A numerically accessible criterion for the breakdown of quasiperiodic solutions and itsrigorous justification
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IOP P
UBLISHING
N
ONLINEARITY
Nonlinearity
23
(2010) 2029–2058doi:10.1088/09517715/23/9/001
A numerically accessible criterion for the breakdownof quasiperiodic solutions and its rigorous justiﬁcation
Renato Calleja
1
,
2
and Rafael de la Llave
1
1
Department of Mathematics, 1 University Station C1200, Austin TX 787120257, USA
2
Department of Mathematics and Statistics, McGill University, Montreal, PQ, CanadaEmail:rcalleja@math.utexas.eduandllave@math.utexas.edu
Received 31 August 2009, in ﬁnal form 24 May 2010Published 29 July 2010Online atstacks.iop.org/Non/23/2029Recommended by K M Khanin
Abstract
We formulate and justify rigorously a numerically efﬁcient criterion for thecomputation of the analyticity breakdown of quasiperiodic solutions in symplecticmaps(anydimension) and1Dstatisticalmechanicsmodels. Dependingon the physical interpretation of the model, the analyticity breakdown may correspond to the onset of mobility of dislocations, or of spin waves (in the 1Dmodels) and to the onset of global transport in symplectic twist maps in 2D.The criterion proposed here is based on the blowup of Sobolev norms of thehullfunctions. Weprovetheoremsthatjustifythecriterion. Thesetheoremsarebasedonanabstractimplicitfunctiontheorem, whichuniﬁesseveralresultsin the literature. The proofs also lead to fast algorithms, which have beenimplementedandusedelsewhere. Themethodcanbeadaptedtoothercontexts.Mathematics Subject Classiﬁcation: 37A60, 70K43, 37M20, 47J08
1. Introduction
The celebrated Kolmogorov–Arnold–Moser (KAM) theory establishes the persistence underperturbation of analytic quasiperiodic solutions in a variety of mathematical contexts whichmodel many different physical systems (e.g. celestial mechanics, solid state physics, etc).Depending on the physical system, the presence or not of these quasiperiodic analyticsolutions has deep physical consequences. In celestial mechanics, it means an abundance of stable orbits; in solid state physics it could mean the presence of extended states and therefore,possibility of transport.Aswellknown,theexistenceornotofthesesolutionsdependsonthevaluesofparameters.Locating the range of parameters where these quasiperiodic solutions exist is a problem (verysimilar to describing a phase diagram) to which considerable attention has been devoted. Seereferences later andappendix B.
09517715/10/092029+30$30.00 © 2010 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA
2029
2030 R Calleja and R de la Llave
The goal of this paper is to present a numerically practical method to compute the valuesof parameters where the KAM theory breaks down which admits a rigorous mathematical justiﬁcation. A more precise description of the method is in section2.We will consider families of problems indexed by a parameter
λ
and seek to identify theset of parameters for which there is an analytic quasiperiodic solution of a ﬁxed Diophantinefrequency
ω
.Roughly speaking the criterion we present (see section2and theorem2.2for a general
overview and theorems5.3and6.1for a justiﬁcation for twist mappings (in any dimension)
and models in 1
−
D
statistical mechanics, respectively) says that (provided some nondegeneracy conditions that can be readily checked) a parameter value
λ
0
is on the boundaryof the parameters with KAM tori of a ﬁxed frequency if and only if, when we consider aSobolev norm of high enough order of the KAM tori for nearby parameters
λ
, it blows up as
λ
approaches
λ
0
.WeanticipatethatthejustiﬁcationisbasedonseveralKAMtheorems:(1)an
‘aposteriori’
KAM theorem for Sobolev regularity with some special features that shows that a computedapproximation with large Sobolev norm implies the existence of a true solution with largeSobolev norm. (2) A bootstrap of regularity theorem that shows that Sobolev solutions of ahigh enough index are analytic.Forourpurposes,itisimportantthattheKAMtheoremswepresentarenotnearintegrable,canbeappliedarbitrarilyclosetobreakdown(weareapproximatingtheboundaryfrominside)andincludeuniqueness. Thisisnotusuallydoneintheliterature. Ontheotherhand,thechoiceof Sobolev norms is not crucial.It is also important that, from the existence of a numerically computed solution of largenorm, we can conclude that this is the only solution which has large norm. This requiresuniqueness. For example, the proofs based in transformation theory (e.g. [Zeh75]does not
have this feature, there are many transformations that give the same torus and one can choosetransformations that blow up even if the torus remains).Of course, for each different context, the proof of the KAM theorems (1) and (2) aboveis very different and requires very different arguments. In this paper we present proofs in twodifferentcontexts:symplectictwistmaps—thatis,mapssuchthatthefrequencydependsontheactions in an invertible way—(see section5) and equilibrium models in statistical mechanicsof 1D systems (see section6).
The different contexts (with very different physical interpretations) have a signiﬁcantoverlap (e.g. Frenkel–Kontorova models are equivalent to twist mappings). Nevertheless,there are models which are in one context but not in the other (see the discussionin [dlL08]). Similar theorems with very different proofs have already been developed
in other contexts such as dissipative systems [CC10,CCdlL09]and for whiskered
tori [FdlLS09,HdlLS09].
The proofs of the KAM theorems presented here also lead to efﬁcient algorithms (seealgorithms6.4and5.5)if one chooses appropriate discretizations and efﬁcient algorithms for
the substeps.Wealsonotethatifalltheerrorsofthenumerics(truncation,roundoff)couldbeestimatedand shown to be small enough with respect to the nondegeneracy assumptions, the KAMtheorems establish rigorously that the tori exist. This is the basis of many
computerassisted
proofs.To make it clear that the method is very general, inappendix A,we present an abstract
Nash–Moser theorem which can be used to prove both theorems (1) and (2). We hope thatthis will help us to show the deep unity and the generality of the methods. The proof of theabstract theorem reduces the proofs of our cases to just a few paragraphs.
Criterion for analyticity breakdown 2031
Inappendix B,weprovideacomparisonamongallthemethodsknowntoustocomputethe
breakdownofinvariantcirclesfortwistmappings. Evenifthissurveyisnecessarilyincomplete,fromthepublicationof[CdlL09b]wereceivedmanyrequestsforsuchacomparison(including
the referees and editors of [CdlL09b]).
2. Criterion for the breakdown
The criterion for the breakdown of quasiperiodic solutions we propose is summarized in thefollowing algorithm. We present a continuation algorithm which relies on an iterative step(e.g. a Newton method) that ﬁnds the solution once one starts close to a solution. We stopwhen the solutions have large Sobolev norm.
Algorithm 2.1.
Choose a path in the parameter space starting in the integrable case.
Initialize
The parameters and the solution at the integrable case
Repeat
Increase the parameters along the path Run the iterative step
If
(Iterations of the Newton step do not converge) Decrease the increment in parameters
Else
(Iteration success) Record the values of the parametersand the Sobolev norm of the solution.
If
Nondegeneracy conditions fail Return ‘inconclusive’
Until
Sobolev norm exceeds a threshold
The metatheorem that guarantees the correctness of the method is
Metatheorem 2.2.
Let
f
λ
beafamilyofanalyticproblems,satisfyingappropriatehypothesis. Assume that
f
λ
0
has a Sobolev regular quasiperiodic solution
u
λ
0
that satisﬁes somenondegeneracy assumptions.Then if

λ
−
λ
0

is small enough depending on the size of the Sobolev norm of
u
λ
0
, then
f
λ
has an analytic solution, which is locally unique.
Ofcourse,dependingontheexactcontextoftheproblem,onehastoproveactualtheoremsthat implement the metatheorem2.2.In this paper, we will present theorems that implement
it for twist maps and for some models of statistical mechanics.Given the metatheorem2.2,it is clear that algorithm2.1will continue progressing until
it gets as close to breakdown as allowed by the resources of the computer.
2.1. Some comments on algorithm2.1
•
It is important that one identiﬁes the exact conditions for the validity of algorithms. Asany numerical algorithm, algorithm2.1returns inconclusive results if the nondegeneracyconditions fail. In the case of twist mappings, a KAM torus may stop satisfying the twistcondition as we continue over parameters.
•
Ontheotherhand, ifthealgorithmhasprogressedandproducedatoruswithasmallerror,reasonablysmallnormandwhichsatisﬁesthenondegeneracyconditions,the
aposteriori
2032 R Calleja and R de la Llave
theorem3.1will prove that such an invariant torus exists, so that if one implements thealgorithm bounding the error in the computation, one obtains rigorous existence of tori.
•
Of course, the rigorous theorem2.2only concludes the fact that the KAM torus cannot becontinuedfromthefactthatanormtends toinﬁnity. Sincethetheoremhas an
aposteriori
format and there is some uniqueness, we can conclude that the only solution has largenorm. From the rigorous point of view the present method gives a sequence of lowerbounds of the breakdown which are guaranteed to converge to the right one.
•
In practice, one can make the results more convincing by observing that the norms blowup according to a power law renormalization group predicts that there is a power lawblow up for each Sobolev norm and that there is a simple relation between the scalingexponents corresponding to Sobolev norm [dlL92,CdlL09b]. These empirically found
(in some cases) scaling relations are consistent with a renormalization group descriptionof the breakdown.
•
algorithm2.1is a continuation algorithm and shares all the shortcomings common tocontinuation algorithms. One can compute the connected component of the set of parameters for which there is an analytic quasiperiodic solution.Indeed, [CdlL09a]presents numerical evidence that the twist maps
T
ε
(p,q)
=
(p
+
ε
[
α
cos
(
2
πx)
+
β
cos
(
4
πx)
]
,q
+
p
+
ε
[
α
cos
(
2
πx)
+
β
cos
(
4
πx)
]
)
(1)the set of parameters with KAM tori has several components when
α
=
1
,β
√
2
−
14
. Thesenonconnected regions were computed using appropriate paths in the two parameter family(1). We think that it is reasonable to conjecture that the set of analytic twist maps with an
analytic KAM torus of frequency
ω
, with
ω
Diophantine, is connected.
2.2. Verifying theorem2.2
The veriﬁcation of theorem2.2can be reduced to two theorems.(1) Showing that given a Sobolev solution, all nearby maps have a locally unique Sobolevsolution.(2) All maps with a Sobolev solution have an analytic solution.Note that, in contrast with many formulations of KAM theory, we are not allowed toconsider only maps which are close to integrable and we have to obtain uniqueness.The key step is a theorem that shows that given an approximate solution (either in analyticsense or in a Sobolev sense) that satisﬁes appropriate nondegeneracy conditions, there is alocallyuniquesolutioninthesamespaceswhichisclosetothecomputedone. Thesetheoremscalled
‘a posteriori’
will be discussed in section3.
3.
A posteriori
KAM estimates
The prototype of an
a posteriori
result is the following Metatheorem [dlLR91]:
Metatheorem 3.1.
Let
X
0
⊂
X
1
be Banach spaces and
U
⊂
X
1
an open set. For certainmaps,
F
:
U
⊂
X
1
→
X
0
,
there exists an explicit function
ε
∗
:
R
+
×
(
R
+
)
n
→
R
+
such that
lim
t
→∞
ε
∗
(t,
·
)
=
0
, explicit functionals
f
1
,...,f
n
:
X
0
→
R
+
and
M
0
,M
1
,...,M
n
∈
R
+
satisfying the following property. Suppose
x
0
∈
X
0
with
x
0
X
0
M
0
, and that
f
1
(x
0
)
M
1
,...,f
n
(x
0
)
M
n
,
F
(x
0
)
X
0
< ε
∗
(M
0
,...,M
n
)
.Then there exists an
x
∗
∈
X
1
such that
F
(x
∗
)
=
0
and
x
0
−
x
∗
X
1
C
M
0
,...,M
n
F
(x
0
)
X
0
.