ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 1–10 (March 12, 1997) S 1079-6762(97)00017-6

LOWER AND UPPER BOUNDS FOR THE SPLITTING OF SEPARATRICES OF THE PENDULUM UNDER A FAST QUASIPERIODIC FORCING

AMADEU DELSHAMS, VASSILI GELFREICH, ÀNGEL JORBA, AND TERE M. SEARA

(Communicated by Jeff Xia)

Abstract. Quasiperiodic perturbations with two frequencies (1/ε, γ/ε) of a pendulum are considered, where γ is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a twodimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for ε small enough. The value of the splitting, that turns out to be O (exp (− const /\sqrt{ε})), is correctly predicted by the Melnikov function.

1. Introduction

The rapidly (and periodically) forced pendulum has been widely used as a model for the motion near a resonance of Hamiltonian systems with two degrees of freedom. As is well known in several situations [DS92], [Gel93], the separatrices of the perturbed system do not coincide, giving rise to the so-called splitting of separatrices, which seems to be the main cause of stochastic behaviour in Hamiltonian systems.

In this announcement we consider a quasiperiodic high-frequency perturbation of the pendulum (it can be regarded as a model near a resonance of a Hamiltonian system with three degrees of freedom), described by the Hamiltonian function

\frac{ω · I}{ε} + h(x, y, θ, ε), (1)

where

ω · I = ω_{1}I_{1} + ω_{2}I_{2}, h(x, y, θ, ε) = \frac{y^{2}}{2} + cos x + ε^{p}m(θ_{1}, θ_{2}) cos x,

with symplectic form dx ∧ dy + dθ_{1} ∧ dI_{1} + dθ_{2} ∧ dI_{2}. We assume that ε is a small positive parameter and that p is a positive parameter. Mainly due to a technical limitation imposed by the Extension Theorem (Theorem 2), we will restrict ourselves to the case p > 3. We also assume that the frequency is of the form ω/ε for

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors July 9, 1996.

1991 Mathematics Subject Classification. Primary 34C37, 58F27, 58F36; Secondary 11J25.

Key words and phrases. Splitting of separatrices, quasiperiodic forcing, homoclinic orbits, normal forms.

ω = (1, γ), where γ = (1 + \sqrt{5})/2 is the golden mean. The equations of motion related to Hamiltonian (1) are:

ẋ = y,  ẏ = (1 + ε^{p}m(θ_{1}, θ_{2})) sin x,
(2)  θ̇_{1} = \frac{1}{ε},  İ_{1} = −ε^{p} cos x\frac{∂m}{∂θ_{1}} (θ_{1}, θ_{2}),
θ̇_{2} = \frac{γ}{ε} ,  İ_{2} = −ε^{p} cos x\frac{∂m}{∂θ_{2}} (θ_{1}, θ_{2}).

The function m is assumed to be a 2π-periodic function of two variables θ_{1} and θ_{2}. Thus it can be represented as a Fourier series:

m(θ_{1}, θ_{2}) = ∑_{k_{1},k_{2}} m_{k_{1}k_{2}} e^{i(k_{1}θ_{1}+k_{2}θ_{2})}.

We assume that, for some positive numbers r_{1} and r_{2},

sup_{k_{1},k_{2}} |m_{k_{1}k_{2}} e^{r_{1}|k_{1}|+r_{2}|k_{2}|}| < ∞, (3)

and that there are positive numbers a and k_{0} such that

|m_{k_{1}k_{2}}| ≥ a e^{−r_{1}|k_{1}|−r_{2}|k_{2}|} (4)

for all k_{1}, k_{2} such that |k_{1}| = F_{n+1} and |k_{2}| = F_{n}, where F_{n} and F_{n+1} are Fibonacci numbers, which are defined by the following recurrent formula:

F_{0} = 1,  F_{1} = 1,  F_{n+1} = F_{n} + F_{n−1}, n ≥ 1. (5)

We call the corresponding terms in the perturbation resonant or Fibonacci terms. For example, the function

m(θ_{1}, θ_{2}) = \frac{cos θ_{1} cos θ_{2}}{(cosh r_{1} − cos θ_{1})(cosh r_{2} − cos θ_{2})}

satisfies these conditions.

The upper bound (3) implies that the function m is analytic on the strip {|ℑθ_{1}| < r_{1}} × {|ℑθ_{2}| < r_{2}}. Equation (4) implies that this function cannot be continued analytically onto a larger strip. Let us select α ∈ (0, 1]. Estimate (3) implies that

|m(θ_{1}, θ_{2})| ≤ Kε^{−2α} (6)

on the strip

|ℑθ_{1}| ≤ r_{1} − ε^{α}, |ℑθ_{2}| ≤ r_{2} − ε^{α}.

Formula (4) implies that the upper bound (6) cannot be improved. It will be seen that the value of the splitting depends essentially on the width of these strips. Moreover, formula (4) will allow us to estimate in the next section the size of the Melnikov function in terms of the separatrix of the unperturbed pendulum. (See Remark 4.) The function m under consideration has a singularity “of the second order”, in the sense that the upper bound (6) for the maximum of the modulus is quadratic with respect to the inverse of the distance to the boundary of the strip. In a similar way the case of a singularity of any “order” q can be considered. In this case m_{k_{1}k_{2}} should be replaced by m_{k_{1}k_{2}}/ |k|^{q−2} in (3) and (4).

The Hamiltonian (1) can be regarded as a singular perturbation of the pendulum

h_{0} = \frac{y^{2}}{2} + cos x. (7)

The unperturbed system has a saddle point (0, 0) and a homoclinic trajectory given by

x_{0}(t) = 4 arctan( e^{t}), y_{0}(t) = ẋ_{0}(t).

The complete system (2) has a whiskered torus T : (0, 0, θ_{1}, θ_{2}). The whiskers are 3D-hypersurfaces in the 4D-dimensional extended phase space (x, y, θ_{1}, θ_{2}). These invariant manifolds are close to the unperturbed pendulum separatrix.

Our main result (Theorem 3) is that, if condition (4) is verified, then for p > 3 and small ε > 0 the invariant manifolds split, and the value of the splitting (i.e., the distance function between these invariant manifolds) is correctly predicted by the Melnikov function, which is O( e^{− const /\sqrt{ε}}).

Remark 1. Our model (1) is based on a previous work by C. Simó [Sim94], where Neishtadt’s Averaging Theorem [Nei84] was generalized to quasiperiodic systems, giving rise to upper estimates of the splitting which are exponentially small with respect to the parameter of perturbation ε. Related upper estimates can be found in [CG94], [Gal94], [BCG95], [Ben96]. In contrast to these results, Theorem 3 provides both lower and upper bounds for our model.

Remark 2. As an example in [DGJS96b] shows, the splitting can be of the order of some power of ε if the function m is not analytic. This makes a first qualitative difference between periodic and quasiperiodic perturbations. Indeed, in the periodic case, only the C^{1} dependence with respect to θ of the perturbed Hamiltonian is needed to prove that the splitting is O( e^{−c/ε}), where c is the width of the analyticity strip of the unperturbed separatrix. (In both cases, the analyticity of the unperturbed system is essential.)

Remark 3. In the case of an entire function m, we think that the method used in the present paper can be modified in order to improve the estimate of the error and to prove that the Melnikov function gives the actual asymptotics at least when the resonant terms decrease not much faster than 1/k!.

2. The Melnikov function

As is well known, the Melnikov function

M(θ_{1}, θ_{2}; ε) = ∫_{−∞}^{∞}{h_{0}, h}(x_{0}(t), y_{0}(t), θ_{1} + t/ε, θ_{2} + γt/ε) dt (8)

gives a first order approximation of the difference between the values of the unperturbed pendulum energy h_{0} on the stable and unstable manifolds. Using the Fourier series of m(θ_{1}, θ_{2}), one can compute the Fourier coefficients of M(θ_{1}, θ_{2}) as

M_{k_{1}k_{2}}(ε) = −\frac{2πiε^{p}(k_{1} + γk_{2})^{2}}{ε^{2} sinh (π(k_{1} + γk_{2})/(2ε))} · m_{k_{1}k_{2}}.

In order to bound the Melnikov function, it is important to know for each fixed ε which are the indices (k_{1}, k_{2}) corresponding to the biggest Fourier coefficient M_{k_{1}k_{2}}(ε). From the expression above, M_{k_{1}k_{2}}(ε) is a product of two factors. For ε fixed and small enough, the first factor of M_{k_{1}k_{2}}(ε) behaves as ε^{p−2}(k_{1} + γk_{2})^{2} × e^{−π(k_{1}+γk_{2})/(2ε)}, and it turns out that it becomes bigger for small k_{1} + γk_{2}, i.e., just for the resonant terms where the second factor m_{k_{1}k_{2}} decreases with respect to (k_{1}, k_{2}) according to the behaviour (3) and (4). Here we have the main difference between the quasiperiodic case and the periodic one: for a periodic perturbation, the first Fourier coefficients M_{±1}(ε) of the Melnikov function M(θ; ε) give generically the main contribution to the Melnikov function M(θ; ε), since M_{k}(ε) = O( e^{−|k|/ε}). However, in the quasiperiodic case the biggest Fourier coefficient depends strongly on the exponent k · (1, γ) = k_{1} + γk_{2} of the Fourier coefficient, i.e., on the rational approximations of γ.

For γ = (1+\sqrt{5})/2, it is very well known that its best approximation by rational numbers is given by the quotient of successive Fibonacci numbers (5). Indeed, it is easy to check that for large values of n one has the following approximation of γ by Fibonacci numbers:

F_{n} − γF_{n−1} = (−1)^{n} \frac{C_{F}}{F_{n−1}} + O (\frac{1}{F_{n−1}^{3}} ) , C_{F} = \frac{1}{γ + γ^{−1}} ,

whereas for the other integers one has the following result.

Lemma 1. If N ∈ ℕ is not a Fibonacci number, then for all integers k

|k − γN| > \frac{γC_{F}}{N} .

Using this lemma one can see that the indices (k_{1}, k_{2}) corresponding to the leading Fourier coefficients M_{k_{1}k_{2}}(ε) depend on ε. In fact, the largest terms correspond to (k_{1}, k_{2}) = ± (F_{n(ε)+1}, −F_{n(ε)}), where F_{n(ε)} is the Fibonacci number closest to F^{∗}(ε) = \sqrt{ϕ_{0}/ε}, where ϕ_{0} = π/(2(γ + γ^{−1})(r_{1}γ + r_{2})). Except for a small neighbourhood of ε = ε^{∗}γ^{−n}, with ε^{∗} given in (10), there is a unique Fibonacci number closest to F^{∗}(ε), and then only the two corresponding terms dominate in the Fourier series. Studying the size of this term (which also depends on r_{1} and r_{2}), one can observe that it is O( e^{−c/\sqrt{ε}}). With a more detailed analysis, one can see that a better estimate is provided by taking c to be not a constant function but a bounded oscillating one:

(9) c(δ) = C_{0} cosh (\frac{δ − δ_{0}}{2} ) for δ ∈ [δ_{0} − log γ, δ_{0} + log γ],

where

C_{0} = \sqrt{\frac{2π(γr_{1} + r_{2})}{γ + γ^{−1}}}, δ_{0} = log ε^{∗}, ε^{∗} = \frac{π(γ + γ^{−1})}{2γ^{2}(r_{1}γ + r_{2})}, (10)

continued periodically onto the whole real axis. In this way, the function c is piecewise-analytic, continuous, and 2 log γ-periodic. This is summarized in the following lemma.

Lemma 2 (Properties of the Melnikov function). The Melnikov function defined by (8) is a 2π-periodic function of θ_{1} and θ_{2}, such that

1) M (θ_{1} − T/ε, θ_{2} − γT/ε; ε) is analytic in the product of strips

{|ℑθ_{1}| < r_{1}} × {|ℑθ_{2}| < r_{2}} × {|ℑT | < π/2};

2) the maximum of the modulus of the Melnikov function taken on real arguments, max_{(θ_{1},θ_{2})∈𝕋^{2}} |M(θ_{1}, θ_{2})|, can be bounded from above and from below

by terms of the form

(11) const ε^{p−1} exp (−\frac{c(log ε)}{\sqrt{ε}})

with different ε-independent constants, where the function c in the exponent is defined by (9);

3) for a fixed small ε only four terms (at most) dominate in the Fourier series for the Melnikov function and the rest can be estimated from above by O( e^{−C_{1}/\sqrt{ε}}), where the constant C_{1} > max c(δ) = C_{0} cosh(log \sqrt{γ}).

Since we have established that for most small values of ε only the terms with (k_{1}, k_{2}) = ±(F_{n(ε)+1}, −F_{n(ε)}) are important, the Melnikov function is essentially

M(θ_{1}, θ_{2}; ε) ≈ 2 |M_{F_{n(ε)+1},−F_{n(ε)}} | sin (F_{n(ε)+1}θ_{1} − F_{n(ε)}θ_{2} + φ(ε)) .

The zeros of the Melnikov function correspond to homoclinic trajectories. The above formula implies that the zeros of the Melnikov function form two lines on the torus. As already noticed by C. Simó [Sim94], the averaged slopes of those lines approach γ when ε → 0.

Remark 4. We note that the Melnikov function is not invariant with respect to canonical changes of variables. After a change, e.g., after a step of the classical averaging procedure, a lot of nonzero harmonics, which were not present in the original system, can appear. If in the original system the Fibonacci terms were not big enough, these new harmonics may give larger contribution to the splitting. This idea was used in [Sim94] to detect the splitting for a system with only four perturbing terms.

Remark 5. The hypothesis that ω in the frequency vector is just (1, γ) can be relaxed. The generalization of the present result to the case when γ is a quadratic number is straightforward, with a similar expression (11) for the size of the Melnikov function. The case in which ω = (ω_{1}, ω_{2}), with the ratio ω_{1}/ω_{2} being of constant type (the continued fraction expansion has bounded coefficients), but not quadratic, can be similarly analyzed, but in this case c(δ) is no longer a periodic function. In some sense one can say, properly speaking, that there are no asymptotics. But it seems that there still exist upper and lower bounds, with the factor \sqrt{ε} in the denominator of the exponential term. The case of two frequencies whose ratio ω_{1}/ω_{2} is not of constant type, as well as the case of more than two perturbing frequencies, is more complicated.

In the following sections we sketch the method used to justify that the prediction given by the Melnikov function is correct. The method used here is a generalization to the quasiperiodic case of the method used in [Laz84], [DS92], [Gel93].

3. Normal form and local manifolds

The first step is to give a description of the dynamics near the 2D-dimensional invariant torus T . So, we will show the existence of a convergent normal form in a neighbourhood of T .

As we have seen during the analysis of the Melnikov function, the size of the splitting depends essentially on the widths of the analyticity strip (r_{1}, r_{2}) of the angular variables θ_{1}, θ_{2}, as well as on the width of the analyticity strip of the separatrix (x_{0}(t), y_{0}(t)). Therefore, to detect the splitting in the quasiperiodic case the loss of domain in the angular variables must be very small (i.e., O(ε^{α}), where α depends on the Diophantine properties of the frequencies). This makes another difference with the periodic case, where the size of the splitting does not depend on the width of the analyticity strip of the angular variable θ, but only on the width of the analyticity strip of the separatrix (x_{0}(t), y_{0}(t)). When dealing with the frequencies (1, γ) one needs a reduction of O(\sqrt{ε}) at most. Hence, during the proof of the convergence of the normal form one has to bound carefully the loss of domain (with respect to the angular variables) in order to achieve such a small reduction.

Finally, we want to stress that if the amount of reduction is something bigger, one can only produce upper bounds for the splitting of separatrices.

Theorem 1 (Normal Form Theorem). Let ε ∈ (0, ε_{0}). In a neighbourhood of the hyperbolic torus T there is a canonical change of variables (x, y) → (X, Y), which depends 2π-periodically on θ_{1} and θ_{2}, such that the Hamiltonian (1) takes the form

H(XY, ε) = H_{0}(XY) + ε^{p−1}H_{1}(XY, ε),

where H_{0} is the normal form Hamiltonian for the unperturbed pendulum. Moreover, the change of variables has the form

\begin{matrix}x = x^{(0)}(X, Y) + ε^{p−1}x^{(1)}(X, Y, θ_{1}, θ_{2}; ε), \\ y = y^{(0)}(X, Y) + ε^{p−1}y^{(1)}(X, Y, θ_{1}, θ_{2}; ε), \end{matrix} (12)

where (x^{(0)}, y^{(0)}) are normal form coordinates for the unperturbed pendulum.

The functions H_{0}, H_{1}, x^{(0)}, y^{(0)}, x^{(1)}, and y^{(1)} are analytic and uniformly bounded in the complex domain defined by

|X|^{2} + |Y |^{2} < r_{0}^{2},  |ℑθ_{1}| < r_{1} − \sqrt{ε},  |ℑθ_{2}| < r_{2} − \sqrt{ε},

for r_{1} and r_{2} in (3) and some positive constant r_{0} > 0.

The proof of this theorem can be found in [DGJS96a].

The Normal Form Theorem provides a convenient parametrization for the local invariant manifolds. Let λ be H′(0, ε). Then,

\begin{matrix}x = x^{s}(T, θ_{1}, θ_{2}) ≡ x(0, e^{−λT} , θ_{1}, θ_{2}), \\ y = y^{s}(T, θ_{1}, θ_{2}) ≡ y(0, e^{−λT} , θ_{1}, θ_{2}), \end{matrix} for T ≥ T_{0}, (13)

and

\begin{matrix}x = x^{u}(T, θ_{1}, θ_{2}) ≡ x( e^{λT} , 0, θ_{1}, θ_{2}), \\ y = y^{u}(T, θ_{1}, θ_{2}) ≡ y( e^{λT} , 0, θ_{1}, θ_{2}), \end{matrix} for T ≤ −T_{0}, (14)

where we have used the change (12). Theorem 1 also implies that, in the domains above,

\begin{matrix}|x^{β}(T, θ_{1}, θ_{2}) − x_{0}(T)| ≤ Cε^{p−1}, \\ |y^{β}(T, θ_{1}, θ_{2}) − y_{0}(T)| ≤ Cε^{p−1}, \end{matrix} for β = s, u.

4. Extension Theorem

The Normal Form Theorem provides a local approximation for the unstable manifold in terms of the unperturbed separatrix, which is O (ε^{p−1}). The following theorem extends this local approximation for solutions of system (2) to a global one. Since the unperturbed separatrix (x_{0}(T), y_{0}(T)) has a singularity on T = ±π/2, we will restrict ourselves to |ℑT | ≤ π/2 − \sqrt{ε}, i.e., up to a distance to the singularity T = ±π/2 of the same order as the loss of domain in the angular variables. Besides, the extension time t + T will be chosen big enough in order that the unperturbed separatrix reaches again the domain of convergence of the normal form. This procedure follows the same ideas as in the Extension Theorem of [DS92], and its complete proof can be also found in [DGJS96a].

Theorem 2 (Extension Theorem). Assume p > 2. Then, there exists ε_{0} > 0 such that the following extension property holds:

For any positive constants C and T_{0} there exists a constant C_{1}, such that for any ε ∈ (0, ε_{0}), every solution of system (2) that satisfies the initial conditions

|x(t_{0}) − x_{0}(t_{0} + T)| ≤ Cε^{p−1}, |y(t_{0}) − y_{0}(t_{0} + T)| ≤ Cε^{p−1},
|ℑθ_{1}(t_{0})| ≤ r_{1} − \sqrt{ε}, |ℑθ_{2}(t_{0})| ≤ r_{2} − \sqrt{ε},

for some T ∈ ℂ, t_{0} ∈ ℝ with

|ℑT | ≤ π/2 − \sqrt{ε}, −T_{0} ≤ t_{0} + ℜT < 0,

can be extended for −T_{0} ≤ t + ℜT ≤ T_{0} satisfying

|x(t) − x_{0}(t + T)| ≤ C_{1}ε^{p−2}, |y(t) − y_{0}(t + T)| ≤ C_{1}ε^{p−2}.

In particular, Theorem 2 can be applied to the local invariant unstable manifold given in (14). As we will see in Lemmas 3 and 4, the above approximation of these invariant manifolds in such a complex domain will allow us to derive suitable bounds of the error on the real axis to detect the splitting. Before closing this section let us note that, as a direct consequence of the Extension Theorem, the difference of unperturbed energies along the invariant manifolds can also be estimated.

Corollary 1. The following estimate holds:

h_{0}(x^{u}, y^{u}) − h_{0}(x^{s}, y^{s}) = M(θ_{1} − T/ε, θ_{2} − γT/ε) + O (ε^{2(p−2)}) ,

where h_{0} is evaluated on the invariant manifolds corresponding to T, θ_{1}, θ_{2},

ℜT ∈ (T_{0} − R, T_{0}), |ℑT | ≤ π/2 − \sqrt{ε}, |ℑθ_{k}| ≤ r_{k} − \sqrt{ε}, k = 1, 2, (15)

for any positive constants T_{0} and R, R < T_{0}.

5. First return

By Theorem 1, the local unstable invariant manifold is ε^{p−1}-close to the unperturbed separatrix. By Theorem 2, it can be continued for −T_{0} ≤ t + ℜT ≤ T_{0}, provided that the parameters (θ_{1}, θ_{2}, T) belong to the complex domain (15), and it remains ε^{p−2}-close to the unperturbed separatrix. Since this unperturbed homoclinic orbit comes back to the domain of the normal form, the same happens to the unstable manifold, which can be compared with the local stable manifold.

In order to describe the difference between the global unstable manifold and the local stable one, it is convenient to take H and T = − log Y/H′(XY) as canonical coordinates near the stable separatrix. The equation of the local stable manifold is then H = 0. In this coordinate system the unstable manifold is the graph of a function H^{u}: H = H^{u}(T, θ_{1}, θ_{2}), which depends 2π-periodically on θ_{1} and θ_{2}, and has zero mean, due to the Hamiltonian character of the perturbation. Using the parametrization provided by the normal form, it turns out that H^{u} is quasiperiodic in T:

H^{u}(T, θ_{1}, θ_{2}) = H^{u}(0, θ_{1} − T/ε, θ_{2} − γT/ε).

Moreover, by the Extension Theorem and its Corollary 1, H^{u} is given in first order by the Melnikov function for (T, θ_{1}, θ_{2}) in the complex domain (15):

H^{u}(T, θ_{1}, θ_{2}) = M(θ_{1} − T/ε, θ_{2} − γT/ε) + O (ε^{2p−4}) . (16)

It is important to notice that the term F = O (ε^{2p−4}) in equation (16) is an analytic function in the complex domain (15) which depends 2π-periodically on θ_{1} and θ_{2}, has zero mean, and is quasiperiodic in T. The following general lemma allows us to bound its Fourier coefficients.

Lemma 3. Let F (θ_{1} + s/ε, θ_{2} + γs/ε) be a 2π-periodic function of the variables θ_{1}, θ_{2}, analytic in the product of strips |ℑθ_{1}| ≤ r_{1}, |ℑθ_{2}| ≤ r_{2}, and |ℑs| ≤ ρ, and |F | ≤ A for these values of the variables. Then for all k_{1}, k_{2} ∈ ℤ

|F_{k_{1}k_{2}}| ≤ A e^{−|k_{1}|r_{1}−|k_{2}|r_{2}} e^{−ρ|k_{1}+γk_{2}|/ε}.

Finally, the next lemma gives the exponentially small upper bound for the function F for real values of the variables.

Lemma 4. Consider the (2 log γ)-periodic function c_{ρ,r_{1},r_{2}}(δ) defined on the interval [log ε^{∗} − log γ, log ε^{∗} + log γ] by

c_{ρ,r_{1},r_{2}}(δ) = C_{0} cosh (\frac{δ − log ε^{∗}}{2} ) ,

ε^{∗} = \frac{ρ(γ + γ^{−1})}{(γr_{1} + r_{2})γ^{2}} , C_{0} = 2\sqrt{\frac{(γr_{1} + r_{2})ρ}{γ + γ^{−1}}},

and continued by 2 log γ-periodicity. Let F satisfy the conditions of Lemma 3. If γ = (1 + \sqrt{5})/2 is the golden mean number and the mean value of the function F is zero, then

(17) |F (θ_{1}, θ_{2})| ≤ const A exp (−\frac{c_{ρ,r_{1},r_{2}}(log ε)}{\sqrt{ε}})

on the real values of its arguments. The constant depends continuously on r_{1} > 0 and r_{2} > 0.

Applying these two lemmas to the error function F = O (ε^{2p−4}) in equation (16), we obtain the desired exponentially small estimates. Now we can summarize the above results on the splitting function H^{u}(T, θ_{1}, θ_{2}) in the following theorem, which is the main result of this paper.

Theorem 3 (Main Theorem). There exist positive constants T_{0} and R, R < T_{0}, such that in the coordinate system (H, T, θ_{1}, θ_{2}) the unstable manifold can be represented as the graph of the function H = H^{u}(T, θ_{1}, θ_{2}; ε), where the function H^{u} depends 2π-periodically on θ_{1} and θ_{2}. In the domain

\begin{matrix}ℜT ∈ (T_{0} − R, T_{0}),  |ℑT | ≤ \frac{π}{2} − \sqrt{ε}, \\ |ℑθ_{1}| < r_{1} − \sqrt{ε},  |ℑθ_{2}| < r_{2} − \sqrt{ε}, \end{matrix}

this function is analytic and close to the Melnikov function:

H^{u}(T, θ_{1}, θ_{2}) = M(θ_{1} − T/ε, θ_{2} − γT/ε) + O (ε^{2p−4}) .

Moreover,

H^{u}(T, θ_{1}, θ_{2}) = H^{u}(0, θ_{1} − T/ε, θ_{2} − γT/ε),

and its mean value is zero:

∫_{𝕋^{2}} H^{u}(0, θ_{1}, θ_{2}) dθ_{1}dθ_{2} = 0. (18)

Furthermore, for p > 3 and real T, θ_{1} and θ_{2},

|H^{u}(T, θ_{1}, θ_{2}) − M (θ_{1} − T/ε, θ_{2} − γT/ε)| ≤ const ε^{2p−4} exp (−\frac{c(log ε)}{\sqrt{ε}}) ,

where c(δ) is defined in (9). If condition (4) is fulfilled, then there exists ε_{0} > 0 such that, for 0 < ε < ε_{0}, the maximum of the modulus of the Melnikov function is larger than the right-hand side of the last upper bound.

Acknowledgments

We are indebted to C. Simó for relevant discussions and remarks. Three of the authors (A. D., A. J. and T. M. S.) have been partially supported by the Spanish grant DGICYT PB94–0215, the EC grant ERBCHRXCT940460, the NATO grant CRG950273, and the Catalan grant CIRIT 1996SGR-00105. One of the authors (V. G.) was supported by a CICYT grant.

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Adv. Sci. Inst. Ser. C Math. Phys. Sci., held in S’Agaró, Spain, 19–30 June 1995, Kluwer Acad. Publ., Dordrecht, to appear in 1997.

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

E-mail address: amadeu@ma1.upc.es

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran via 585, 08007 Barcelona, Spain

Current address: Chair of Applied Mathematics, St.Petersburg Academy of Aerospace Instrumentation, Bolshaya Morskaya 67, 190000, St. Petersburg, Russia

E-mail address: gelf@maia.ub.es, gelf@misha.usr.saai.ru

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

E-mail address: angel@tere.upc.es

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

E-mail address: tere@ma1.upc.es