Singularity Theory for Non-twist Tori (I) - Methodology

Transcription

Singularity Theory for Non-twist Tori (I)
Methodology
A. González A. Haro R. de la Llave
Universitat de Barcelona & University of Texas at Austin
Coherent Structures in DS
Lorentz Center 2011
A. Haro (UB)
Leiden 2011
1 / 38
Index
1
Setting of the problem
2
Motivating examples
3
Methodology
4
Bifurcation theory for non-twist tori
5
Numerical examples
6
Conclusions
A. Haro (UB)
Leiden 2011
2 / 38
A. Haro (UB)
Leiden 2011
3 / 38
KAM Theory for non-degenerate Hamiltonian systems
Existence and persistence of quasi-periodic motions in Hamiltonian systems
Twist conditions (' diffeomorphic frequency map):
Persistence of quasi-periodic motions
Kolmogorov, Arnold, Moser, ...
KAM results based on reducibility
Celletti-Chierchia, Salamon-Zehnder, González-de la
Llave-Jorba-Villanueva, ...
Rüssmann conditions (' weakly non-degenerate frequency map):
Persistence of Cantor families of invariant tori
Rüssmann, Sevryuk, Xu, You, Zhang, ...
A. Haro (UB)
Leiden 2011
4 / 38
KAM Theory for degenerate Hamiltonian systems
Existence and persistence of degenerate invariant tori with prescribed frequency
Existence and persistence of non-twist curves in 2-parameter
families of apm. Delshams-de la Llave (2002)
Normal forms for the fold and cusp singularities for 4D maps.
Dullin-Ivanov-Meiss (2006)
Existence of meandering curves in area preserving non-twist
maps (althought the curves are twist!). Simó (1998)
Persistence of tori under non-zero Brouwer degree condition (no
control on degeneracies!) Xu-You (2010)
Computation of normal forms for non-twist tori. Haro (2002)
A. Haro (UB)
Leiden 2011
5 / 38
Some problems for non-twist tori
This talk
Persistence of non-twist tori with prescribed Diophantine
frequency.
Bifurcations of non-twist tori with prescribed Diophantine
frequency.
Numerical algorithms to compute non-twist tori.
The geometrical framework of the theory will be considered in Alejandra
González’s talk.
A. Haro (UB)
Leiden 2011
6 / 38
Main contribution
A methodology
to study
bifurcations of KAM tori
with
fixed Diophantine frequency
for
Hamiltonian systems.
A. Haro (UB)
Leiden 2011
7 / 38
Motivating examples
Motivating examples
A. Haro (UB)
Leiden 2011
8 / 38
Motivating examples
Example I: A collision of invariant tori
A family of non-twist apm
Consider the quadratic standard family of apm in T × R:

2

 x̄ = 0.375 + x + ȳ ,

 ȳ
= y−
κ
sin(2πx) .
2π
Problem:
Study the bifurcations of invariant tori with frequency ω =
with respect to κ.
A. Haro (UB)
√
3− 5
,
2
Leiden 2011
9 / 38
Motivating examples
Example I: A collision of invariant tori
Numerical observations: a fold singularity
K= 1.000000
0.2
0.15
0.15
0.1
0.1
0.05
0.05
y
y
K= 0.000000
0.2
0
-0.05
0
-0.05
-0.1
-0.1
-0.15
-0.15
-0.2
0
0.2
0.4
x
0.6
0.8
-0.2
1
0
0.2
0.4
0.2
0.15
0.1
0.1
0.05
0.05
0
-0.05
0.8
1
0
-0.05
-0.1
-0.1
-0.15
-0.15
-0.2
0.6
K= 1.500000
0.2
0.15
y
y
K= 1.361408
x
0
0.2
0.4
A. Haro (UB)
x
0.6
0.8
1
-0.2
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Leiden 2011
10 / 38
Motivating examples
Example II: Integrable systems
Frequency map of an integrable system in Tn × Rn
Let
x + ∇W (y )
f0 (x, y ) =
y
be an integrable system, where W : Rn → R.
The frequency map is ω̂ = ∇W : Rn → Rn .
θ
n
∀p ∈ R , Kp (θ) =
is f0 -invariant with frequency ω̂(p):
p
f0 (Kp (θ)) = Kp (θ + ω̂(p)) .
Kp is twist iff the torsion
T̄ (p) = D ω̂(p) = Hess W (p)
is non-degenerate. Otherwise, Kp is non-twist.
A. Haro (UB)
Leiden 2011
11 / 38
Motivating examples
Potential for frequency ω of an integrable system in Tn × Rn
Consider the modified family (Moser, Herman):
x + ∇W (y )
λ
f0,λ (x, y ) =
+
.
y
0
∀p ∈ Rn , Kp (θ) =
θ
is f0,λ(p) -invariant with frequency ω:
p
f0,λ(p) (Kp (θ)) = Kp (θ + ω) ,
where λ(p) = ω − ∇W (p).
The potential A : Rn → R, defined by
A(p) = W (p) − p> ω ,
satisfies
λ(p) = −∇p A(p) , T̄ (p) = Hess A(p).
A. Haro (UB)
Leiden 2011
12 / 38
Motivating examples
Invariant tori as critical points of the potential
Conclusions:
1
2
Kp∗ is f0 -invariant with frequency ω if and only if p∗ is a
critical point of the potential A(p).
Kp∗ is a twist f0 -invariant torus with frequency ω if and only p∗
is a non-degenerate critical point of A(p).
3
Kp∗ is a non-twist f0 -invariant torus with frequency ω if and
only p∗ is a degenerate critical point of A(p).
A. Haro (UB)
Leiden 2011
13 / 38
Motivating examples
A degenerate case in T × R
Let fµ be family of integrable symplectomorphisms in T × R with:
frequency map ω̂(y ; µ) = ω + µ + y 2 .
potential (for ω): A(p; µ) = µp +
p3
.
3
Then (for fixed frequency ω):
K0 is a non-twist torus for f0 .
There are two invariant tori for µ < 0.
There are no invariant tori for µ > 0.
A. Haro (UB)
Leiden 2011
14 / 38
Motivating examples
A degenerate case in T × R
Observations:
f0 is Rüssmann non-degenerate, but invariant tori with prescribed
frequency ω can be destroyed.
Since ω lies in the boundary of the image of the frequency map of
f0 , Brouwer degree theory can not be applied.
µ is the parameter of unfolding of the singularity 0 for A(p; 0),
hence Singularity Theory can be applied.
A. Haro (UB)
Leiden 2011
15 / 38
Motivating examples
Natural questions
(From the motivating examples)
For a fixed Diophantine frequency ω:
1
Is the potential and its properties persistent under perturbations of
an integrable system?
2
Given a non-integrable system, are there sufficient conditions that
guarantee the existence of a suitable potential?
3
Can the potential be used to develop a bifurcation theory of
invariant tori?
A. Haro (UB)
Leiden 2011
16 / 38
Methodology
Methodology
A. Haro (UB)
Leiden 2011
17 / 38
Methodology
Methodology
Setting
Let Ω =
On −In
In On
be the standard symplectic matrix in Tn × Rn .
Let P ⊂ Rm be an open subset of parameters κ.
Let fκ : Tn × Rn → Tn × Rn a smooth family of real-analytic exact
symplectomorphisms, with primitive functions Sκ : Tn × Rn → R:
dSκ = fκ∗ (y dx) − y dx.
Let ω ∈ Rn be fixed and Diophantine:
>
τ
∀` ∈ Zn \ {0}, m ∈ Z.
` ω − m ≥ γ |`|−
1 ,
Goal: Finding fκ -invariant tori, with frequency ω.
A. Haro (UB)
Leiden 2011
18 / 38
Methodology
Methodology
Step 1: KAM theory for families of invariant tori
Find a family
(p; κ) ∈ U × P → K(p; κ) = (λ(p; κ), K(p;κ) )
such that
λ(p; κ)
fκ ◦ K(p;κ) (θ) +
= K(p;κ) (θ + ω) ,
0
hKp;κ (θ) − Kp (θ)i = 0 .
For each (p, κ), the torsion of K(p,κ) is the n × n symmetric matrix
D
E
T̄ (p; κ) = N(p;κ) (θ + ω)> Ω Dz fκ (K(p;κ) (θ)) N(p;κ) (θ) ,
−1
where N(p;κ) (θ) = J Dθ K(p;κ) (θ) Dθ K(p;κ) (θ)> Dθ K(p;κ) (θ)
.
A. Haro (UB)
Leiden 2011
19 / 38
Methodology
Methodology
Step 2: Symplectic Geometry of families of invariant tori
Define the potential of the family K
D
E
A(p; κ) = −p> λ(p; κ) − Sκ ◦ K(p;κ) (θ) .
Properties of the potential:
λ(p; κ) = −∇p A(p; κ).
T̄ (p; κ) W1 (p; κ) = W2 (p; κ) HessA(p; κ)
(for suitable matrices W1 (p; κ), W2 (p; κ)).
A. Haro (UB)
Leiden 2011
20 / 38
Methodology
Methodology
Step 3: Singularity Theory for critical points of the potential
For κ∗ fixed:
K(p∗ ;κ∗ ) is fκ∗ -invariant with frequency ω iff p∗ is a critical point of
the potential A(p; κ∗ ).
If W1 (p∗ ; κ∗ ) and W2 (p∗ ; κ∗ ) are invertible:
dim ker T̄ (p∗ , κ∗ ) = dim ker Hessp A(p∗ , κ∗ ).
A. Haro (UB)
Leiden 2011
21 / 38
Methodology
Applications
Close-to-integrable systems:
1
Persistence of invariant tori with fixed frequency under weak nondegeneracy conditions.
2
Persistence of invariant tori for perturbations of isochronous
systems.
3
Robust bifurcation theory for non-twist tori.
Numerical computations in far-to-integrable systems:
1
A fold singularity.
2
Birth of meandering.
A. Haro (UB)
Leiden 2011
22 / 38
A. Haro (UB)
Leiden 2011
23 / 38
A persistence result of non-twist tori
Hypotheses
Theorem (Persistence theorem of non-twist tori)
Given:
ω ∈ Rn , a Diophantine frequency vector.
f0 , a real-analytic integrable simplectic map with frequency map
ω̂(y ) = ω + ∇A(y ), s.t. y ∗ = 0 is a finitely determined singularity
of A(y ).
fµ , a µ-family of real-analytic integrable simplectic map with
frequency map ω̂µ (y ) = ω + ∇Aµ (y ), s.t. Aµ (y ) is a stable
unfolding of the singularity at zero.
fµ,ε , a perturbed family of exact symplectomorphims:
x + ω + ∇Aµ (y )
fµ,ε (x, y ) =
+ εg(x, y , ε).
y
A. Haro (UB)
Leiden 2011
24 / 38
A persistence result of non-twist tori
Theses
Theorem (Theses)
Then:
(KAM) For (µ, ε) sufficiently small, there exists a potential Aµ,ε (p)
defined around 0 ∈ Rn and an n-dimensional torus
Kp,µ,ε : Tn → Tn × Rn , depending smoothly on (p, µ, ε), such that:
∇p Aµ,ε (p)
fµ,ε ◦ Kp,µ,ε (θ) = Kp,µ,ε (θ + ω) +
.
0
(ST) For ε sufficiently small, there exist µ(ε) and p(ε) such that
Aµ(ε),ε has a singularity at p(ε) that is of the same type (in the
sense of Singularity Theory) of the singularity of A0 at zero.
A. Haro (UB)
Leiden 2011
25 / 38
Numerical examples
Numerical examples
A. Haro (UB)
Leiden 2011
26 / 38
Numerical examples
A fold singularity
Applying the methodology
Modified family:
f(κ,λ) (x, y ) =
!
2
0.375 + x + y − sin(2πx) + λ
.
κ
y − 2π
sin(2πx)
κ
2π
The primitive function of fκ,λ is
Sκ (x, y ) =
3
κ
κ
2 y
−
cos(2πx)
+
sin(2πx)
.
3
2π
4 π2
The co-rank of the torsion of a torus is either 0 (twist) or 1 (nontwist).
A. Haro (UB)
Leiden 2011
27 / 38
Numerical examples
A fold singularity
A collision of invariant tori
K= 0.000000
0.2
0.15
0.1
y
0.05
T = 0.1669253
T = −0.1669253
0
-0.05
-0.1
-0.15
-0.2
0
0.2
0.4
0.6
x
0.8
1
0.01
0.0004
0.0002
A
<h
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
A. Haro (UB)
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Leiden 2011
28 / 38
Numerical examples
A fold singularity
K= 1.000000
0.2
0.15
0.1
y
0.05
T = 0.1123464
T = −0.1123464
0
-0.05
-0.1
-0.15
-0.2
0
0.2
0.4
0.6
x
0.8
1
0.01
0.0004
0.0002
A
<h
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
A. Haro (UB)
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Leiden 2011
29 / 38
Numerical examples
A fold singularity
K= 1.361408
0.2
0.15
0.1
T = 0.6558 · 10−8
T = −0.6558 · 10−8
y
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.2
0.4
0.6
x
0.8
1
0.01
0.0004
0.0002
A
<h
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
A. Haro (UB)
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Leiden 2011
30 / 38
Numerical examples
A fold singularity
K= 1.500000
0.2
0.15
0.1
y
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
x
0.7
0.8
0.9
1
0.01
0.0004
0.0002
A
<h
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
A. Haro (UB)
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Leiden 2011
31 / 38
Numerical examples
The birth of meandering tori
A numerical example
Consider a quadratic standard family of symplectomorphisms
fκ : T × R → T × R, defined by:


 x̄ = x + (ȳ + 0.1)(ȳ − 0.2) ,

 ȳ
= y−
κ
sin(2πx) .
2π
√
Problem: Look for invariant tori with frequency ω =
to parameter κ.
A. Haro (UB)
5−1
32 ,
with respect
Leiden 2011
32 / 38
Numerical examples
A fold singularity
K= 0.420000
0.5
0.4
0.3
0.2
y
0.1
0
-0.1
-0.2
-0.3
-0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
x
0.7
0.8
0.9
1
-0.0016
0.00
-0.0018
A
<h
0.01
-0.01
-0.02
-0.0020
0
0.02
A. Haro (UB)
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Leiden 2011
33 / 38
Numerical examples
A fold singularity
K= 0.430396
0.5
0.4
0.3
0.2
T = −1.6261 · 10−7
T = 1.6261 · 10−7
y
0.1
0
-0.1
-0.2
-0.3
-0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
x
0.7
0.8
0.9
1
-0.0016
0.00
-0.0018
A
<h
0.01
-0.01
-0.02
-0.0020
0
0.02
A. Haro (UB)
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Leiden 2011
34 / 38
Numerical examples
A fold singularity
K= 0.440000
0.5
0.4
0.3
0.2
T = −0.0211805
T = 0.0211805
y
0.1
0
-0.1
-0.2
-0.3
-0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
x
0.7
0.8
0.9
1
-0.0016
0.00
-0.0018
A
<h
0.01
-0.01
-0.02
-0.0020
0
0.02
A. Haro (UB)
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Leiden 2011
35 / 38
Numerical examples
A fold singularity
K= 0.450000
0.5
0.4
0.3
0.2
T = −0.0309003
T = 0.0309003
y
0.1
0
-0.1
-0.2
-0.3
-0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
x
0.7
0.8
0.9
1
-0.0016
0.00
-0.0018
A
<h
0.01
-0.01
-0.02
-0.0020
0
0.02
A. Haro (UB)
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Leiden 2011
36 / 38
Conclusions
Conclusions
A. Haro (UB)
Leiden 2011
37 / 38
Conclusions
Conclusions
Main contributions
The infinite dimensional problem of finding invariant tori for
Hamiltonian systems is reduced to the finite dimensional problem
of finding critical points of the potential.
Singularity Theory for invariant tori
Non-twist tori correspond to degenerate critical points.
Applicable to any finite-determined singularity of the frequency
map.
Efficient numerical algorithms to compute and validate invariant
tori and its bifurcations.
Persistence results for small twist systems.
Applicable to perturbation of isochronous systems.
A. Haro (UB)
Leiden 2011
38 / 38

Singularity Theory for Non-twist Tori (I) - Methodology

Transcription

Similar documents

Singularity Theory for Non-twist Tori - MAiA