THE SUBHARMONIC MELNIKOV METHOD

CLARK ROBINSON

We consider a Hamiltonian system with two degrees of freedom which completely decouples for ϵ = 0. Therefore we assume there is a Hamiltonian

H^{ϵ}(z_{1}, z_{2}) = F_{1}(z_{1}) + F_{2}(z_{2}) + ϵH_{1}(z_{1}, z_{2}) + O(ϵ^{2}).

We also write H_{0}(z_{1}, z_{2}) = F_{1}(z_{1})+F_{2}(z_{2}). This induces a system of Hamiltonian differential equations which we write as

(*) (ż_{1} ż_{2}) = X^{ϵ}(z_{1}, z_{2}) = X(z_{1}, z_{2}) + ϵY (z_{1}, z_{2}) + O(ϵ^{2}).

For ϵ = 0, we write X(z_{1}, z_{2}) = (\begin{matrix}X_{1}(z_{1}) \\ X_{2}(z_{2}) \end{matrix}) so the differential equations divide into two

subequations, ż_{1} = X_{1}(z_{1}) and ż_{2} = X_{2}(z_{2}). We write ϕ(t, z, ϵ) for the flow for X^{ϵ} with

ϕ(0, z, ϵ) = z and \frac{d}{dt}ϕ(t, z, ϵ) = X^{ϵ} ∘ ϕ(t, z, ϵ).

We assume further that the unperturbed system for ϵ = 0 has the following properties: A1 The subsystem ż_{1} = X_{1}(z_{1}) has a fixed point at z_{1,0} and a homoclinic orbit to z_{1,0}

enclosing a region filled with periodic orbits of the unperturbed system. Let Σ_{1} be a single transversal for the subsystem ż_{1} = X_{1}(z_{1}). For the subsystem ż_{1} = X_{1}(z_{1}), let ẑ_{1}^{μ}(t) be the family of periodic orbits with ẑ_{1}^{μ}(0) ∈ Σ_{1}.

A2 The subsystem ż_{2} = X_{2}(z_{2}) has a region filled with periodic orbits. Let ẑ_{2}^{λ}(t, t_{0}) be this family of periodic orbits F_{2} with two parameters: the parameter λ varies the orbit and ẑ_{2}^{λ}(t, t_{0}) = ẑ_{2}^{λ}(t + t_{0}, 0) so t_{0} indicates the phase of the periodic orbit.

Let

ẑ(t; μ, λ, t_{0}) = (ẑ_{1}^{μ}(t), ẑ_{2}^{λ}(t, t_{0})).

Because the system decouples for ϵ = 0, the function ẑ(0; μ, λ, t_{0}) makes (μ, λ, t_{0}) into coordinates on Σ_{1}, i.e., ẑ(0; μ, λ, t_{0}) uniquely determines a point on Σ_{1}.

In the situation for the usual Melnikov function for a homoclinic orbit, the transverse homoclinic orbit of the purturbation must lie on the perturbed stable and unstable manifolds; this determines one dimension in the phase space. In the current situation, we must determine the points which persist in a resonance between the two oscillation. To define a function which give this resonance, we first define the return times of the Poincaré maps.

1991 Mathematics Subject Classification. 34C35, 34C27, 54H20.

Key words and phrases. dynamical systems, subharmonic, Melnikov.

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

Using the transversal Σ_{1}, let τ_{1}(z_{1}) be the return times of the Poincaré map for the periodic orbits for the subsystem ż_{1} = X_{1}(z_{1}). For ϵ ≠ 0, Σ_{1} can be considered a transversal for the full system by letting the z_{2} be arbitrary. Using this extended transversal Σ_{1}, let τ_{1}(z_{1}, z_{2}, ϵ) be the return times of the Poincaré map for Σ_{1} for the full system ż = X^{ϵ}(z). Notice that for ϵ = 0, τ_{1} can be defined for all points, but that τ_{1} is only defined on Σ_{1} for nonzero ϵ.

For the subsystem ż_{2} = X_{2}(z_{2}), take a family of transversals, Σ_{2,z_{2}}: if the system is written in action angle coordinates (I, θ) then the transversals can be taken to be θ = θ_{0}. Again Σ_{2,z_{2}} can be considered a family of transversals in the full system. Let τ_{2}(z_{1}, z_{2}, ϵ) be the return times of the Poincaré map for Σ_{2,z_{2}} with initial conditions (z_{1}, z_{2}).

We are interested in finding periodic orbits which are close to unperturbed orbits which are in resonance between the two subsystems. Let (m, n) be a fixed set of positive integers. For points on Σ_{1}, we consider the difference of the multiples of the periods,

Ψ^{m,n}(z_{1}, z_{2}, ϵ) = nτ_{1}(z_{1}, z_{2}, ϵ) − mτ_{2}(z_{1}, z_{2}, ϵ).

Zeroes of Ψ^{m,n} correspond to points where the return times are in a m to n resonance. The orbits which are in resonace are isolated; to this end we need to assume that the periods vary within the energy surface, i.e., that the gradent of H_{0} and Ψ^{m,n} are independent.

A3 Let (μ_{0}, λ_{0}) = (μ(m, n, h_{0}), λ(m, n, h_{0})) be parameters of the two families of orbits for ϵ = 0 for which Ψ^{m,n}(ẑ(0; μ_{0}, λ_{0}, t_{0})) = 0 and H_{0}(ẑ(0; μ_{0}, λ_{0}, t_{0})) = h_{0}. (Note that both Ψ^{m,n} and H_{0} are independent of t_{0}.) Assume that the gradients of H_{0} and Ψ^{m,n}, as functions on Σ_{1} are independent at ẑ(0; μ_{0}, λ_{0}, t_{0}). In terms of the parameterization ẑ(0; μ, λ, t_{0}), this assumption can be expressed as follows. Since the orbits ẑ_{2}^{λ}(t, t_{0}) are periodic, the gradient of F_{2} is nonzero and we can solve for λ = λ(μ, h_{0}) such that

H_{0} ∘ ẑ(0; μ, λ(μ, h_{0}), t_{0}) = F_{1} ∘ ẑ_{1}^{μ}(0)) + F_{2} ∘ ẑ_{2}^{λ}(0, t_{0}) = h_{0}.

Using this notation, the independence of the gradients can be expressed by

\frac{∂}{∂μ}Ψ^{m,n}(ẑ(0; μ, λ(μ, h_{0}), t_{0}), 0)|_{μ_{0}} ≠ 0.

Notice that Σ_{1}∩{t_{0} = 0} is two dimensional, and H_{0}^{−1}(h_{0})∩Σ_{1}∩{t_{0} = 0} is one dimensional with a point determined by the parameter μ. Since

\frac{∂}{∂μ}Ψ^{m,n}(ẑ(0; μ, λ(μ, h_{0}), 0), 0)|_{μ_{0}} ≠ 0,

the set of solutions μ(m, n, h_{0}) and λ(m, n, h_{0}) = λ(μ(m, n, h_{0}), h_{0}) is a set of locally isolated parameters.

Next we want to define the subharmonic Melnikov function which can be used to determine a value of t_{0} for which the resonance persists for ϵ ≠ 0. For z ∈ Σ_{1}, let

Ĝ^{m,n}(z, ϵ) = F_{1} ∘ ϕ(nτ_{1}(z, ϵ), z, ϵ) − F_{1}(z).

(It works just as well to use time mτ_{2}(z, ϵ) instead of nτ_{1}(z, ϵ).) For ϵ = 0, Ĝ^{m,n}(z, 0) ≡ 0, so we can write

Ĝ^{m,n}(z, ϵ) = ϵG^{m,n}(z, ϵ).

Using these choice of parameters, define the subharmonic Melnikov function as follows:

M^{m,n}(t_{0}, h_{0}) = \frac{∂}{∂ϵ}Ĝ^{m,n}(ẑ(0; μ_{0}, λ_{0}, t_{0}), ϵ)|_{ϵ=0} = G^{m,n}(ẑ(0; μ_{0}, λ_{0}, t_{0}), 0).

where μ_{0} = μ(m, n, h_{0}) and λ_{0} = λ(m, n, h_{0}) are the parameters of the two families of orbits for ϵ = 0 given above.

There are two main theorem. The first theorem relates nondegenerate zeroes of M^{m,n}(t_{0}, h_{0}) with periodic orbits. The second theorem relates M^{m,n}(t_{0}, h_{0}) to an integral.

Theorem 1. Make assumptions (A1-A3) and define M^{m,n} as above. Assume there is a

(t_{0}^{∗}, h_{0}^{∗}) such that M^{m,n}(t_{0}^{∗}, h_{0}^{∗}) = 0 and \frac{∂M^{m,n}}{∂t_{0}} (t_{0}^{∗}, h_{0}^{∗}) ≠ 0. Then there exists a positive

number ϵ_{0}(m, n, h_{0}^{∗}) > 0 such that for |ϵ| ≤ ϵ_{0}(m, n, h_{0}^{∗}), the perturbed system (∗) has a periodic orbit with period approximately

nτ_{1}(ẑ_{1}^{μ(m,n,h_{0}^{∗})}(0), 0) = mτ_{2}(ẑ_{2}^{λ(m,n,h_{0}^{∗})}(0, t_{0}^{∗}), 0)

with initial conditions close to ẑ(0; μ(m, n, h_{0}^{∗}), λ(m, n, h_{0}^{∗}), t_{0}^{∗}) and which stays within O(ϵ) of

ẑ(t; μ(m, n, h_{0}^{∗}), λ(m, n, h_{0}^{∗}), t_{0}^{∗})

for all time.

Theorem 2. Using these choice of parameters, the subharmonic Melnikov function is given by the following integral:

M^{m,n}(t_{0}, h_{0}) = ∫_{0}^{nτ_{1}}(DF_{1} · Y)_{ẑ(t;μ_{0},λ_{0},t_{0}^{∗})} dt

where μ_{0} = μ(m, n, h_{0}), λ_{0} = λ(m, n, h_{0}), and nτ_{1} = nτ_{1}(ẑ_{1}^{μ_{0}}(0)).

Proof of Theorem 1. We consider the function Θ^{m,n} : Σ_{1} × [−1, 1] → ℝ^{3} given by

Θ^{m,n}(z, ϵ) = (\begin{matrix}H^{ϵ}(z) \\ Ψ^{m,n}(z, ϵ) \\ G^{m,n}(z, ϵ) \end{matrix}) .

The function ẑ(0; μ, λ, t_{0}) = (ẑ_{1}^{μ}(0), ẑ_{2}^{λ}(0, t_{0})), induces the coordinates (μ, λ, t_{0}) on Σ_{1}. Let

(μ_{0}^{∗}, λ_{0}^{∗}) = (μ(m, n, h_{0}^{∗}), λ(m, n, h_{0}^{∗})).

Since G^{m,n}(ẑ(0; μ_{0}^{∗}, λ_{0}^{∗}, t_{0}), 0) = M^{m,n}(t_{0}, h_{0}^{∗}), the derivative of Θ^{m,n}(ẑ(0; μ, λ, t_{0}), ϵ) with respect to (μ, λ, t_{0}) for t = 0, ϵ = 0, μ = μ_{0}^{∗}, λ = λ_{0}^{∗}, and t_{0} = t_{0}^{∗} is

D_{(μ,λ,t_{0})}(Θ^{m,n} ∘ ẑ)_{((0;μ_{0}^{∗},λ_{0}^{∗},t_{0}^{∗},0)}

= (\begin{matrix}\frac{∂(H_{0} ∘ ẑ)}{∂μ} & \frac{∂(H_{0} ∘ ẑ)}{∂λ} & 0 \\ \frac{∂(Ψ^{m,n} ∘ ẑ)}{∂μ} & \frac{∂(Ψ^{m,n} ∘ ẑ)}{∂λ} & 0 \\ ∗ & ∗ & \frac{∂M^{m,n}}{∂t_{0}} \end{matrix} )_{(0;μ^{∗},λ^{∗},t_{0}^{∗},0),h_{0}^{∗}} .

Because (i) the gradients of H_{0} and Ψ^{m,n} are independent and (ii)

\frac{∂M^{m,n}}{∂t_{0}} (t_{0}^{∗}, h_{0}^{∗}) ≠ 0,

this matrix is nonsingular. By the Implicit Function Theorem, the variables μ, λ, and t_{0} can be solved for in terms of ϵ, (μ(ϵ), λ(ϵ), t_{0}(ϵ)), such that

Θ^{m,n}((μ(ϵ), λ(ϵ), t_{0}(ϵ), ϵ) ≡ (\begin{matrix}h_{0} \\ 0 \\ 0 \end{matrix}) .

Let z^{∗}(ϵ) = ẑ(0; μ(ϵ), λ(ϵ), t_{0}(ϵ)). Because Ψ^{m,n}(z^{∗}(ϵ), ϵ) = 0, nτ_{1}(z^{∗}(ϵ), ϵ) = mτ_{2}(z^{∗}(ϵ), ϵ), and z^{∗}(ϵ) and ϕ(nτ_{1}(z^{∗}(ϵ), ϵ), z^{∗}(ϵ), ϵ) are both points in the same Σ_{2,z^{∗}(ϵ)}. (This means they have the same t_{0} coordinate.) Also, they are both points in Σ_{1} by the definition of τ_{1}. Because Ĝ^{m,n}(z^{∗}(ϵ), ϵ) = ϵG^{m,n}(z^{∗}(ϵ), ϵ) = 0, F_{1} has the same value at z^{∗}(ϵ) and ϕ(nτ_{1}(z^{∗}(ϵ), ϵ), z^{∗}(ϵ), ϵ). Because H^{ϵ} is preserved, its value is the same at these two points. Because H_{0} and F_{1} are independent, for small ϵ, H^{ϵ} and F_{1} are independent on Σ_{1} ∩ Σ_{2,z^{∗}(ϵ)} and it follows that

ϕ(nτ_{1}(z^{∗}(ϵ), ϵ), z^{∗}(ϵ), ϵ) = z^{∗}(ϵ)

and z^{∗}(ϵ) is a periodic orbit for X^{ϵ} with period nτ_{1}(z^{∗}(ϵ), ϵ). (Notice that F_{1} is independent from H^{ϵ} on a fixed Σ_{1} ∩ Σ_{2,z^{∗}(ϵ)} which is two dimensional.) Because all the variables are determined by the Implicit Function Theorem, the period, the initial conditions, and the orbits are within O(ϵ) for all time.

Proof of Theorem 2. We consider

\frac{d}{dt} \frac{∂}{∂ϵ}F_{1} ∘ ϕ(t, z, ϵ)|_{ϵ=0} = \frac{∂}{∂ϵ} \frac{d}{dt}F_{1} ∘ ϕ(t, z, ϵ)|_{ϵ=0}

= \frac{∂}{∂ϵ}[DF_{1} · (X + ϵY)]_{ϕ(t,z,ϵ)}|_{ϵ=0}

= [DF_{1} · Y]_{ϕ(t,z,0)} + \frac{∂}{∂ϵ}[DF_{1} · X]_{ϕ(t,z,ϵ)}|_{ϵ=0}

= [DF_{1} · Y]_{ϕ(t,z,0)}.

The last equality follows because [DF_{1} · X]_{z} = 0 at all points z. Letting

z_{m,n} = ẑ(0; μ(m, n, h_{0}), λ(μ(m, n, h_{0})), t_{0})

and integrating from t = 0 to nτ_{1} = nτ_{1}(z_{m,n}, 0) we get

M^{m,n}(t_{0}, h_{0}) = \frac{∂}{∂ϵ}F_{1} ∘ ϕ(nτ_{1}, z_{m,n}, ϵ)|_{ϵ=0} − \frac{∂}{∂ϵ}F_{1}(z_{m,n})|_{ϵ=0}

= ∫_{0}^{nτ_{1}} \frac{d}{dt} \frac{∂}{∂ϵ}F_{1} ∘ ϕ(t, z_{m,n}, ϵ)|_{ϵ=0} dt

= ∫_{0}^{nτ_{1}}[DF_{1} · Y]_{ϕ(t,z_{m,n},0)} dt

= ∫_{0}^{nτ_{1}}[DF_{1} · Y]_{ẑ(t;μ(m,n,h_{0}),λ(m,n,h_{0}),t_{0})} dt.

This gives the desired result.

References

[1] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York and Berlin, 1983.

[2] C. Robinson, Horseshoes for autonomous Hamiltonian systems using the Melnikov integral, Ergodic Theory and Dynamical Systems 1988, 395–409.

[3] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York and Berlin, 1990

Department of Mathematics, Northwestern University, Evanston IL 60201

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