Singularity Theory for Non-twist Tori - MAiA

Transcription

Singularity Theory for Non-twist Tori
A. González
A. Haro
R. de la Llave
Universitat de Barcelona & University of Texas at Austin
1st Swedish-Catalan Conference, Barcelona 2010
Alex Haro (Univ. Barcelona)
Barcelona 2010
1 / 28
Index
1
Outline
2
Motivating example 1: A collision of invariant tori
3
Motivating example 2: Integrable symplectomorphisms
4
Methodology
5
Example 1: A collision of invariant tori (cont.)
6
Example 2: The birth of meandering tori
7
Conclusions
Barcelona 2010
2 / 28
Outline
KAM Theory
Existence and persistence of quasi-periodic motions in Hamiltonian systems
Under twist conditions (' diffeomorphic frequency map):
Persistence of quasi-periodic motions (Kolmogorov, Arnold,
Moser...)
A posteriori results of existence of invariant tori with fixed
frequency, based on reducibility (de la Llave, González,
Jorba, Villanueva)
Existence of Cantor families of tori under weaker conditions:
Unfoldings of quasi-periodic tori (Broer, Huitema, Moser,
Pöschel, Sevryuk, Takens, ...)
Persistence of invariant tori under Rüssmann condition
(Sevryuk, Rüssmann, Xu, You, Zhang, ...)
Barcelona 2010
3 / 28
Outline
Non-twist tori with fixed frequency
Previous results
Numerical results in non-twist area preserving maps:
del Castillo-Negrete, Greene, Morrison (1996, 1997):
periodic orbits and transition to chaos (Apte, Wurm, Petrisor,
Shinohara, Aizawa, ...)
Haro (2002): computation of non-twist tori and normal forms.
Rigorous results:
Simó (1998): existence of meandering curves in area
preserving non-twist maps (but the curves are twist!).
Delshams, de la Llave (2002): existence and persistence of
non-twist curves in 2-parameter families of apm.
Dullin, Ivanov, Meiss (2006): normal forms for the fold and
cusp singularities for 4D maps.
Barcelona 2010
4 / 28
Outline
Non-twist tori with fixed frequency
This talk
Singularity Theory for non-twist tori
(arbitrary dimension and any finite-determined singularity)
Bifurcations of invariant tori, with fixed frequency.
At the bifurcation points, the invariant tori are non-twist.
Classification of the possible degeneracies of invariant tori.
Unfolding Hamiltonian systems around non-twist tori.
Persistence of non-twist invariant tori.
Efficient numerical methods to study bifurcations.
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Motivating example 1
A family of quadratic standard maps
Consider the quadratic standard family of symplectomorphisms
fκ : T × R → T × R, defined by:

2

 x̄ = 0.375 + x + ȳ ,

 ȳ = y − κ sin(2πx) .
2π
Problem: Look for invariant tori with frequency ω =
respect to parameter κ.
√
3− 5
,
2
with
Barcelona 2010
6 / 28
A collision of invariant tori
Numerical observations
K= 1.000000
0.2
0.15
0.1
0.1
0.05
0.05
y
y
K= 0.000000
0.2
0.15
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
0.15
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
y
y
K= 1.361408
x
0
0.2
0.4
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
Barcelona 2010
7 / 28
Motivating example 2
Frequency map of an integrable symplectomorphism
Consider the integrable system f0 : Tn × Rn → Tn × Rn ,
x + ∇W (y )
f0 (x, y ) =
,
y
where W : Rn → R.
Observations:
θ
For p ∈ Rn , the torus Kp (θ) =
is invariant for f0 with
p
frequency ω̂(p) = ∇W (p):
f0 (Kp (θ)) = Kp (θ + ω̂(p)) .
Kp is twist if Dω̂(p) = Hess W (p) is non-degenerate.
Barcelona 2010
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Integrable symplectomorphisms
Invariant tori with frequency ω
Problem: Look for invariant tori with frequency ω.
Define the counterterm λ(p) = ∇W (p) − ω.
Define the modified family
λ(p)
fλ(p) (x, y ) = f0 (x, y ) −
.
0
θ
For p ∈ R , the torus Kp (θ) =
is invariant for fλ(p) with
p
frequency ω:
λ(p)
f0 (Kp (θ)) −
= Kp (θ + ω) .
0
n
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Invariant tori as critical points of the potential A
Define the potential A(p) = W (p) − p> ω.
Obviously: λ(p) = ∇p A(p).
Then:
Kp∗ is an f0 -invariant torus 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 the potential A(p).
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The simplest degenerate example
Assume A(p) =
p3
3
(ω̂(y ) = ω + y 2 )
p = 0 is a degenerate critical point of A(p).
θ
The torus K0 (θ) =
with frequency ω is non-twist.
0
Unfolding: A(p; µ) = µp +
p3
3
Counterterm: λ(p; µ) = µ + p2 .
(ω̂(y ; µ) = ω + µ + y 2 )
λ(p; µ)
µ>0
µ=0
p
µ<0
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Methodology
Methodology
Setting
Tn × U is an annulus, where U ⊂ Rn is open simply
connected.
P ⊂ Rm is an open subset of parameters.
fκ : Tn × U → Tn × Rn is a smooth family of exact
symplectomorphisms, with κ ∈ P:
Dfκ (x, y )> J Dfκ (x, y ) = J, where J =
On −In
;
In On
There exists a primitive function Sκ : Tn × U → R, such that
dSκ = fκ∗ (y dx) − y dx.
Let ω ∈ Rn be fixed and Diophantine:
>
` ω − m ≥ γ |`|− τ ,
∀` ∈ Zn \ {0}, m ∈ Z.
1
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Methodology
Methodology
Modified family and invariant Lagrangian deformations
1) Consider the modified family
λ
f(κ,λ) (x, y ) = fκ (x, y ) −
,
0
where λ ∈ Rn is the counterterm.
2) Use KAM techniques to find a family
(p; κ) ∈ U × P → (λ(p; κ), K(p;κ) )
such that:
f(κ,λ(p;κ)) ◦ K(p;κ) (θ) = K(p;κ) (θ + ω) ,
θ
Kp;κ (θ) −
= 0.
p
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Methodology
Methodology
Potential and countertem
3) Define the potential
D
E
A(p; κ) = p> λ(p; κ) + Sκ ◦ K(p;κ) (θ) ,
where Sκ is the primitive function of fκ .
The following holds:
λ(p; κ) = ∇p A(p; κ)
Hence, fixed κ = κ∗ :
K(p∗ ;κ∗ ) is fκ∗ -invariant with frequency ω if and only if p∗ is a
critical point of the potential A(p; κ∗ ).
Barcelona 2010
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Methodology
Methodology
Potential and torsion
4) The torsion of K(p;κ) is the n × n symmetric matrix
D
E
>
T̄ (p; κ) = N(p;κ) (θ + ω) J Dz f(κ,λ(p;κ)) (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)
.
There exist n × n matrices W1 (p; κ) and W2 (p; κ) such that:
T̄ (p; κ) W1 (p; κ) = W2 (p; κ) HessA(p; κ) .
Hence, if W1 (p∗ ; κ∗ ) and W2 (p∗ ; κ∗ ) are invertible:
The co-rank of p∗ as a critical point of A(p; κ∗ ) equals the
co-rank of the torsion T̄ (p∗ ; κ∗ ).
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Example 1
Applying the methodology
Modified family:
2
κ
0.375 + x + y − 2π
sin(2πx) − λ
f(κ,λ) (x, y ) =
.
κ
sin(2πx)
y − 2π
The primitive function of fκ is
Sκ (x, y ) =
3
κ
2 κ
y
−
cos(2πx)
+
sin(2πx)
.
4 π2
3
2π
The co-rank of the torsion of a torus is either 0 (twist) or 1
(non-twist).
Barcelona 2010
16 / 28
A fold singularity
K= 0.000000
0.2
0.15
0.1
T = 0.1669253
T = −0.1669253
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
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Barcelona 2010
17 / 28
A fold singularity
K= 1.000000
0.2
0.15
0.1
T = 0.1123464
T = −0.1123464
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
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Barcelona 2010
18 / 28
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
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Barcelona 2010
19 / 28
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
0.005
0
0
-0.0002
-0.005
-0.0004
-0.1
-0.05
0
p
0.05
0.1
-0.1
-0.05
0
p
0.05
0.1
Barcelona 2010
20 / 28
Example 2
Another family of quadratic standard maps
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 =
respect to parameter κ.
5−1
,
32
with
Barcelona 2010
21 / 28
A fold singularity
The birth of meandering tori
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
0.01
-0.01
-0.02
-0.0020
0
0.02
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Barcelona 2010
22 / 28
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
0.01
-0.01
-0.02
-0.0020
0
0.02
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Barcelona 2010
23 / 28
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
0.01
-0.01
-0.02
-0.0020
0
0.02
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Barcelona 2010
24 / 28
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
0.01
-0.01
-0.02
-0.0020
0
0.02
0.04
p
0.06
0.08
0.1
-0.0022
0
0.02
0.04
p
0.06
0.08
0.1
Barcelona 2010
25 / 28
Conclusions
Conclusions
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.
Non-twist tori correspond to degenerate critical points of the
potential.
The Singularity Theory for invariant tori is provided by the
Singularity Theory of the critical points of the potential.
Barcelona 2010
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Conclusions
Conclusions
The method yields very efficient numerical algorithms to
compute invariant tori and its bifurcations.
The method is suitable to validate numerical computations;
We are able to study any finite-determined singularity of the
frequency map.
The method can be also applied for obtaining small twist
theorems.
Our method is designed for the cases on which numerical
evidence of bifurcations is known but the system is not
close to integrable.
Barcelona 2010
27 / 28
Thanks!
Barcelona 2010
28 / 28

Singularity Theory for Non-twist Tori - MAiA

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