Contemporary Mathematics Volume 00, 19xx

Melnikov Method for Autonomous Hamiltonians

Clark Robinson

Abstract. This paper presents the method of applying the Melnikov method to autonomous Hamiltonian systems in dimension four. Besides giving an application to Celestial Mechanics, it discusses the problem of convergence of the Melnikov function and the derivative of the Melnikov function.

1. Introduction

The original Melnikov method applies to periodically forced Hamiltonian systems in dimension two (one degree of freedom). In this paper, our main emphasis is to show how a modification of this method applies directly to autonomous Hamiltonian systems in dimension four (two degrees of freedom). These original unperturbed system needs to be completely integrable. By restricting to the energy surface, it is possible to convert the autonomous system in dimension four into a periodically forced Hamiltonian systems in dimension two and then apply the usual Melnikov method. It is our contention that this is not the natural method to treat this type of system. In fact the calculations are much more direct and natural when done directly with the autonomous system. Thus in this paper we advocate applying the Melnikov method directly to the autonomous Hamiltonian system. These ideas go back at least to the papers of P. Holmes and J. Marsden, [3] and [4]. Also see [9] and [6]. In Section 3 we discuss the main theorems in the theory of autonomous systems and their proofs. In Section 4 we indicate how these theorems can be applied to the circular restricted three body problem.

Because we want to connect our treatment with the material for periodically forced systems, we start by describing the periodically forced situation using our approach in Section 2.

This paper also points out the connection between the convergence of the improper integral giving the Melnikov function and the smoothness and dependence on parameters of the stable manifold of a periodic orbit. (See the discussion of Lemma 3.8 and the remarks which follow it.) This connection has also been made recently in the context of Celestial Mechanics by J. Casasayas, E. Fontich, and A. Nunes, [2].

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0271-4132/00 $1.00 + $.25 per page

1991 Mathematics Subject Classification. 34C35, 70F07.

Key words and phrases. dynamical systems, horseshoes, Melnikov method.

This research partially supported by grants from the National Science Foundation.

A final point made in this paper is the possibility of differentiating the Melnikov function by differentiating the integrand of its representation as a improper integral. See Theorem 3.5 in Section 3.

2. Periodically force systems

Consider a Hamiltonian H : ℝ^{2} → ℝ, with associated Hamiltonian vector field X = X_{H}. Next, consider the periodically force perturbation,

\begin{aligned}ż & = X(z) + ϵY (z, τ) \\ τ̇ & = 1 \end{aligned}

where τ is considered as a periodic variable of some period T. We assume that X has a homoclinic orbit to a hyperbolic fixed point (in the z-space). Let γ_{0} be the associated periodic orbit in the (z, τ)-space, S̃, and

Γ_{0} = {(z_{0}, τ_{0}) z_{0} is homoclinic to γ}

be the set of homoclinic orbits in the extended phase space S̃.

Since γ_{0} is hyperbolic, it can be continued for ϵ ≠ 0 to a periodic orbit γ_{ϵ} in the extended phase space S̃. For σ equal s and u, let W^{σ}(γ_{ϵ}, X +ϵY) be the stable and unstable manifolds of γ_{ϵ} respectively. In order to determine the orbits on Γ_{0} to a particular orbit on W^{σ}(γ_{ϵ}, X +ϵY) for ϵ ≠ 0, we let N_{(z_{0},τ_{0})} be a (one dimensional) transversal to Γ_{0} at a point (z_{0}, τ_{0}) ∈ Γ_{0}. Then for σ = s, u, set

ζ^{σ}(z_{0}, τ_{0}, ϵ) = W^{σ}(γ_{ϵ}, X + ϵY) ∩ N_{(z_{0},τ_{0})}.

In our development, we use strongly the fact that these stable and unstable manifold depend differentiably on the parameter and position on the manifold, so are smooth in the space S̃, i.e., for σ = s and u, ζ^{σ}(z_{0}, τ_{0}, ϵ) is a smooth function of all its variables.

The Hamiltonian function is constant on Γ_{0} and has nonzero gradient, so it is a good measure of the displacement perpendicular to Γ_{0}. With this motivation, we define the Melnikov function by

M(z_{0}, τ_{0}) = \frac{∂}{∂ϵ}[H ∘ ζ^{u}(z_{0}, τ_{0}, ϵ) − H ∘ ζ^{s}(z_{0}, τ_{0}, ϵ)]|_{ϵ=0}.

Thus, M is a measurement of the infinitesimal separation of the stable and unstable manifolds as measured by the Hamiltonian function H. By parameterizing the solutions of the equations and restricting to a transverse cross section, M can be considered a function of a real variable (as it usually is). Notice that we use the fact that the stable and unstable manifolds depend smoothly on the parameter to define M.

There are two theorems related to M. The first states that a nondegenerate zero of M corresponds to a transverse homoclinic orbit of γ_{ϵ} for ϵ ≠ 0 but small enough. We refer to [9], [10], or other standard references for this theorem. (Also see Theorem 3.3 in the next section.) The second theorem relates M to an improper integral. We state this second theorem in order to emphasis the fact that with the proper statement, it can be calculated in any coordinate system (not just symplectic coordinates).

Theorem 2.1.

(*)  M(z_{0}, τ_{0}) = ∫_{−∞}^{∞} k · (X × Y)_{(ϕ_{0}^{t}(z_{0}),t+τ_{0})} dt

= ∫_{−∞}^{∞}(DH · Y)_{(ϕ_{0}^{t}(z_{0}),t+τ_{0})} dt. (**)

We remark that the integral in equality (*) is the usual one derived for the Melnikov function: the integrand is the scalar cross product (component in the k direction) or wedge product of the two vector fields. This integrand must be calculated in symplectic coordinates. The the integral in equality (**) is valid in any set of coordinates, even those which are not symplectic. See [9] or [10] for a proof of the second form of the integral. Also in the discussion below of the autonomous case, we indicate a sketch of the proof why the above theorem is true. The proof in [9] uses the smoothness of the stable and unstable manifolds and not the uniform estimate for 0 ≤ t < ∞ on the difference of solutions for ϵ ≠ 0 and ϵ = 0 as many proofs do. (See e.g. [1] and [2].)

3. Autonomous Hamiltonians in four dimensions

In this section, the phase space is some four dimensional space which we label as S^{4}: in Celestial Mechanics applications S^{4} is usually ℝ^{4} or some product of copies of the circle (periodic variables) and copies of the reals. The autonomous Hamiltonian H^{ϵ} : S^{4} → ℝ is given as

H^{ϵ}(z) = H_{0}(z) + ϵH_{1}(z) + O(ϵ^{2}).

The corresponding Hamiltonian vector field is given as

X^{ϵ}(z) = X_{0}(z) + ϵY (z) + O(ϵ^{2}).

The basic assumptions are as follows. (A1) We assume that X_{0} is completely integrable, with a second independent

integral F. (The function F is mostly likely not an integral of X^{ϵ} for ϵ ≠ 0.) (A2) We assume that X_{0} has a hyperbolic periodic orbit γ_{0} with set of homoclinic

orbits

Γ_{0} = comp[W^{u}(γ_{0}, X_{0}) \ γ_{0}]
= comp[W^{s}(γ_{0}, X_{0}) \ γ_{0}].

Here Γ_{0} is one of the components of the stable and unstable manifolds indicated and has dimension two because of the second integral.

Example 3.1. The easiest example of the above type of system is one which decouples for ϵ = 0 into a subsystem with a homoclinic orbit and another That is oscillator: e.g., assume H_{0}(z, w) = F (z) + G(w) where

F (z_{1}, z_{2}) = \frac{z_{2}^{2}}{2} − \frac{z_{1}^{2}}{2} + \frac{z_{1}^{3}}{3}

has a homoclinic orbit to the fixed point at 0, and

G(w_{1}, w_{2}) = \frac{α}{2}(w_{1}^{2} + w_{2}^{2})

is an oscillator. A simple coupling is of the type H_{1}(z, w) = z_{1}w_{1}.

Example 3.2. P. Holmes and J. Marsden [4] gave an example of a system which does not decouple. (Compare this example with the closely related one from Celestial Mechanics given in [1].) Let

H^{ϵ}(p, q, θ, I) = \frac{p^{2}}{2} + U(q) + \frac{I^{2}}{2q^{2}} + ϵ sin(θ),

so

İ = −ϵ cos(θ)

and F = I is a second integral for ϵ = 0. The potential function U(q) must be chosen so that there is a homoclinic orbit in the (p, q)-subsystem for ϵ = 0 and I fixed.

We set up the definition of the Melnikov function using transversals much as we did above for the periodically forced case. For each point z_{0} ∈ Γ_{0}, let N_{z_{0}} be a two dimensional plane perpendicular to Γ_{0} at z_{0}. Let γ_{ϵ} be the continuation of the periodic orbit γ_{0} for ϵ ≠ 0 with γ_{ϵ} ⊂ (H^{ϵ})^{−1}(h_{0}) where h_{0} = H_{0}(γ_{0}) is the energy of the unperturbed orbit. Then for σ = s and u and ϵ ≠ 0, let

ζ^{σ}(z_{0}, h_{0}, ϵ) = N_{z_{0}} ∩ W^{σ}(γ_{ϵ}, X^{ϵ})

be the continuation of z_{0}. Notice that both the stable and the unstable manifolds continue to lie in the same energy surface (H^{ϵ})^{−1}(h_{0}), so H_{0} is not a good measure of the separation; on the other hand, the second integral F is constant on Γ_{0} and is independent of H_{0} so it is a good measure of the distance perpendicular to Γ_{0} within this energy surface. The Melnikov function, M : Γ_{0} → ℝ is then defined as the infinitesimal displacement of the stable and unstable manifolds as measured by the second integral F:

M(z_{0}, h_{0}) = \frac{∂}{∂ϵ}[F ∘ ζ^{u}(z_{0}, h_{0}, ϵ) − F ∘ ζ^{s}(z_{0}, h_{0}, ϵ)]|_{ϵ=0}.

By expressing M in terms of specific coordinates and restricting to a transverse section, it is possible to give it as a function of a single variable (plus the parameter h_{0}). See the example in the next section.

Again, there are two theorems about the Melnikov function. The first states that a nondegenerate zero of M corresponds to a transverse homoclinic point. The second theorem indicates how to calculate M in terms of an improper integral. In general for this case, the integral representation of M is only conditionally convergent, so the times must be chosen correctly in the improper integral. The third theorem below justifies differentiating under the integral sign to calculate the derivative of the Melnikov function.

Theorem 3.3. Make the basic assumptions A1-A2). If M(·, h_{0}) : Γ_{0} → ℝ has a nondegenerate zero at z_{0} ∈ Γ_{0}, then X^{ϵ} has a transverse homoclinic orbit z_{ϵ} nearby for small enough nonzero ϵ.

Theorem 3.4. Make the basic assumptions (A1-A2). Assume that T_{j} and −T_{j}^{∗} are chosen so that the points ϕ_{0}(T_{j}, z_{0}) and ϕ_{0}(−T_{j}^{∗}, z_{0}) converge to the same point z_{∞} ∈ γ_{0}. Then

M(z_{0}, h_{0}) = lim_{j→∞} ∫_{−T_{j}^{∗}}^{T_{j}} (DF · Y)_{ϕ_{0}(t,z_{0})} dt.

Theorem 3.5. Make the basic assumptions (A1-A2) as in Theorem 3.4 but assume that the integral converges absolutely (not conditional convergence). (See remarks following Lemma 3.8 for conditions which imply this assumption.) Then

D_{z}M_{(z,h_{0})}v = ∫_{−∞}^{∞} D_{z}((DF · Y)_{ϕ_{0}(t,z)})v dt.

In the above equation, D_{z} is differentiation with respect to the initial spatial coordinate z, and is calculated in the direction of the vector v.

Remark 3.6. The integral in Theorems 3.4 and 3.5 can be calculated in any coordinates, even non-symplectic coordinates. This is relevant to applications in Celestial Mechanics because McGehee coordinates are non-symplectic coordinates. See Section 4.

Remark 3.7. Although the convergence is only conditional in general, there are several circumstances in which the convergence is absolute. We make this more explicit below.

The proof of Theorem 3.3 follows as usual from the Implicit Function Theorem, so we do not give any details. See [10].

Idea of the proof of Theorem 3.4. As in the usual proof,

M(z_{0}, h_{0}) = ∫_{−T_{j}^{∗}}^{T_{j}} (DF · Y)_{ϕ_{0}(t,z_{0})} dt + R(−T_{j}^{∗}, T_{j})

where the remainder term

\begin{aligned}R(−T_{j}^{∗}, T_{j}) = & \frac{∂}{∂ϵ}[F ∘ ϕ(−T_{j}^{∗}, ζ^{u}(z_{0}, ϵ), ϵ)]|_{ϵ=0} \\ & − \frac{∂}{∂ϵ}[F ∘ ϕ(T_{j}, ζ^{s}(z_{0}, ϵ), ϵ)]|_{ϵ=0}. \end{aligned}

In [9], we prove a lemma about the convergence of the remainder. The statement of Lemma 2.1 in [9] only includes the final statement of the following lemma but the proof in that paper implies the following extended statement. Clearly Lemma 3.8 implies the Theorem 3.4.

Lemma 3.8. Assume that T_{j} and −T_{j}^{∗} are chosen so that ϕ_{0}(−T_{j}^{∗}, z_{0}) converges to z_{∞}^{−} and ϕ_{0}(T_{j}, z_{0}) converges to z_{∞}^{+} as j → ∞. Then

\frac{∂}{∂ϵ}[F ∘ ϕ(−T_{j}^{∗}, ζ^{u}(z_{0}, ϵ), ϵ)]|_{ϵ=0}

converges to

\frac{∂}{∂ϵ}[F ∘ γ(z_{∞}^{−}, ϵ)]|_{ϵ=0}

and

\frac{∂}{∂ϵ}[F ∘ ϕ(T_{j}, ζ^{s}(z_{0}, ϵ), ϵ)]|_{ϵ=0}

converges to

\frac{∂}{∂ϵ}[F ∘ γ(z_{∞}^{+}, ϵ)]|_{ϵ=0}.

Here γ(z, ϵ) is the function which gives the perturbed orbit γ_{ϵ} for ϵ ≠ 0 as a graph over the unperturbed orbit. In particular, if z_{∞}^{−} = z_{∞}^{+} then R(−T_{j}^{∗}, T_{j}) goes to zero as j goes to infinity.

Remark 3.9. By Lemma 3.8, the improper integral in Theorem 3.4 always gives M(z_{0}, h_{0}) provided T_{j} and −T_{j}^{∗} are chosen so that z_{∞}^{−} = z_{∞}^{+}.

Remark 3.10. If DF_{(z)} ≡ 0 for all z ∈ γ_{0}, then the convergence is not conditional and the integral is just the improper integral from −∞ to ∞. This feature is essentially what is true in the proof for the usual Melnikov function when using the function H which vanishes at the fixed point.

Remark 3.11. If the periodic orbit does not move with ϵ, γ_{ϵ} ≡ γ_{0}, then the convergence is also a usual improper integral (and not conditional) since the limiting values are zero and do not need to cancel each other. When applying this theorem to parabolic orbits which are asymptotic to periodic orbits at infinity, this is often the case since the orbits at infinity are not affected by the small coupling terms. See the example of the circular restricted three body problem in Section 4.

Remark 3.12. What makes the convergence work in the proof of Lemma 3.8 is the stable manifold theory. Casasayas, Fontich, and Nunes proved in [2] that this type of result is often true even when the periodic orbit is weakly hyperbolic (from higher order terms). Notice that they emphasize that it is possible to prove a stable manifold theorem in the parabolic case with a particular smoothness. Then then prove that this smoothness of the stable manifold is sufficient to imply a convergence of the above type to give the Melnikov function as an improper integral. They treat the usual time periodic perturbations but their results go over to the above context of autonomous systems as well.

Proof of Theorem 3.5. The fact that we can differentiate under the integral sign follows from a theorem in analysis about differentiating improper integrals. What is needed to differentiate under the integral sign of an improper integral is that the integrals for M(z, h_{0}) and D_{z}M_{(z,h_{0})}v converge uniformly in z. See [5] or [7]. In our situation the convergence of both integrals is uniform for the same reason that the integral converges in the proof of Lemma 3.8 as given in [9].

4. Planar Restricted Three Body Problem Example

Both J. Llibre and C. Simo in [8] and Z. Xia in [11] applied the usual Melnikov method to the circular planar three body problem. In this section, we indicate how the same results can be obtain (possibly) more directly by using the method for the autonomous systems indicated in the last section. Our remarks about the applicability in non-symplectic coordinates also justifies making the calculations in McGehee coordinates.

Below we sketch the introduction of the coordinates into the problem. See the above references for more detailed explanations and treatment. Our treatment most closely follows that in [11].

In this situation, the third mass m_{3} ≡ 0, m_{1} = μ is a small parameter, and m_{2} = 1−μ. Jacobi coordinates are used as a starting place. The problem is circular so the variable

Q = (cos(t), sin(t))

gives the motion of the vector from m_{2} to m_{1}. The position of m_{3} relative to m_{1} and m_{2} is given by q.

Next, McGehee coordinates are introduced. The variable x measures the distance from infinity,

x^{−2} = |q|,

so x goes to zero as |q| goes to infinity. The direction to m_{3} is given by θ,

\frac{q}{|q|} = (cos θ, sin θ).

This angle is measured in rotating coordinates by introducing the 2π-periodic variable s = t − θ. The momentum variable for m_{3}, p is broken down into radial and angular components by introducing the variables y and ρ respectively by the equation

p = [y + x^{2}ρi][cos θ + i sin θ].

(This last equations uses the notation of complex variables for points in ℝ^{2}.) The energy equation in these variables becomes

H^{μ} = \frac{1}{2}y^{2} + \frac{1}{4}x^{4}ρ^{2} − U^{μ}

where U^{0} = x^{2}.

When μ = 0, ρ̇ = 0, so ρ, which is the angular momentum of m_{3} (the Jacobi constant), is a second integral. After a change of the time variable, the equations become

x′ = −\frac{1}{2}x^{3}y
y′ = −x^{4} + x^{2}ρ^{2} + μg_{1}(x, s) + O(μ^{2})
ρ′ = μg_{2}(x, s) + O(μ^{2})
s′ = 1 − x^{4}ρ.

Here s is considered a 2π periodic variable.

When μ = 0, (\begin{matrix}x \\ y \end{matrix}) = (\begin{matrix}0 \\ 0 \end{matrix}) is a fixed point in (\begin{matrix}x \\ y \end{matrix})-space. This fixed point

corresponds to a periodic orbit in the total space (with s the periodic variable). This periodic orbit is weakly hyperbolic but it has smooth stable and unstable manifolds which correspond to the parabolic orbits of the original problem. We parameterize the homoclinic orbits

x = ξ_{0}(t, ρ_{0}, s_{0})
y = η(t, ρ_{0}, s_{0})
s = s_{0} + σ(t, ρ_{0})

where we take the initial conditions so that η(0, ρ_{0}, s_{0}) = 0, i.e. the orbits cross the transverse section y = 0 at t = 0. Fixing ρ_{0} at t = 0 is like fixing the energy of the system.

For μ = 0, ρ is constant and so is a second integral of the system. Thus the Melnikov function is

M(s_{0}, ρ_{0}) = ∫_{−∞}^{∞} (Dρ) · Y_{ϕ_{0}(t,ρ_{0},s_{0})} dt

= ∫_{−∞}^{∞} g_{2} ∘ ϕ_{0}(t, ρ_{0}, s_{0}) dt

= ∫_{−∞}^{∞} ξ_{0}^{4}(t) sin(s_{0} + σ(t))S(t, ρ_{0}, s_{0},) dt,

where

S(t, ρ_{0}, s_{0},) = [1 − \frac{1}{[1 + 2ξ_{0}^{2}(t) cos(s_{0} + σ(t)) + ξ_{0}^{4}(t)]^{3/2}]}.

The above form of the Melnikov function is derived in [11] by starting with the form arising from a periodically forced system and then integrating by parts. This type of derivation is not surprising since the equivalence of tbe usual Melnikov integral and that obtained for autonomous sytems is proved in [9] using integration by parts. The point that is being made here is the above treatment is very direct and seems less ad hoc then what is found in [11]. (I am certain that Z. Xia was very aware of the representation of the Melnikov integral by means of a second integral and that he just preferred using the more familiar form.) Notice that the orbit at infinity does not vary with μ so the integral converges absolutely.

Once the above form is obtained, symmetry considerations imply that there is a zero of M for s_{0} = 0 and π, M(0, ρ_{0}) = M(π, ρ_{0}) = 0.

The orbit at infinity does not vary with μ, so the above integral representation

of M is not conditional. Therefore by Theorem 3.5, \frac{∂M}{∂s_{0}} (s_{0}, ρ_{0}) can be calculated

by differentiating under the integral sign with respect to s_{0} which appears in a very simple way. (The integral of this derivative with respect to s_{0} also converges uniformly in s_{0}.) Special arguments are needed to get that the improper integral

representing \frac{∂M}{∂s_{0}} (s_{0}, ρ_{0}) for s_{0} = 0 or π is nonzero, and so the corresponding zero

of M(·, ρ_{0}) is a nondegenerate zero as a function of s_{0}. These arguments which verify this fact done very differently in [11] and [8].

References

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2. , Homoclinic orbits to parabolic points, preprint, 1994.

3. P. Holmes and J. Marsden, Horseshoes in perturbations of hamiltonian systems with two degrees of freedom, Commun. Math. Phys. 82 (1982), 523–544.

4. , Horseshoes and arnold diffusion for hamiltonian systems on lie groups, Indiana Univ. Math. J. 32 (1983), 273–309.

5. S. Lang, Analysis i, Addison-Wesley Publ. Co., Reading Mass., 1969.

6. L. Lerman and Ia. Umanskii, On the existence of separatrix loops in four-dimensional systems similar to the integrable hamiltonian systems, Appl. Math. Mech. 47 (1984), 335–340.

7. J. Lewin and M. Lewin, An introduction to mathematical analysis, McGraw-Hill, New York, 1993.

8. J. Llibre and C. Simo, Oscillatory solutions in the planar restricted three-body problem, Math. Annalen 248 (1980), 153–184.

9. C. Robinson, Horseshoes for autonomous hamiltonian systems using the melnikov integral, Ergod. Th. & Dynam. Sys. 8^{∗} (1988), 395–409.

10. S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer-Verlag, New York-Berlin-Heidelberg, 1990.

11. Z. Xia, Melnikov method and transversal homoclinic orbits in the restricted three-body problem, J. Diff. Equat. 96 (1992), 170–184.

Department of Mathematics, Northwestern University, Evanston IL 60201

 E-mail address: clark@math.nwu.edu