Singularly perturbed Hamiltonian systems:the dynamics near slow manifold∗

L. Lerman

Institute for Applied Math. & Cybernetics and University of Nizhny Novgorod10 Ul’yanov St., Nizhny Novgorod, 603005 Russia

E-mail: lermanl@unn.ac.ru

Abstract. When studying a slow-fast (singularly perturbed) Hamiltonian system in twodegrees of freedom one can say unexpectedly much about its dynamics near its slow mani-fold. We demonstrate some details of this picture assuming the phase portrait of the slowsystem be known and generic. Since this latter system is in one degree of freedom, theproblem in question is about the structure near the annulus on the slow manifold filledwith periodic orbits, near a center singular point, and near a separatrix loop. In all thesecase 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 thesingular perturbation theory [1, 3, 26]. In the most explicit form it can be described bythe so-called slow-fast or singularly perturbed system (see, e.g. [3, 26])

εdX

dτ= P (X, Y, ε),

dY

dτ= Q(X, Y, ε), (1)

where X ∈ Rn, Y ∈ Rm, and ε > 0 is a small real positive parameter. Variables X areusually called fast variables, and Y slow variables. For ε > 0 it is convenient to introducethe 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 boththese forms of the system.

If we formally set ε = 0 in (1), (2), then we obtain two limit systems. It is clear fromthe 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 differentialsystem, where X is considered as a function of Y , X = ϕ(Y ). The function ϕ is notnecessarily univalued. The set of points (X, Y ) determined by the equation P (X, Y, 0) = 0is usually called slow manifold , and the corresponding system Y = Q(ϕ(Y ), Y, 0) is calledslow 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 slowmanifold 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 propertiesof the system (1) or (2) for ε positive, knowing properties of both limit systems. The givengeometric interpretation suggests that the form (2) is more flexible for the qualitativestudy. In order to analyse the slow-fast system for ε > 0, we will take into account thebehavior 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 systemwhich loses one degree of freedom at ε = 0,

εx = ∂H/∂y, u = ∂H/∂v,

εy = −∂H/∂x, v = ∂H/∂u,(3)

or, with the fast timex′ = ∂H/∂y, u′ = ε∂H/∂v,

y′ = −∂H/∂x, v′ = −ε∂H/∂u, (4)

where H = H(x, y, u, v, ε) is an analytic function in all its variables. The latter system isHamiltonian with respect to the symplectic form Ω = dy∧dx+ ε−1 dv ∧du, and so is theformer one with respect to the form εΩ.

Later on we assume that the slow manifold is represented by an univalued functionof the slow variables (u, v) in some region of these variables. We will suppose the fastvariables (x, y) to be elliptic near slow manifold, that is, all equilibria of the fast systemcorresponding to the points on slow manifold are centers.

Since slow system is in one degree of freedom, its behavior is, in principle, well known, ifwe 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 arecompact sets, then the phase portrait of slow system in an invariant compact region is thefollowing one. There are a finite number of saddle equilibria, their separatrices divide thisregion into a finite number of cells filled with periodic orbits which are contracted insideone cell to a center equilibrium. Passing to the structure of the whole system near ellipticslow manifold, we then need to understand what orbit structure will be near an annulusfilled with periodic orbits of slow system, near a center equilibrium, and, finally, near theobject 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 atall orders of power expansions in ε > 0. All this was studied in [14, 15, 16] and here wepresent an overview of these results.

One of the results obtained is that in a neighborhood of the “ghost” separatrix there areone-dimensional stable and unstable semi-separatrices of a singular point of saddle-centertype [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 followa different approach (see, [13, 12]) which are independent of the particular form of theHamiltonian (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 anotherdirection and consider the case, when this manifold is elliptic. This leads to a quite differ-ent picture of the dynamics. In fact, the approach we follow is close to the lines of papersby 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)

can be solved with respect to x, y

x = f(u, v), y = g(u, v) (6)

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

∆(u, v) = det

∂2

xxH ∂2xyH

∂2yxH ∂2

yyH

∣∣∣∣∣∣x=f(u,v),y=g(u,v),ε=0

> 0 (7)

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 Functiontheorem. Recall that the manifold given by (6) is usually called slow manifold and thesystem 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). (8)

Let (u0, v0) be an equilibrium of the slow system, then the point

O = (f(u0, v0), g(u0, v0), u0, v0)

3

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 canbe 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 resultingsystem 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). Furthe-more, it is easily checked that (f(u(ε), v(ε), ε), g(u(ε), v(ε), ε), u(ε), v(ε)) is an equilibriumfor (3), we will denote it O(ε).

Remark 1 . It is evident that O(ε) depends on ε analytically, so we can shift this pointto the origin. We suppose this is done henceforth. It means, in particular, that H doesnot 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 particularO, are saddles for the fast system. ∆ is supposed to be negative for equilibria from aconnected component of D, which contains O, and, in particular, in a neighborhood ofthe ghost separatrix. In this case the slow manifold S0 for the system (4) consists ofnormally hyperbolic singular points, since ∆ is merely a product of fast eigenvalues forε = 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 issymplectic with respect to the singular 2-form, and the restriction of (3) to this manifoldgives a Hamiltonian system which is close to the slow system, therefore, it has a saddleequilibrium 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 thelevel containing O) of two-dimensional global stable and unstable manifolds of O, thatfollows immediately from normal hyperbolicity of Sε. Results of [32, 31] says that thereis no nontrivial dynamics in a small enough neighborhood of Γ(ε), only a one-parameterfamily of periodic orbits accumulation at Γ(ε) from one side in terms of values of theHamiltonian. All other trajectories leave a small neighborhood of Γ(ε).

Remark 2 This result holds even for the case of multidimensional fast system providedthat 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 in-variant 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

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the slow system is transverse, that is, stable and unstable manifolds of the saddle point Obelonging both to the level Hslow = Hslow(O) intersect transversally along the homoclinicorbit (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 eigenval-ues are pure imaginary. Then the equilibrium O(ε) is of saddle-center type for positiveε [6, 23]. Such a singular point has one-dimensional stable and unstable manifolds, atwo-dimensional center manifold filled with Lyapunov saddle periodic orbits, and three-dimensional center stable and center unstable manifolds. The latters are formed by theunion 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 forall small ε but one can expect the existence of a countable set of values of ε for whichso-called multi-bump (or multi-round) homoclinic orbits exist provided the system is re-versible Hamiltonian for any ε with respect to some involution G [25, 17]. In the case ofa 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 theexistence of countable sets of parameter values at which there are multi-round homoclinicorbits to O [22, 21].

1.4 Changing stability

There is an important situation, when the fast system changes the type of the equilibriawhen varying (u0, v0). Then the fast system generically undergoes a bifurcation, whena center and a saddle collide and disappear through a parabolic point (sometimes thisbifurcation 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-centertype [23]). Usually, slow manifold near the curve C in (u, v)-plane, where the change ofstability occurs, consists of two sheets, and projection of slow manifold onto slow variableshas a fold singularity along C. Both these sheets lie on one side from C with respect tothe projection onto the slow variables. On one sheet of this manifold the equilibria ofthe fast system are saddles, for the fixed slow variables (u0, v0) the disk u = u0, v = v0contains usually a homoclinic orbit (a loop) of the saddle which enclose the center lying onthe 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 anexample.

Example. The slow-fast Hamiltonian system is governed by the Hamilton functionH = y2/2 + x3/3 + xu + v with the equations

εx = ∂H/∂y = y, u = ∂H/∂v = 1,

εy = −∂H/∂x = −x2 − u, v = ∂H/∂u = −x, (9)

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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. Todescribe the motion between u ∈ (−d, d), d > 0, we can reduce the system to a one degreeof freedom nonautonomous system with a Hamiltonian V = v0 − xu − y2/2 − x3/3 andthe ”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 cometo

x′′ + ετ + x2 = 0, (11)

that is exactly the first Painleve equation. Thus, the model problem of understandingdifferent phenomena in the case under consideration, in particular, passage through sep-aratrices 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 functionsf and g in (6) are analytic in a complex δ-neighborhood D0 + iδ of the real domain D0.Consider the slow manifold

W slow0 = (x, y, u, v) : x = f(u, v), y = g(u, v), (u, v) ∈ D0

Denote D1(δ) = W slow0 + 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 canonicalchange of coordinates Φ : (x, y, u, v) → (X, Y, U, V ) and constants δ, c > 0 such that inthe new coordinates the Hamiltonian H takes the form

H Φ−1 = H0(I, U, V ; ε) +R(X, Y, U, V ; ε), (12)

where I = 12(X2 + Y 2) and

R = O(e−

cε

). (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 in-tegrable up to an exponentially small error, the additional integral for the system withHamiltonianH0 is I, this integrable system has true invariant slow manifoldX = 0, Y = 0,on which the restriction of the vector field XH0 gives a one-degree-of-freedom system withthe Hamiltonian h = H0(0, U, V, ε) being ε-close to the slow system, since the transfor-mation sending the initial Hamiltonian to that of the form (12) is ε-close to the identityin (u, v) variables.

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3 Over an annulus filled with periodic orbits

The persistence (or non-persistence) of a true slow elliptic manifold depends on the dy-namics of the slow system described by the Hamiltonian Hslow (8). This system is ofone degree of freedom, so it is natural to assume that it has an annulus K0 ⊂ D0 filledwith periodic orbits. We assume the annulus K0 to be closed. Each orbit belongs to alevel Hslow = E and its frequency is a function ω0(E). We say that K0 is filled with anon-degenerate family of periodic orbits if there are positive constants c1 and c2 such that

1) |ω0(E)| > c1;2) |ω′

0(E)| > c2.

Since ω0 is defined on a connected set K0, the inequalities mean that each of the func-tions ω0(E) and ω′

0(E) is either strictly positive or strictly negative. All four combinationsof signs are possible.

Our main analytical result is formulated in the form of the following theorem. Weassume henceforth that 0 < ε < ε0, where ε0 is a sufficiently small positive constant.

Theorem 4 Let the system (3) have a slow manifold W 0slow defined by (6) and assume

the Hamilton function H satisfies the following properties:

[H1] Analyticity: H(x, y, u, v, ε) is analytic in all its variables in a neighborhood of W 0slow;

[H2] Normal ellipticity: det(∂2x,yH) > c3 > 0 at (x, y, u, v) ∈W 0

slow;

[H3] the slow system has an annulus K0 ⊂ D0 filled with a non-degenerate family ofperiodic orbits.

Then there is an ε0 > 0 such that for ε ∈ (0, ε0) the system (3) has an invariant manifoldWε filled with periodic orbits; Wε is diffeomorphic Kε = K0 \Rε, where Rε is a union ofnε 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 showthat 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 anylevel H = E.

The annulus K0 is filled with a non-degenerate family of periodic orbits of the slowsystem with the Hamiltonian Hslow(u, v). The (u, v) component of the coordinate change

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

∂ρh0(ρ, ε) > c1, |∂2ρh0(ρ, ε)| > c2 ρ ∈ (ρ1, ρ2).

It is clear that the assumption of positivity of ∂ρh0(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 thefull system we come to the Hamiltonian defined in a region D2 ×K, K = T1 × [ρ1, ρ2].The new Hamiltonian is 2π−periodic in the angle variable s. It is convenient to use thesame letters for functions after the change of variables, since later on we use only thisform 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∂Ih1) y + ∂yr, s = ∂ρh0 + I∂ρh1 + ∂ρr,

εy = − (h1 + I∂Ih1) x− ∂xr, ρ = −I∂sh1 − ∂sr.(15)

The last system has an important property: the section Π0 = s = 0 is transversal tothe flow provided |I| < δ for some δ > 0.

The section Π0 is 3-dimensional disk foliated by levels H = E. A position of a pointon the section is uniquely defined by its coordinates (x, y) and E, due to Implicit FunctionTheorem. 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 ∂ρh0 = 0 in the annulus K. Observe that the function U has a repre-sentation

U(x, y, s, ε, E) = U0(I, s, ε, E) + U1(x, y, s, ε, E), (17)

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

U1 = O(e−

cε

).

Differentiating with respect to I, we obtain

∂IU0 = −h1(I, s, U0, ε) + I∂Ih1

∂ρh0 + I∂ρh1.

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This implies∂IU0 = 0 (18)

for I small enough. Thus, the invariant 3-dimensional submanifold H = E is representedas the graph of the function U .

It is convenient to use (x, y, E) as local coordinates on Π0. Since E is preserved by theflow it can be considered as a parameter. So we obtain a family FE,ε of area preservingPoincare 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 inaccordance with (18).

Lemma 5 The Poincare map FE,ε : x → x1 is defined in a small ε-independent neigh-borhood of the origin. This area preserving map has the following form:

x1 = R(2πε−1A0(I, E, ε)) x+ P (x,E, ε),

where x = (x, y)T , I = 12(x2 + y2); R(α) is the matrix of rotation by an angle α,

|A0(0, E, ε)| > c > 0, |∂EA0(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 sys-tem (15) on the submanifold H = E can be rewritten as a singularly perturbed non-autonomous one-degree-of-freedom 2π-periodic in s system:

εdx

ds=

(h1 + I∂Ih1)y + ε∂yr

∂ρh0 + I∂ρh1 + ε∂ρr= −∂U

∂y, ε

dy

ds= −(h1 + I∂Ih1)x+ ε∂xr

∂ρh0 + I∂ρh1 + ε∂ρr=∂U

∂x. (19)

Let us introduce a new independent variable τ = s/ε. Then we get the system that canbe considered as that with slow varying parameter s = ετ . Transform the system toaction-angle variables (x, y) → (I, θ), x =

√2I cos θ, y =

√2I sin θ. In these variables

the system reads

I = −∂U1

∂θ, θ =

∂U0

∂I+∂U1

∂I, (20)

where U1 = U1(√

2I cos θ,√

2I sin θ, E, ε). Both derivatives in the right hand sides of(20) are analytic in (I, θ) and bounded outside of some neighborhood of I = 0, sincethey depend analytically on

√I, and I−1/2 enters as a factor in front of ∂U1/∂I. The

derivatives of U1 are exponentially small, therefore one can take exponentially thin in εneighborhood of I = 0, I > C2 exp[−c1/ε], c1 ≤ c, with some positive constant C2, suchthat 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 intervalwhere the annulus K0 was given, say, if E0 < E < E1, then for E0 > E0, E

1 < E1,E0 < E1, 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].

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Theorem 6 Let for system (20) with one degree of freedom A0(I, E, ε) = 0. Then thereare positive constants ε1, δ1, c2 and E0, E1 such that for all 0 < ε < ε1, exp[−c1/ε] <I < I0, E

0 < E < E1 the action variable I is the eternal adiabatic invariant. For valuesof E, I, ε under consideration this part of the phase space, except for a set of the measurenot 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 slowmanifold. Generically the equilibrium for a one degree of freedom system can be of twotypes, center or saddle. The linear type of the equilibrium for the whole system will bethen (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, forinstance, [4]) says that there exists an analytic two-dimensional symplectic submanifoldthrough O being tangent to the symplectic plane corresponding eigenvalues ±iΩ (i.e.,being transversal to the slow manifold), filled up with periodic orbits. When approachingO the period of such the periodic orbit tends 2π/Ω.

The existence of another family of periodic orbits adjacent to equilibrium, correspond-ing to eigenvalues ±iεω, depends strongly on the value of ε. Indeed, the ratio Ω/εω, asε → 0, passes countably many times through the resonant values εn = Ω/nω, at theseεn the Lyapunov center theorem near the equilibrium fails to work. The behavior of thesystem for ε near such resonant value is described well by the related integrable resonantnormal form (see [4] and references therein). Branching families of periodic orbits of theopposite stability when passing through these values of ε is the common feature.

To utilize the KAM theorem for proving invariant tori carrying quasi-periodic orbitsnear an elliptic singular point, one needs to check the condition for the point to be of thegeneral elliptic type [2, 4], i.e., the non-degeneracy of the corresponding Birkhoff normalform. But the condition for this is given for a Hamiltonian and a related system given instandard symplectic coordinates. To transform our system and Hamiltonian to this formwe make a scaling of variables u =

√εu1, v =

√εv1. Then our singular 2-form becomes

standard one, and Hamiltonian takes the form

H(x, y, u1, v1) = H(x, y,√εu1,

√εv1),

thus, system becomes standard Hamiltonian system with this Hamiltonian. Further, wetransform this Hamiltonian to the normal form. In order do not complicate notations wehold the same letters for new Hamiltonian and variables.

We will transform Hamiltonian to the form

H =Ω

2(x2+y2)+

εω

2(u2+v2)+

1

2(A(x2+y2)2+2εB(x2+y2)(u2+v2)+ε2C(u2+v2)2)+· · · .

(21)

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The condition of general ellipticity is expressed in one of two forms [2, 4]:

1. ε2(B2 −AC) = 0 (nondegenericity)

2. ε2 det

A B ΩB C ωΩ ω 0

= 0 (isoenergetical nondegenericity).

(22)

Using the ordinary method of generating function for the normalization, we calculatedcoefficients

A = H(new)2200 = H

(old)2200 + 3

(|H(old)

1200 |2 + |H(old)3000 |2

)/Ω + |H(old)

1110 |2/ω+

ε(|H(old)

2010 |22Ω + εω

− |H(old)2001 |2

2Ω − εω ),

εB = εH(new)1111 = εH

(old)1111 − 2ε(H

(old)1200 H

(old)1011 − H(old)

1200 H(old)1011 )/iΩ−

2ε(H(old)0012 H

(old)1110 − H(old)

0012 H(old)1110 )/iω+

4ε(|H(old)

2001 |22Ω − εω +

|H(old)2010 |2

2Ω + εω) + 4ε2(

|H(old)1020 |2

Ω + 2εω− |H(old)

1002 |2Ω − 2εω

),

ε2C = ε2H(new)0022 = ε2H

(old)0022 + ε2

[|H(old)

1002 |2/(Ω − 2εω) + |H(old)1020 |2/(Ω + 2εω)+

|H(old)0111 |2/Ω + 3|H(old)

0012 |2/ω + 3|H(old)0003 |2/ω

],

(23)

where H(old)pqrs , H

(new)pqrs are coefficients in the expansion of H written in complex symplectic

coordinates to diagonalize the linear parts. Thus, we get the following theorem.

Theorem 7 Let B2 −AC does not vanish at ε = 0. Then O is the nondegenerate ellipticpoint. If the inequality Aω2 − 2BωΩ + CΩ2 = 0 is fulfilled, then O is isoenergeticallynondegenerate elliptic point. In both cases there is an O(ε)−neighborhood U of O suchthat KAM theorem is valid there.

This theorem ensures the set of all Lagrangian tori in U has the measure, which isclose to the relatively full measure of U .

For the case of a saddle on the slow manifold we assume (it is a genericity conditionfor one degree of freedom Hamiltonians)

Assumption 2. For ε = 0, the slow system in D0 has a saddle equilibrium p with ahomoclinic orbit Γ associated to it.

The problem we are going to discuss is the dynamics of the system (3) or (4) in someneighborhood of the ghost separatrix loop Γ on the slow manifold. The main focus will begiven 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, homoclinicorbits to Lyapunov periodic orbits, in the spirit of [23, 22], require a more advanced toolthat 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. Theonly thing which prevents to a direct usage of results [27] is a singular entering of theparameter ε, and so we need to estimate the size with respect to ε of a neighborhood ofO, where the transformation to the normal form is analytic. Using the approach of [7] weproved [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 transformationconverges 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 fieldwith an equilibrium O having eigenvalues (λ1, λ2,−λ1,−λ2), λ1/λ2 is nonreal, gives someanalytic symplectic coordinates (x1, x2, y1, y2) (complex ones, generally speaking, thoughthere is a related real normal form) in which the Hamiltonian takes the form

H(x1, x2, y1, y2) = λ1x1y1 + λ2x2y2 + h3(x1y1, x2y2). (24)

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

H =ω(ε)

2(x2 + y2) + λ(ε)uv + h3

(x2+y2

2, uv, ε

). (25)

4.2 Local map

Here, using the normal form obtained, we derive the mapping from the cross-section N s tothe stable separatrix in O(ε)-neighborhood of O onto the cross-section Nu to the unstableseparatrix in O(ε)-neighborhood of O. In the Moser coordinates the Hamiltonian vectorfield with Hamilton function (25) takes the form

εx = (ω +∂h3

∂η)y, u = (λ+

∂h3

∂ξ)u,

εy = −(ω +∂h3

∂η)x, v = −(λ +

∂h3

∂ξ)v,

(26)

where η = (x2 + y2)/2, ξ = uv. Functions ξ, η are local integrals, so for every initialpoint (x0, y0, u0, v0) the system can be easily integrated as a linear system with constantcoefficients 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 Nu is one of two small 3-disksu = ±εd with some d > 0. The choice of signs depends on the ghost separatrix Γ: weassume 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:

x(t) = x0 cos[(ω +∂h0

3

∂η)t/ε] + y0 sin[(ω +

∂h03

∂η)t/ε], u(t) = u0 exp[(λ+

∂h03

∂ξ)t]

y(t) = −x0 sin[(ω +∂h0

3

∂ηt/ε] + y0 cos[(ω +

∂h03

∂ηt/ε], v(t) = v0 exp[−(λ +

∂h03

∂ξ)t],

(27)

here upperscript zero means that one needs to evaluate the expression at ξ0 = u0v0, η0 =(x2

0 + y20)/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(η2).In N s ∩ H = 0 = N s

0 we obtain v = εd, and therefore, u0 = a(η0)/εd < 0. So,as coordinates in N s

0 (and Nu0 ) one can take (x, y). Now we are ready to calculate the

passage time from N s till Nu : u = −εd) in the zero level of Hamiltonian. As is easilyverified this time is equal to

tp =1

λ+∂h0

3

∂ξ

ln

(−dεu0

).

Thus, the local map T : N s0 → Nu

0 is:

x1 = x0 cos[(ω +∂h0

3

∂η)tp/ε] + y0 sin[(ω +

∂h03

∂η)tp/ε],

y1 = −x0 sin[(ω +∂h0

3

∂η)tp/ε] + y0 cos[(ω +

∂h03

∂η)tp/ε].

(28)

One can pass to symplectic polar coordinates on N s0 and Nu

0 : x =√

2η cos θ, y =√2η sin θ. Then the local map takes the form

η1 = η, θ1 = θ − (ω +∂h0

3

∂η)tp/ε = θ + ε−1a′(η) ln(−ε2d2/a(η)) (29)

So, this map is analytic in a punctured neighborhood of the origin since it has a logarithmicsingularity at (0, 0), but it can be extended till the homeomorphism of some disk centeredat the origin. Observe, that out of exponentially thin neighborhood of separatrices (η >C0e

−c/ε), the time of passage from N s till Nu is bounded from below by some constantproportional to ε−1.

5 Separatrix splitting and invariant tori

Here we study properties of the global map and discuss separatrix splitting and theexistence of the invariant tori in the degenerate level of Hamiltonian. We shall outlineonly 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-caseletters 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 isa Hamilton function for a one degree of freedom system on the almost invariant slowmanifold Wslow(ε) given by x = y = 0.

Without loss of generality one can assume that the saddle equilibrium of the slowsystem is located at (u, v) = (0, 0) for all ε. Consider first the structure of an integrablesystem with Hamiltonian H0 = h(u, v; ε)+ Ih1(I, u, v; ε). This system has an equilibriumP of the center-saddle type at the point (0, 0, 0, 0). Its one-dimensional stable and unstablesemi-separatrices form the loop Γ which belongs to the invariant plane x = y = 0 carryingan one-degree-of-freedom Hamiltonian system with Hamiltonian h. The structure of anintegrable Hamiltonian system near Γ is well known [24]. Let us observe that the formof this slow-fast system is such that both integrals, the Hamiltonian and the additionalintegral, are regular analytic functions, so their joint levels, being invariant sets for theflow, are the same as for a nonsingular case. The singular nature of the problem isdisplayed 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 Hamiltoniansystem near a homoclinic orbit to a saddle-center singular point. To facilitate the ex-position, we suppose the system on the invariant plane x = y = 0 to have a saddleP whose stable and unstable manifolds coalesce forming two homoclinic loops (”figureeight”), one of which is Γ. We shall discuss here only the behavior within the degeneratelevel containing the equilibrium P .

The degenerate level H0 = 0 can beeither 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 man-ifold obtained has a smooth boundary – two disjoint smooth tori. Inside of these gluedsolid tori there is a one-dimensional ”figure eight” being the union of two homoclinicloops, lying each in its own torus, one of loops is Γ. Each solid torus is foliated intotwo-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 pointsidentified, after gluing they also represent equilibrium P . Inside this torus also ”figureeight” 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 somepositive I0. The cone-like structure of the level H0 = 0 near P permits Γ to have an anglebetween 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 theintegrable 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 ofP 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 analyticcoordinates (X, Y, U, V ) (depending C∞ on ε) such that H takes the form of an analyticfunction of J = (X2 + Y 2)/2 and Ξ = UV . Due to the exponential smallness of analyticfunction 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 intersectstransversely for ε > 0 both cross-sections N s : V = dε, Nu : U = dε. One-dimensionalstable, W s, and unstable, W u, manifolds, taking their compact pieces, depend C∞ on theperturbation, therefore, traces of Γ and W u on Nu and traces of Γ and W s on N s areexponentially close to each other.

Now let us estimate a possible travel time from Nu till N s along the global part ofΓ. For the initial system (3), if we take some cross-sections to ingoing and outgoingparts 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 totransform the system to the normal form only in O(ε)-neighborhood, therefore, we haveto estimate the time of passage from Nu to N s from such a neighborhood. This time canbe represented as a sum Tout + T + Tin, where Tout is the passage time of an orbit fromNu in O(ε)-neighborhood to the cross-section in O(1)-neighborhood, and Tin defines thesimilar time for ingoing separatrix. Both times are estimated in the same manner, so weshall estimate Tout only.

It is clear that this problem is the same as if we would have a differential equationx = λx+ ϕ(x)x, ϕ(0) = 0, with some function ϕ(x), and one would need to estimate thetime 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, integratingfrom 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 scalethis time is of the order −Cε−1 ln ε.

Thus, we see that that the deviation of unstable separatrix of O from Γ on Nu isexponentially small, the same is true for N s. So, one needs only to prove that thisunstable separatrix deviates exponentially small from Γ during its travel along the globalpart of Γ, then it comes to N s being exponentially close to the stable separatrix. Thisfollows 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 enoughthe original slow-fast analytic Hamiltonian system (3) has near the saddle on the slowmanifold a true equilibrium O of the saddle-center type whose one-dimensional stable andunstable 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 singularityparameter ε.

Remark 10 More interesting and much more complicated problem is to give an asymp-totic representation for this separatrix splitting permitting to say when this splitting reallyexists. The work in this direction is in progress.

15

Let us remark that the level H = H(O) is topologically (and differentially) the sameas H0 = 0. Consider the region Vε of the level H = H(O) out of some exponentiallythin (in ε) neighborhood of the ”figure eight” where splitted separatrices of O lie. Fordefiniteness, 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 intoinvariant tori. Thus, one can reduce the system to a noautonomous Hamiltonian systemwith one degree of freedom depending slowly periodically on the longitudinal variable ssimilar to that, as we did in [15]. Then, the existence of invariant tori follows from resultsof [2]. The necessary condition of nonlinearity follows from the form of the local map nearthe saddle-center. Here the logarithmic singularity plays the main role. It allows one toderive the nonlinearity condition out of some exponentially thin neighborhood of splittedseparatrices.

Let us present an example where separatrices is indeed splitted.Example. Consider a Hamiltonian:

H =x2 + y2

2+v2

2− u

2

2+u3

3+ µxu2.

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 correspondto 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 slowsystem is

u = v, v = u− u2 + 2µ2u3.

This system preserves the homoclinic loop for all |µ| < 1/3, at |µ| = 1/3 a homocliniccontour is formed, since there is another saddle for µ = 0. We show that the whole systemdoes not possess a close 1-round loop nearby.

Following Poincare, we fix the parameter ε and assume the auxiliary parameter µ tobe sufficiently small. For µ = 0 the homoclinic loop exists, since slow and fast variablesare decoupled. Let us compute one-dimensional stable and unstable invariant manifoldsof the complete system with the error of the order O(µ2). We will see that they do notcoincide in the first order in µ.

The first two equations of the Hamiltonian system are linear in x and y, so we cansolve 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∫ τ

±∞sin(τ − τ ′)u2

0(τ′) dτ ′,

y±0 (τ) = −µ∫ τ

±∞cos(τ − τ ′)u2

0(τ′) dτ ′,

and computing the differences:

x+0 (τ) − x−0 (τ) = µ

∫ ∞

−∞sin(τ − τ ′)u2

0(τ′) dτ ′,

y+0 (τ) − y−0 (τ) = µ∫ ∞

−∞cos(τ − τ ′)u2

0(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(y+

0 (τ) − y−0 (τ)) = iµ

∫ ∞

−∞e−i(τ−τ ′)u2

0(τ′) dτ ′.

The last integral is easily computed by means of residues:

∆0(τ) = iµe−iτ∫ ∞

−∞eiτ

cosh4(ετ)dτ.

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. Thenonexistence of 1-round homoclinic loop does not mean that no homoclinic orbits for Oexist at all, namely, multi-round homoclinic orbits can exist, when ε → 0. Indeed, thesystem under consideration is, in addition, reversible one (y → −y, v → −v, t→ −t doesnow change the vector field). Then results [25, 17] suggest (though they cannot be applieddirectly, since the limit ε = 0 is singular) that such multi-round homoclinic orbits mayappear (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 thanksfor a support the Russian Foundation of Basic Research (grant 01-01-00905) and theprogram “Universities of Russia” (project N 1905).

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