A Numerically Accessible Criterion for the Breakdown of Quasi-Periodic Solutions and Its Rigorous Justification

A Numerically Accessible Criterion for the Breakdown of Quasi-Periodic Solutions and Its Rigorous Justification
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  A numerically accessible criterion for the breakdown of quasi-periodic solutions and itsrigorous justification This article has been downloaded from IOPscience. Please scroll down to see the full text article.2010 Nonlinearity 23 2029(http://iopscience.iop.org/0951-7715/23/9/001)Download details:IP Address: article was downloaded on 10/08/2010 at 23:39Please note thatterms and conditions apply.Viewthe table of contents for this issue, or go to the journal homepagefor more HomeSearchCollectionsJournalsAboutContact usMy IOPscience  IOP P UBLISHING N ONLINEARITY Nonlinearity 23 (2010) 2029–2058doi:10.1088/0951-7715/23/9/001 A numerically accessible criterion for the breakdownof quasi-periodic 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 78712-0257, USA 2 Department of Mathematics and Statistics, McGill University, Montreal, PQ, CanadaE-mail: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 quasi-periodic solutions in sym-plecticmaps(anydimension) and1Dstatisticalmechanicsmodels. Dependingon the physical interpretation of the model, the analyticity breakdown may cor-respond 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 blow-up 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 quasi-periodic 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 quasi-periodic 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 quasi-periodic solutions exist is a problem (verysimilar to describing a phase diagram) to which considerable attention has been devoted. Seereferences later andappendix B. 0951-7715/10/092029+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 quasi-periodic 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 non-degeneracy 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 1-D 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 sub-steps.Wealsonotethatifalltheerrorsofthenumerics(truncation,round-off)couldbeestimatedand shown to be small enough with respect to the non-degeneracy assumptions, the KAMtheorems establish rigorously that the tori exist. This is the basis of many computer-assisted  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 quasi-periodic 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  Non-degeneracy conditions fail Return ‘inconclusive’ Until  Sobolev norm exceeds a threshold  The meta-theorem that guarantees the correctness of the method is Meta-theorem 2.2. Let  f  λ beafamilyofanalyticproblems,satisfyingappropriatehypothesis. Assume that  f  λ 0 has a Sobolev regular quasi-periodic solution u λ 0 that satisfies somenon-degeneracy 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 meta-theorem2.2.In this paper, we will present theorems that implement it for twist maps and for some models of statistical mechanics.Given the meta-theorem2.2,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 non-degeneracyconditions fail. In the case of twist mappings, a KAM torus may stop satisfying the twistcondition as we continue over parameters. • Ontheotherhand, ifthealgorithmhasprogressedandproducedatoruswithasmallerror,reasonablysmallnormandwhichsatisfiesthenon-degeneracyconditions,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 quasi-periodic 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 . Thesenon-connected 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 non-degeneracy 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 Meta-theorem [dlLR91]: Meta-theorem 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 .
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