Lecture 3: Exponentially small splitting of separatrices Marcel Guardia Universitat Polit` ecnica de Catalunya February 8, 2017 M. Guardia (UPC) Lecture 3 February 8, 2017 1 / 26
Lecture 3: Exponentially small splitting of
separatrices
Marcel Guardia
Universitat Politecnica de Catalunya
February 8, 2017
M. Guardia (UPC) Lecture 3 February 8, 2017 1 / 26
Exponentially small splitting of separatrices
For the RPC3BP we want to prove transversality of the invariant
manifolds for fixed µ ∈ (0,1/2] and G0 ≫ 1.
The distance between the invariant manifolds is exponentially
small in G0.
Melnikov Theory does not apply unless µ≪ e−G3
0
3 .
To prove splitting for any µ ∈ (0,1/2] is a difficult problem since
classical perturbative techniques do not apply.
We have to deal with exponentially small splitting of separatrices.
We show how to deal with this problem in a simpler model.
M. Guardia (UPC) Lecture 3 February 8, 2017 2 / 26
The perturbed pendulum
Hamiltonian
H
(
y , x ,t
ε
)
=y2
2+ (cos x − 1) + δ(cos x − 1) sin
t
ε
Equations
x = y
y = sin x + δ sin x sint
ε
This is a model of the resonances of nearly completely integrable
Hamiltonian systems.
Tipically in this problems δ is a fixed non small constant.
In the RPC3BP one has: ε ∼ G−30 and δ ∼ µG−2
0 .
Luckily in the RPC3BP ressembles this model with δ ∼ ε2/3.
This makes a big difference for analyzing the splitting.
M. Guardia (UPC) Lecture 3 February 8, 2017 3 / 26
The perturbed pendulum
From now on:
H
(
y , x ,t
ε
)
=y2
2+ (cos x − 1) + ε(cos x − 1) sin
t
ε
Equations
x = y
y = sin x + ε sin x sint
ε
t = 1
(x , y) = (0,0) is a hyperbolic periodic orbit for this system.
We want to compute the splitting between its invariant manifolds.
M. Guardia (UPC) Lecture 3 February 8, 2017 4 / 26
First rigorous result dealing with this problem:
Theorem (Neishtadt 84)
Let us fix ε0 > 0. Then, for ε ∈ (0, ε0), there exists an ε-close to the
identity canonical transformation that transforms system
H
(
y , x ,t
ε
)
=y2
2+ (cos x − 1) + H1
(
x , y ,t
ε
)
with H1 analytic into
H
(
y , x ,t
ε
)
=y2
2+ (cos x − 1) + εH0(y , x , ε) + P
(
y , x ,t
ε
)
and P satisfies
|P| ≤ c2e−c1ε
for certain constants c1, c2 > 0.
M. Guardia (UPC) Lecture 3 February 8, 2017 5 / 26
This theorem implies that:
The system is exponentially small close to integrable.
The invariant manifolds are exponentially close.
It does not give any information about the splitting of the separatrix
We do not know whether they coincide or not.
We need to compute the first order of the distance between the
invariant manifolds.
M. Guardia (UPC) Lecture 3 February 8, 2017 6 / 26
Plan
Look for suitable parameterizations of the invariant manifolds.
Extend these parameterizations to certain regions of the complex
plane.
Analyze the difference of the parameterizations in these regions.
Deduce from this analysis the first order of the difference in the
reals.
Simplifications of our example: the unperturbed separatrix is a
graph over the base
M. Guardia (UPC) Lecture 3 February 8, 2017 7 / 26
Parameterizations of the invariant manifolds
We look for parameterizations of the invariant manifolds as graphs
y = ∂xSu,s(x , t/ε)
The generating functions Su,s satisfies the Hamilton-Jacobi
equation.
ε−1∂τS (x , τ) + H (x , ∂x S (x , τ) , τ) = 0
Pendulum example
ε−1∂τS +(∂xS)2
2+ cos x − 1 + ε(cos x − 1) sin τ = 0
Asymptotic conditions:
limx→0
∂xSu (x , t/ε) = 0
limx→2π
∂xSs (x , t/ε) = 0
M. Guardia (UPC) Lecture 3 February 8, 2017 8 / 26
Reparameterization using the time over the separatrix
Parameterization of the pendulum separatrix:
x0(v) = 4 arctan(ev ), y0(v) =2
cosh v.
Reparametrize x = x0(v), T u,s(v , τ) = Su,s(x0(v), τ)
Equation
ε−1∂τS +(∂xS)2
2+ cos x − 1 + ε(cos x − 1) sin τ = 0
becomes
ε−1∂τT +cosh2 v
8(∂v T )2 −
2
cosh2 v− ε
2
cosh2 vsin τ = 0
Asymptotic conditions
limRe v→−∞
cosh v ∂v T u (v , τ) = 0
limRe v→+∞
cosh v ∂vT s (v , τ) = 0.
M. Guardia (UPC) Lecture 3 February 8, 2017 9 / 26
New parameterizations
ε−1∂τT +cosh2 v
8(∂v T )2 −
2
cosh2 v− ε
2
cosh2 vsin τ = 0
Parameterizations of the form
x = 4 arctan(ev ), y =cosh v
2∂v T u,s (v , τ) .
For the unperturbed pendulum: ∂v T0(v) =4
cosh2 v.
We can consider a formal power series expansion
T u,s(v , τ) = T0(v) +∑
k≥1
εk Tu,sk (v , τ)
M. Guardia (UPC) Lecture 3 February 8, 2017 10 / 26
Formal power series
We have: T uk (v , τ) = T s
k (v , τ) ∀k ∈ N
Namely: T u(v , τ) − T s(v , τ) = O(εk ) ∀k ∈ N
Proceeding formally, their difference is beyond all orders.
What is happening?
1 The power series is convergent: both manifolds coincide in theperturbed case
2 The power series in ε is divergent: the manifolds do not coincide
and their difference is flat with respect ε.
We will see that is happening the second option
M. Guardia (UPC) Lecture 3 February 8, 2017 11 / 26
New parameterizations
ε−1∂τT +cosh2 v
8(∂v T )2 −
2
cosh2 v− ε
2
cosh2 vsin τ = 0
Unperturbed pendulum: ∂vT0(v) =4
cosh2 v.
Perturbed pendulum: T u,s = T0 + T u,s.
Parameterizations of the invariant manifolds:
x = 4 arctan(ev )
y =2
cosh v+
cosh v
2∂vT
u,s (v , τ)
To study the difference between manifolds: analyze T u − T s.
M. Guardia (UPC) Lecture 3 February 8, 2017 12 / 26
ε−1∂τT +cosh2 v
8(∂v T )2 −
2
cosh2 v− ε
2
cosh2 vsin τ = 0
Plug T u,s = T0 + T u,s into the Hamilton-Jacobi equation.
Recall ∂vT0(v) =4
cosh2 v.
New equation:
ε−1∂τT + ∂vT +cosh2 v
8(∂vT )2 − ε
2
cosh2 vsin τ = 0
T u,s are expected to be small.
We can solve this equation by inverting the linear differential
operator.
M. Guardia (UPC) Lecture 3 February 8, 2017 13 / 26
The integral operators
Inverses of L0 = ε−1∂τ + ∂v .
For functions which decay to zero as Re v −→ −∞,
Gu(h)(v , τ) =
∫ 0
−∞
h(v + t , τ + ε−1t)dt .
For functions which decay to zero as Re v −→ +∞,
Gs(h)(v , τ) =
∫ 0
+∞
h(v + t , τ + ε−1t)dt .
M. Guardia (UPC) Lecture 3 February 8, 2017 14 / 26
The fixed point equation
T u is a solution of
T u = Gu
(
−cosh2 v
8(∂vT
u)2 + ε2
cosh2 vsin τ
)
with
Gu(h)(v , τ) =
∫ 0
−∞
h(v + t , τ + ε−1t)dt .
Look for fixed points of
Fu(T ) = Gu
(
−cosh2 v
8(∂vT )2 + ε
2
cosh2 vsin τ
)
T s is a fixed point of
Fs(T ) = Gs
(
−cosh2 v
8(∂vT )2 + ε
2
cosh2 vsin τ
)
M. Guardia (UPC) Lecture 3 February 8, 2017 15 / 26
The fixed point argument in the reals
T u,s = Fu,s(T u,s) = Fu,s(0) +O(ε2).
Fu(0) = ε
∫ 0
−∞
2
cosh2(v + t)sin(
τ + ε−1t)
dt
Fs(0) = ε
∫ 0
+∞
2
cosh2(v + t)sin(
τ + ε−1t)
dt
The difference: T u − T s = Fu(0)−Fs(0) +O(ε2).
Difference between first iteration is the Melnikov potential
Fu(0)−Fs(0) = ε
∫ +∞
−∞
2
cosh2(v + t)sin(
τ + ε−1t)
dt .
We have recovered Melnikov theory through the Hamilton-Jacobi
equation.
M. Guardia (UPC) Lecture 3 February 8, 2017 16 / 26
First order (Melnikov potential):
Fu(0)−Fs(0) = 4πe−π2ε sin(τ − ε−1v)
Melnikov is exponentially small and the error is O(ε2).
We need better estimates for the error.
We consider the fixed point equation for complex values of v ∈ C.
To compute the Melnikov function, we have analyzed the
singularities of the Melnikov integrand at v = ±iπ
2.
We extend the parameterizations of the invariant manifolds up to a
distance O(ε) of these singularities.
M. Guardia (UPC) Lecture 3 February 8, 2017 17 / 26
Complex domains:
In τ : Tσ = {τ ∈ C/(2πZ) : |Im τ | ≤ σ}.
In v :
M. Guardia (UPC) Lecture 3 February 8, 2017 18 / 26
Close to the singularities of the unperturbed separatrix
By the fixed point argument T u,s are defined in these complex
domains and satisfy:
|T u,s(v , τ)| .ε2
(v ± iπ/2)2.
Unperturbed separatrix
T0(v) ∼1
v ± iπ/2
Both become larger close to the singularity.
T0 is always larger than T u,s (otherwise the fixed point argument
would not work!).
M. Guardia (UPC) Lecture 3 February 8, 2017 19 / 26
Difference between manifolds
Parameterizations of the invariant manifolds:
x = 4 arctan(ev )
y =2
cosh v+
cosh2 v
2∂vT
u (v , τ)
We study ∆ = T u − T s.
Subtract the equation
ε−1∂τT + ∂vT +cosh2 v
8(∂vT )2 − ε
2
cosh2 vsin τ = 0
for T u,s.
∆ satisfies L∆ = 0 for the linear operator
L = ε−1∂τ + ∂v +cosh2 v
8
(
∂vTu(v , τ) + ∂vT
s(v , τ))
∂v
L is close to L0 = ε−1∂τ + ∂v .
M. Guardia (UPC) Lecture 3 February 8, 2017 20 / 26
Difference between manifolds
Assume that ∆ ∈ KerL0, L0 = ε−1∂τ + ∂v .
Main idea: functions in KerL0 bounded in a complex strip are
exponentially small in the reals.
Proposition
Let ψ(v , τ) ∈ KerL0 be an analytic in τ function in [−ir , ir ] × Tσ. Then,
ψ can be extended analytically to {|Im v | ≤ r} × Tσ
Define M = max(v ,τ)∈[−ir ,ir ]×Tσ
|∂vψ(v , τ)|
Then, for ε≪ 1,
∀(v , τ) ∈ R× T, |∂vψ(v , τ)| ≤ 2Me−ε−1r
M. Guardia (UPC) Lecture 3 February 8, 2017 21 / 26
Why?
ε−1∂τψ + ∂vψ = 0 implies ψ(v , τ) = Λ(τ − ε−1v) for some Λ.
We can extend analytically ψ to {|Im v | ≤ r} × Tσ.
Let us take and example ψ(v , τ) = A sin(τ − ε−1v) for some
constant A > 0.
We have defined M = max(v ,τ)∈[−ir ,ir ]×Tσ
|∂vψ(v , τ)|.
Take (v , τ) = (ir ,0),
M ≥ Aε−1 cosh(ε−1r).
So A ≤ 2Mεe−rε .
For real values of (v , τ): |ψ(v , τ)| ≤ 2Mεe− rε .
M. Guardia (UPC) Lecture 3 February 8, 2017 22 / 26
Better estimates for the error
We apply this lemma to ∆− Melnikov = T u − T s − Melnikov:
(Approximately) ∆− Melnikov ∈ KerL0
The fixed point argument gives bounds for ∆− Melnikov for
(v , τ) ∈ [−ir , ir ]× Tσ with r =π
2− ε.
The bounds of ∆− Melnikov are smaller than the size of Melnikov
(in the complex).
We deduce exponentially small bounds for real values of the
variables.
M. Guardia (UPC) Lecture 3 February 8, 2017 23 / 26
Conclusion
Parameterizations
x = x0(v)
y = y0(v) + y−10 (v)∂vT
u
(
v ,t
ε
)
We obtain
∂v∆(v , τ) = ∂vTu(v , τ) − ∂vT
s(v , τ)
= −4πε−1e−π2ε
(
sin(τ − ε−1v) +O(ε))
.
We have a first order of the difference between the invariant
manifolds.
M. Guardia (UPC) Lecture 3 February 8, 2017 24 / 26
Splitting of separatrices
We have proved splitting of separatrices for
H
(
y , x ,t
ε
)
=y2
2+ (cos x − 1) + ε(cos x − 1) sin
t
ε
Typically in resonances of nearly integrable systems
H
(
y , x ,t
ε
)
=y2
2+ (cos x − 1) + δ(cos x − 1) sin
t
ε
for fixed δ ∈ R.
This case is much harder: Melnikov does not give the first order.
Analysis of the invariant manifolds close to the singularities gives
a new first order.
M. Guardia (UPC) Lecture 3 February 8, 2017 25 / 26
Back to the circular problem
One can apply the explained ideas to the circular problem
Then, we have an asymptotic formula for the distance between the
invariant manifolds for any µ ∈ (0,1/2) and G0 ≫ 1
∂vTu(v , ξ)−∂vT
s(v , ξ) = G32
0 e−G3
0
3
(
C(µ) sin(ξ + G30v) +O(G
− 12
0 )
)
where
C(µ) =1
2
√
π
2µ(1 − µ)(1 − 2µ).
This allows us to prove their transversality in this range of
parameters.
M. Guardia (UPC) Lecture 3 February 8, 2017 26 / 26