρ, ε)| > c˜2

∂ρ h0 (ρ, ε) > c˜1 ,

ρ ∈ (ρ1 , ρ2 ).

It is clear that the assumption of positivity of ∂ρ h0 (ρ, ε) does not restrict the generality, since if this derivative is negative, one can change t → −t in order to make it positive. Applying the transformation to action-angle variables (being identical in (x, y)) to the full system we come to the Hamiltonian defined in a region D 2 × K, K = T1 × [ρ1 , ρ2 ]. The new Hamiltonian is 2π−periodic in the angle variable s. It is convenient to use the same letters for functions after the change of variables, since later on we use only this form of the Hamiltonian. In the new variables the Hamiltonian takes the form: H = h0 (ρ, ε) + Ih1 (I, s, ρ, ε) + r(x, y, s, ρ; ε)

(14)

and the system (3) is transformed into the following εx˙ =

(h1 + I∂I h1 ) y + ∂y r,

s˙ =

∂ρ h0 + I∂ρ h1 + ∂ρ r,

εy˙

− (h1 + I∂I h1 ) x − ∂x r,

ρ˙

−I∂s h1 − ∂s r.

=

=

(15)

The last system has an important property: the section Π0 = { s = 0 } is transversal to the flow provided |I| < δ for some δ > 0. The section Π0 is 3-dimensional disk foliated by levels H = E. A position of a point on the section is uniquely defined by its coordinates (x, y) and E, due to Implicit Function Theorem. Indeed, the equation H(x, y, s, ρ, ε) = E with the Hamiltonian function (14) has a solution ρ = U(x, y, s, ε, E), (16) since we assumed ∂ρ h0 = 0 in the annulus K. Observe that the function U has a representation (17) U(x, y, s, ε, E) = U0 (I, s, ε, E) + U1 (x, y, s, ε, E), where U0 is a solution to the equation h0 (U0 , ε) + Ih1 (I, s, U0 , ε) = E, and U1 := U − U0 . Similar to r, the function U1 has an exponential upper bound 

c



U1 = O e− ε . Differentiating with respect to I, we obtain ∂I U0 = −

h1 (I, s, U0 , ε) + I∂I h1 . ∂ρ h0 + I∂ρ h1 8

This implies ∂I U0 = 0

(18)

for I small enough. Thus, the invariant 3-dimensional submanifold H = E is represented as the graph of the function U. It is convenient to use (x, y, E) as local coordinates on Π0 . Since E is preserved by the flow it can be considered as a parameter. So we obtain a family FE,ε of area preserving Poincar´e maps parameterized by E and ε. Denote A0 (I, E, ε) the mean value over the period of ∂U0 /∂I, then A0 (I, E, ε) = 0 in accordance with (18). Lemma 5 The Poincar´e map FE,ε : x → x1 is defined in a small ε-independent neighborhood of the origin. This area preserving map has the following form: x1 = R(2πε−1 A0 (I, E, ε)) x + P (x, E, ε), where x = (x, y)T , I = 12 (x2 + y 2 ); R(α) is the matrix of rotation by an angle α, |∂E A0 (0, E, ε)| > c > 0,

|A0 (0, E, ε)| > c > 0, and P (x, E, ε) = O(e−c/ε ).

Let us use the angle variable s as the new time. In this way the restriction of system (15) on the submanifold H = E can be rewritten as a singularly perturbed nonautonomous one-degree-of-freedom 2π-periodic in s system: ε

dx (h1 + I∂I h1 )y + ε∂y r ∂U = =− , ds ∂ρ h0 + I∂ρ h1 + ε∂ρ r ∂y

ε

dy (h1 + I∂I h1 )x + ε∂x r ∂U =− = . ds ∂ρ h0 + I∂ρ h1 + ε∂ρ r ∂x

(19)

Let us introduce a new independent variable τ = s/ε. Then we get the system that can be considered as that with slow varying parameter s = ετ√. Transform the system to √ action-angle variables (x, y) → (I, θ), x = 2I cos θ, y = 2I sin θ. In these variables the system reads ∂U0 ∂ Uˆ1 ∂ Uˆ1 , θ˙ = + , (20) I˙ = − ∂θ ∂I ∂I √ √ ˆ1 = U1 ( 2I cos θ, 2I sin θ, E, ε). Both derivatives in the right hand sides of where U (20) are analytic in (I, θ) and outside of some neighborhood of I = 0, since √ bounded−1/2 they depend analytically on I, and I enters as a factor in front of ∂ Uˆ1 /∂I. The ˆ derivatives of U1 are exponentially small, therefore one can take exponentially thin in ε neighborhood of I = 0, I > C2 exp[−c1 /ε], c1 ≤ c, with some positive constant C2 , such that for C2 exp[−c1 /ε] < I < I0 functions in the right hand sides of (20) are analytic in (I, θ), periodic in θ and uniformly bounded. The range of E is defined by an interval where the annulus K0 was given, say, if E0 < E < E1 , then for E 0 > E0 , E 1 < E1 , E 0 < E 1 , there are ε1 and δ1 such that for 0 < ε < ε1 , I < δ1 all our considerations hold, in particular, for the system (20) the following theorem is valid [2, 30]. 9

Theorem 6 Let for system (20) with one degree of freedom A0 (I, E, ε) = 0. Then there are positive constants ε1 , δ1 , c2 and E 0 , E 1 such that for all 0 < ε < ε1 , exp[−c1 /ε] < I < I0 , E 0 < E < E 1 the action variable I is the eternal adiabatic invariant. For values of E, I, ε under consideration this part of the phase space, except for a set of the measure not greater than exp(−c2 /ε), is filled with invariant tori being close to tori I = const. These tori are filled with quasi-periodic solutions with Diophantine rotation numbers.

4

Near an equilibrium

In this section we consider the behavior of the system near an equilibrium on the slow manifold. Generically the equilibrium for a one degree of freedom system can be of two types, center or saddle. The linear type of the equilibrium for the whole system will be then (due to inequality ∆ > 0) either elliptic or saddle-center (with slow real eigenvalues). For the first case let us discuss the existence of periodic orbits near O for the system (4). Since for ε small enough one has εω = nΩ, the Lyapunov center theorem (see, for instance, [4]) says that there exists an analytic two-dimensional symplectic submanifold through O being tangent to the symplectic plane corresponding eigenvalues ±iΩ (i.e., being transversal to the slow manifold), filled up with periodic orbits. When approaching O the period of such the periodic orbit tends 2π/Ω. The existence of another family of periodic orbits adjacent to equilibrium, corresponding to eigenvalues ±iεω, depends strongly on the value of ε. Indeed, the ratio Ω/εω, as ε → 0, passes countably many times through the resonant values εn = Ω/nω, at these εn the Lyapunov center theorem near the equilibrium fails to work. The behavior of the system for ε near such resonant value is described well by the related integrable resonant normal form (see [4] and references therein). Branching families of periodic orbits of the opposite stability when passing through these values of ε is the common feature. To utilize the KAM theorem for proving invariant tori carrying quasi-periodic orbits near an elliptic singular point, one needs to check the condition for the point to be of the general elliptic type [2, 4], i.e., the non-degeneracy of the corresponding Birkhoff normal form. But the condition for this is given for a Hamiltonian and a related system given in standard symplectic coordinates. To√transform√ our system and Hamiltonian to this form we make a scaling of variables u = εu1 , v = εv1 . Then our singular 2-form becomes standard one, and Hamiltonian takes the form √ √ ˆ H(x, y, u1, v1 ) = H(x, y, εu1 , εv1 ), thus, system becomes standard Hamiltonian system with this Hamiltonian. Further, we transform this Hamiltonian to the normal form. In order do not complicate notations we hold the same letters for new Hamiltonian and variables. We will transform Hamiltonian to the form H=

Ω 2 2 εω 2 2 1 (x +y )+ (u +v )+ (A(x2 +y 2)2 +2εB(x2 +y 2)(u2 +v 2 )+ε2C(u2 +v 2 )2 )+· · · . 2 2 2 (21) 10

The condition of general ellipticity is expressed in one of two forms [2, 4]: 1. ε2 (B 2  − AC) = 0  A B Ω   2. ε2 det  B C ω  =  0 Ω ω 0

(nondegenericity) (isoenergetical nondegenericity).

(22)

Using the ordinary method of generating function for the normalization, we calculated coefficients 



(new) (old) (old) 2 (old) 2 (old) 2 ˆ 2200 ˆ 2200 ˆ 1200 ˆ 3000 ˆ 1110 A=H =H + 3 |H | + |H | /Ω + |H | /ω+ (old) (old) ˆ ˆ 2001 |2 |2 |H |H ε( 2010 − ), 2Ω + εω 2Ω − εω (new) (old) (old) ˆ (old) (old) ˆ (old) ˆ 1200 ˆ 1111 ˆ 1200 ˆ 1111 H1011 − H H1011 )/iΩ− = εH − 2ε(H εB = εH (old) (old) (old) (old) ˆ 1110 − H ˆ 0012 H ˆ 1110 )/iω+ ˆ 0012 H 2ε(H (old) (old) (old) 2 ˆ (old) 2 ˆ 2001 |2 ˆ 2010 |2 ˆ 1002 | |H |H |H 2 |H1020 | 4ε( + ) + 4ε ( − ), 2Ω − εω 2Ω + εω Ω + 2εω Ω − 2εω

(23)



(new) (old) (old) 2 (old) 2 ˆ 0022 ˆ 0022 ˆ 1002 ˆ 1020 = ε2 H + ε2 |H | /(Ω − 2εω) + |H | /(Ω + 2εω)+ ε2 C = ε2 H  (old) (old) (old) 2 2 2 ˆ 0111 | /Ω + 3|H ˆ 0012 | /ω + 3|H ˆ 0003 | /ω , |H

ˆ (new) are coefficients in the expansion of H written in complex symplectic ˆ (old) , H where H pqrs pqrs coordinates to diagonalize the linear parts. Thus, we get the following theorem. Theorem 7 Let B 2 − AC does not vanish at ε = 0. Then O is the nondegenerate elliptic point. If the inequality Aω 2 − 2BωΩ + CΩ2 = 0 is fulfilled, then O is isoenergetically nondegenerate elliptic point. In both cases there is an O(ε)−neighborhood U of O such that KAM theorem is valid there. This theorem ensures the set of all Lagrangian tori in U has the measure, which is close to the relatively full measure of U. For the case of a saddle on the slow manifold we assume (it is a genericity condition for one degree of freedom Hamiltonians) Assumption 2. For ε = 0, the slow system in D0 has a saddle equilibrium p with a homoclinic orbit Γ associated to it. The problem we are going to discuss is the dynamics of the system (3) or (4) in some neighborhood of the ghost separatrix loop Γ on the slow manifold. The main focus will be given to the study of separatrix splitting and the existence of invariant tori near the loop. For a slow-fast Hamiltonian system, the study of families of periodic orbits, homoclinic orbits to Lyapunov periodic orbits, in the spirit of [23, 22], require a more advanced tool that we hope to develop elsewhere. 11

4.1

Moser normal form near the equilibrium

In some neighborhood of O we need to transform the system to the normal form. The only thing which prevents to a direct usage of results [27] is a singular entering of the parameter ε, and so we need to estimate the size with respect to ε of a neighborhood of O, where the transformation to the normal form is analytic. Using the approach of [7] we proved [16] that the normal form is valid in a O(ε)-neighborhood of O. More specfically, the following theorem is valid. Theorem 8 There are positive constants ε0 , small enough, and C such that in R4 ×(0, ε0 ) a region given by the inequality x < Cε exists, where the normalizing transformation converges for any ε = const. Recall that in our case for ε > 0 the equilibrium O is of saddle-center type (see above). The Moser normal form theorem for an analytic four-dimensional Hamiltonian vector field with an equilibrium O having eigenvalues (λ1 , λ2 , −λ1 , −λ2 ), λ1 /λ2 is nonreal, gives some analytic symplectic coordinates (x1 , x2 , y1 , y2 ) (complex ones, generally speaking, though there is a related real normal form) in which the Hamiltonian takes the form H(x1 , x2 , y1 , y2) = λ1 x1 y1 + λ2 x2 y2 + h3 (x1 y1 , x2 y2 ).

(24)

The real Hamiltonian to which we transform the initial Hamiltonian has the form H=

4.2

 2 2  ω(ε) 2 (x + y 2 ) + λ(ε)uv + h3 x +y , uv, ε . 2 2

(25)

Local map

Here, using the normal form obtained, we derive the mapping from the cross-section N s to the stable separatrix in O(ε)-neighborhood of O onto the cross-section N u to the unstable separatrix in O(ε)-neighborhood of O. In the Moser coordinates the Hamiltonian vector field with Hamilton function (25) takes the form ∂h3 ∂h3 )y, u˙ = (λ + )u, ∂η ∂ξ ∂h3 ∂h3 )x, v˙ = −(λ + )v, εy˙ = −(ω + ∂η ∂ξ

εx˙ = (ω +

(26)

where η = (x2 + y 2 )/2, ξ = uv. Functions ξ, η are local integrals, so for every initial point (x0 , y0 , u0, v0 ) the system can be easily integrated as a linear system with constant coefficients depending on ξ0 , η0 . Without loss of generality we suppose λ to be positive, then stable manifold locally coincides with the v-axis, and unstable one with the u-axis. As N s we take one of two small 3-disks v = ±εd, and N u is one of two small 3-disks u = ±εd with some d > 0. The choice of signs depends on the ghost separatrix Γ: we assume that Γ comes coinciding with semi-axis v > 0, that is, N s : v = εd with d positive.

12

Thus, we can derive the formulae for the local flow: ∂h03 ∂h0 ∂h0 )t/ε] + y0 sin[(ω + 3 )t/ε], u(t) = u0 exp[(λ + 3 )t] ∂η ∂η ∂ξ ∂h03 ∂h03 ∂h0 t/ε] + y0 cos[(ω + t/ε], v(t) = v0 exp[−(λ + 3 )t], y(t) = −x0 sin[(ω + ∂η ∂η ∂ξ x(t) = x0 cos[(ω +

(27)

here upperscript zero means that one needs to evaluate the expression at ξ0 = u0 v0 , η0 = (x20 + y02)/2. In the level H = 0 in some neighborhood of O (may be smaller than the initial U) one can solve the equation H = 0 with respect to variable ξ, ξ = a(η) = −(ω/λ)η − O(η 2). In N s ∩ {H = 0} = N0s we obtain v = εd, and therefore, u0 = a(η0 )/εd < 0. So, as coordinates in N0s (and N0u ) one can take (x, y). Now we are ready to calculate the passage time from N s till N u : u = −εd) in the zero level of Hamiltonian. As is easily verified this time is equal to

1 dε tp = . 0 ln − ∂h3 u0 λ+ ∂ξ Thus, the local map T : N0s → N0u is: ∂h03 ∂h0 )tp /ε] + y0 sin[(ω + 3 )tp /ε], ∂η ∂η ∂h03 ∂h0 )tp /ε] + y0 cos[(ω + 3 )tp /ε]. y1 = −x0 sin[(ω + ∂η ∂η x1 = x0 cos[(ω +

One can pass to symplectic polar coordinates on N0s and N0u : x = √ 2η sin θ. Then the local map takes the form η1 = η,

θ1 = θ − (ω +

(28) √

2η cos θ, y =

∂h03 )tp /ε = θ + ε−1 a
(η) ln(−ε2 d2 /a(η)) ∂η

(29)

So, this map is analytic in a punctured neighborhood of the origin since it has a logarithmic singularity at (0, 0), but it can be extended till the homeomorphism of some disk centered at the origin. Observe, that out of exponentially thin neighborhood of separatrices (η > C0 e−c/ε ), the time of passage from N s till N u is bounded from below by some constant proportional to ε−1 .

5

Separatrix splitting and invariant tori

Here we study properties of the global map and discuss separatrix splitting and the existence of the invariant tori in the degenerate level of Hamiltonian. We shall outline only the principal lines of considerations, the details will be given elsewhere. 13

We begin with the Hamiltonian in the form as in Theorem 3 (we will write lower-case letters for the variables that has not to lead to a confusion) H = H0 (I, u, v; ε) + r(x, y, u, v; ε) = h(0, u, v; ε) + Ih1 (I, u, v; ε) + r(x, y, u, v; ε)

(30)

with exponentially small function r, and all functions being analytic in position variables (x, y, u, v) and C ∞ -smooth in ε. Denote h(u, v; ε) = h(0, u, v; ε). This function h is a Hamilton function for a one degree of freedom system on the almost invariant slow manifold Wslow (ε) given by x = y = 0. Without loss of generality one can assume that the saddle equilibrium of the slow system is located at (u, v) = (0, 0) for all ε. Consider first the structure of an integrable system with Hamiltonian H0 = h(u, v; ε) + Ih1 (I, u, v; ε). This system has an equilibrium P of the center-saddle type at the point (0, 0, 0, 0). Its one-dimensional stable and unstable semi-separatrices form the loop Γ which belongs to the invariant plane x = y = 0 carrying an one-degree-of-freedom Hamiltonian system with Hamiltonian h. The structure of an integrable Hamiltonian system near Γ is well known [24]. Let us observe that the form of this slow-fast system is such that both integrals, the Hamiltonian and the additional integral, are regular analytic functions, so their joint levels, being invariant sets for the flow, are the same as for a nonsingular case. The singular nature of the problem is displayed only in the fast (∼ ε−1 ) rotation in the angle variable conjugated to I. Recall some results from [24] concerning the behavior of an integrable Hamiltonian system near a homoclinic orbit to a saddle-center singular point. To facilitate the exposition, we suppose the system on the invariant plane x = y = 0 to have a saddle P whose stable and unstable manifolds coalesce forming two homoclinic loops (”figure eight”), one of which is Γ. We shall discuss here only the behavior within the degenerate level containing the equilibrium P . The degenerate level H0 = 0 can be either A) a union of two disjoint solid tori with distinguished inner points (B1 , p1 ), (B2 , p2 ) glued at these points. After gluing the point obtained represents equilibrium P . The manifold obtained has a smooth boundary – two disjoint smooth tori. Inside of these glued solid tori there is a one-dimensional ”figure eight” being the union of two homoclinic loops, lying each in its own torus, one of loops is Γ. Each solid torus is foliated into two-dimensional Lagrangian tori shrinking at its own loop as I → 0; or B) one solid torus with two distinguished inner points p1 , p2 with these points identified, after gluing they also represent equilibrium P . Inside this torus also ”figure eight” lies, all other orbits belong to invariant Lagrangian tori foliating the solid torus, the boundary here is one smooth torus. In both cases the boundary is the intersection of the level H0 = 0 and I = I0 for some positive I0 . The cone-like structure of the level H0 = 0 near P permits Γ to have an angle between stable and unstable manifolds at P (they locally form ”the cross”). Now let us turn to the initial system being an exponentially small perturbation of the integrable one just described. The equations for finding singular points are regular, so, due to nondegeneracy of P , we conclude that in an exponentially small neighborhood of P there is an equilibrium O of the same type (saddle-center) as P . Due to Moser theorem 14

of subsec. 4.1, in some O(ε)-neighborhood of O one can introduce symplectic analytic coordinates (X, Y, U, V ) (depending C ∞ on ε) such that H takes the form of an analytic function of J = (X 2 + Y 2 )/2 and Ξ = UV . Due to the exponential smallness of analytic function r, the transformation will be exponentially close to the identity in some O(ε)neighborhood of O. It implies that Γ and stable and unstable separatrices of O intersects transversely for ε > 0 both cross-sections N s : V = dε, N u : U = dε. One-dimensional stable, W s , and unstable, W u , manifolds, taking their compact pieces, depend C ∞ on the perturbation, therefore, traces of Γ and W u on N u and traces of Γ and W s on N s are exponentially close to each other. Now let us estimate a possible travel time from N u till N s along the global part of Γ. For the initial system (3), if we take some cross-sections to ingoing and outgoing parts of the ghost separatrix in O(1)-neighborhood of the singular point O for the system (3), then this time is of order 1 with respect to ε, say T + O(ε). But we are able to transform the system to the normal form only in O(ε)-neighborhood, therefore, we have to estimate the time of passage from N u to N s from such a neighborhood. This time can be represented as a sum Tout + T + Tin , where Tout is the passage time of an orbit from N u in O(ε)-neighborhood to the cross-section in O(1)-neighborhood, and Tin defines the similar time for ingoing separatrix. Both times are estimated in the same manner, so we shall estimate Tout only. It is clear that this problem is the same as if we would have a differential equation x˙ = λx + ϕ(x)x, ϕ(0) = 0, with some function ϕ(x), and one would need to estimate the time passage from x = dε, till x = d, where d is taken in such a way that |ϕ(d)| < λ/2. Then, for x > 0, λ/2 ≤ λ + ϕ(x) ≤ 3λ/2 we have λ/2 ≤ d ln |x|/dt ≤ 3λ/2, or, integrating from x0 = dε at t0 = 0 till x1 = d at t = t1 we get λt1 /2 ≤ − ln ε ≤ 3λt1 /2 or −2 ln ε/3λ ≤ t1 ≤ −2 ln ε/λ. Thus, Tout , Tin are of order −C ln ε. In the fast time scale this time is of the order −Cε−1 ln ε. Thus, we see that that the deviation of unstable separatrix of O from Γ on N u is exponentially small, the same is true for N s . So, one needs only to prove that this unstable separatrix deviates exponentially small from Γ during its travel along the global part of Γ, then it comes to N s being exponentially close to the stable separatrix. This follows from the arguments of the existence of adiabatic invariants during this travel (see [28, 29, 4]). Thus, we get Proposition 9 Let us Assumptions 1,2 are fulfilled. Then for any positive ε small enough the original slow-fast analytic Hamiltonian system (3) has near the saddle on the slow manifold a true equilibrium O of the saddle-center type whose one-dimensional stable and unstable manifolds lie in an exponentially small neighborhood of the ghost separatrix loop, so, they are splitted, then this splitting is exponentially small with respect to the singularity parameter ε. Remark 10 More interesting and much more complicated problem is to give an asymptotic representation for this separatrix splitting permitting to say when this splitting really exists. The work in this direction is in progress. 15

Let us remark that the level H = H(O) is topologically (and differentially) the same as H0 = 0. Consider the region Vε of the level H = H(O) out of some exponentially thin (in ε) neighborhood of the ”figure eight” where splitted separatrices of O lie. For definiteness, we suppose we deal with the case A, and consider a solid torus containing Γ. In Vε the vector field XH is exponentially close to the integrable one foliated into invariant tori. Thus, one can reduce the system to a noautonomous Hamiltonian system with one degree of freedom depending slowly periodically on the longitudinal variable s similar to that, as we did in [15]. Then, the existence of invariant tori follows from results of [2]. The necessary condition of nonlinearity follows from the form of the local map near the saddle-center. Here the logarithmic singularity plays the main role. It allows one to derive the nonlinearity condition out of some exponentially thin neighborhood of splitted separatrices. Let us present an example where separatrices is indeed splitted. Example. Consider a Hamiltonian: H=

x2 + y 2 v 2 u2 u3 + − + + µxu2 . 2 2 2 3

The corresponding equations of motion have the form εx˙ = y,

u˙ = v,

εy˙ = −x − µu2

v˙ = u − u2 − 2µxu.

(31)

The origin is a saddle-center equilibrium of the system. The rapid variables correspond to the elliptic direction. On the unperturbed (ε = µ = 0) slow manifold W slow = { x = y = 0 } the slow system has a homoclinic trajectory: x0 = y0 = 0,

u0 =

6 , cosh2 t

v0 = −

12 sinh t , cosh3 t

which makes up a closed loop. For µ = 0 the slow manifold is x = −µu2 , y = 0, and slow system is u˙ = v, v˙ = u − u2 + 2µ2 u3 . This system preserves the homoclinic loop for all |µ| < 1/3, at |µ| = 1/3 a homoclinic contour is formed, since there is another saddle for µ = 0. We show that the whole system does not possess a close 1-round loop nearby. Following Poincar´e, we fix the parameter ε and assume the auxiliary parameter µ to be sufficiently small. For µ = 0 the homoclinic loop exists, since slow and fast variables are decoupled. Let us compute one-dimensional stable and unstable invariant manifolds of the complete system with the error of the order O(µ2 ). We will see that they do not coincide in the first order in µ. The first two equations of the Hamiltonian system are linear in x and y, so we can solve them explicitely: x± (τ ) = −µ

 τ ±∞

sin(τ − τ
)u2(τ
) dτ
,

16

y ± (τ ) = −µ

 τ ±∞

cos(τ − τ
)u2 (τ
) dτ
,

We substitute the zero order approximation for the homoclinic loop and obtain x± 0 (τ ) = −µ y0± (τ )

= −µ

 τ ±∞

 τ

±∞

sin(τ − τ
)u20(τ
) dτ
, cos(τ − τ
)u20 (τ
) dτ
,

and computing the differences: x+ 0 (τ )

−

x− 0 (τ )

= µ

y0+ (τ ) − y0− (τ ) = µ

 ∞ −∞

 ∞

−∞

sin(τ − τ
)u20(τ
) dτ
, cos(τ − τ
)u20 (τ
) dτ
.

Instead of these 2 real integrals it is convenient to compute a single complex one: ∆0 (τ ) =

(x+ 0 (τ )

−

x− 0 (τ ))

+

i(y0+ (τ )

−

y0− (τ ))

= iµ

 ∞ −∞




e−i(τ −τ ) u20 (τ
) dτ
.

The last integral is easily computed by means of residues: −iτ

∆0 (τ ) = iµe

 ∞ −∞

eiτ dτ. cosh4 (ετ )

Since the integral does not vanish, the stable and unstable manifolds are splitted. Remark 11 This result says that a 1-round homoclinic orbit does not exist here. The nonexistence of 1-round homoclinic loop does not mean that no homoclinic orbits for O exist at all, namely, multi-round homoclinic orbits can exist, when ε → 0. Indeed, the system under consideration is, in addition, reversible one (y → −y, v → −v, t → −t does now change the vector field). Then results [25, 17] suggest (though they cannot be applied directly, since the limit ε = 0 is singular) that such multi-round homoclinic orbits may appear (see, also [22, 21] for a two-parameter nonreversible case).

6

Acknowledgment

The author acknowledges a partial support of INTAS (the grant 00-221) and also thanks for a support the Russian Foundation of Basic Research (grant 01-01-00905) and the program “Universities of Russia” (project N 1905).

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