melnikov theory to all orders and puiseux series for subharmonic solutions we study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time in the case in which we only assume that the subharmonic melnikov function has at least one zero if the order of zero is odd then there is always at least one subharmonic solution whereas if the order is even in general other conditions have to be assumed to guarantee the existence of subharmonic solutions even when such solutions exist in general they are not analytic in the perturbation parameter we show that they are analytic in a fractional power of the perturbation parameter to obtain a fully constructive algorithm which allows us not only to prove existence but also to obtain bounds on the radius of analyticity and to approximate the solutions within any fixed accuracy we need further assumptions the method we use to construct the solution when this is possible is based on a combination of the newton puiseux algorithm and the tree formalism this leads to a graphical representation of the solution in terms of diagrams finally if the subharmonic melnikov function is identically zero we show that it is possible to introduce higher order generalizations for which the same kind of analysis can be carried out

stability for quasi periodically perturbed hill s equations we consider a perturbed hill s equation of the form open image in new window p0 t p1 t 0 where p0 is real analytic and periodic p1 is real analytic and quasi periodic and is small assuming diophantine conditions on the frequencies of the decoupled system i e the frequencies of the external potentials p0 and p1 and the proper frequency of the unperturbed 0 hill s equation but without making any assumptions on the perturbing potential p1 other than analyticity we prove that quasi periodic solutions of the unperturbed equation can be continued into quasi periodic solutions if lies in a cantor set of relatively large measure in open image in new window where 0 is small enough our method is based on a resummation procedure of a formal lindstedt series obtained as a solution of a generalized riccati equation associated to hill s problem

positive lyapunov exponents for schrodinger operators with quasi periodic potentials we present a new simple way to estimate the rate of exponential growth lyapunov exponent of solutions of the finite difference schrodinger equation h e psi n mathop limits def psi n 1 psi n 1 lambda f alpha n theta psi n heref is a non constant real analytic function of period 1 and is irrational for large we prove that the lyapunov exponent is positive for every energye in the spectrum ofh and a e in particular the absolutely continuous spectrum ofh is empty in the continuum we study the quasi periodic operator onl2 r h frac d 2 dx 2 k 2 cos x cos alpha x theta for largek and show that for wide intervals of low energies the lyapunov exponent is positive the main idea which originated from m herman s subharmonic argument 11 is to deform the phase to the complex plane this enables us to avoid small denominator problems by moving them off the axis making estimates much easier to perform we recover the information for real using an elementary extension of jensen s formula subharmonicity

an optimization based framework for anisotropic hp adaptation of high order discretizations abstract this paper presents a novel framework for hp adaptation of high order discontinuous finite element discretizations for compressible flow simulation using the sensitivities of an adjoint based error estimate our method seeks element size shape and polynomial degree distributions which minimize an adjoint based error estimate for a specified number of degrees of freedom this approach results in an optimized hp mesh tailored to yield the most accurate prediction of an output quantity of interest such as aerodynamic coefficients at a given computational cost the proposed approach features a reduced dependence on user defined parameters compared to established fixed fraction adjoint based adaptive methods it provides a unifying framework where adaptation decisions such as isotropic anisotropic h p refinement coarsening do not only rely on local arbitrary measures of solution anisotropy and smoothness but rather where a globally optimal distribution of degrees of freedom is sought to minimize the error in the chosen quantity of interest

quasi periodic solutions for two level systems we consider the schrodinger equation for a class of two level atoms in a quasi periodic external field in the case in which the spacing 2 between the two unperturbed energy levels is small and we study the problem of finding quasi periodic solutions of a related generalized riccati equation we prove the existence of quasi periodic solutions of the latter equation for a cantor set of values of around the origin which is of positive lebesgue measure such solutions can be obtained from the formal power series by a suitable resummation procedure the set can be characterized by requesting infinitely many diophantine conditions of mel nikov type

existence of quasi periodic solutions of fast excited van der pol mathieu duffing equation the van der pol mathieu duffing equation x 02 h1cos 1t h2cos 2t x x2 x h3x3 h4 32cosxcos 3t is considered in this paper where h1 h2 h3 h4 1 2 are small parameters 0 the frequency 3 is large compared to 1 and 2 the above parameters are real for 0 we use kam kolmogorov arnold moser theory to prove that the van der pol mathieu duffing equation possesses quasi periodic solutions for most of the parameters 0 1 2 3 it verifies some phenomenon of fahsi and belhaq commun nonlinear sci 14 244 253 2009 and can be regarded as a extension of abouhazim et al nonlinear dyn 39 395 409 2005

the distributional borel summability and the large coupling φ4 lattice fields we complete and correct some proofs of an earlier paper on distributional borel summability and we add an application which can be useful in the discussion of semiclassical problems

renormalization group and the melnikov problem for pde s we give a new proof of persistence of quasi periodic low dimensional elliptic tori in infinite dimensional systems the proof is based on a renormalization group iteration that was developed recently in bgk to address the standard kam problem namely persistence of invariant tori of maximal dimension in finite dimensional near integrable systems our result covers situations in which the so called normal frequencies are multiple in particular it provides a new proof of the existence of small amplitude quasi periodic solutions of nonlinear wave equations with periodic boundary conditions

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

reconstructing a three dimensional model with arbitrary errors a number of current technologies allow for the determination of interatomic distance information in structures such as proteins and rna thus the reconstruction of a three dimensional set of points using information about its interpoint distances has become a task of basic importance in determining molecular structure the distance measurements one obtains from techniques such as nmr are typically sparse and error prone greatly complicating the reconstruction task many of these errors result in distance measurements that can be safely assumed to lie within certain fixed tolerances but a number of sources of systematic error in these experiments lead to inaccuracies in the data that are very hard to quantify in effect one must treat certain entries of the measured distance matrix as being arbitrarily corrupted the existence of arbitrary errors leads to an interesting sort of error correction problem how many corrupted entries in a distance matrix can be efficiently corrected to produce a consistent three dimensional structure for the case of an n n matrix in which every entry is specified we provide a randomized algorithm running in time o n log n that enumerates all structures consistent with at most 1 2 e n errors per row with high probability in the case of randomly located errors we can correct errors of the same density in a sparse matrix one in which only a fraction of the entries in each row are given for any constant gt 0

adjoint recovery of superconvergent functionals from pde approximations motivated by applications in computational fluid dynamics a method is presented for obtaining estimates of integral functionals such as lift or drag that have twice the order of accuracy of the computed flow solution on which they are based this is achieved through error analysis that uses an adjoint pde to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy numerical results are presented for the poisson equation in one and two dimensions and for the nonlinear quasi one dimensional euler equations the theory is equally applicable to nonlinear equations in complex multi dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

elliptic type degenerate invariant tori for quasi periodically forced four dimensional non conservative systems abstract in this paper we will focus on the persistence of elliptic type degenerate invariant tori with the same frequency vector r d as the forcing in a class of quasi periodically forced four dimensional non conservative systems it is shown that under suitable hypothesis of smoothness with respect to and the brjuno russmann s non resonant condition instead of ordinary diophantine condition with respect to the frequency vector r d the invariant tori persist under small perturbations for most of the sufficiently small parameters in the sense of the lebesgue measure hence the system has a quasi periodic solution with the same frequency vector as the forcing the proof is based on the poschel russmann kam method which is a kam type method in which one uses the polynomial structure of function to truncate introduce a parameter q and make the steps of kam iteration infinitely small in the speed of function q n 0 q 1 rather than super exponential function like traditional kam method

lower dimensional invariant tori for quasiperiodically forced circle diffeomorphisms abstract for any analytic quasiperiodically forced circle diffeomorphisms p q e f where f is fixed and e is small we show that if is diophantine and the fibred rotation number of the diffeomorphism remains constant in a unilateral neighborhood of e 0 i e there is a unilateral phase locking at e 0 then the diffeomorphism has at least one analytic q invariant torus provided e is small enough

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

lower dimensional invariant tori in the regions of instability for nearly integrable hamiltonian systems consider a hamiltonian system of kam type h p q n p p p q with n degrees of freedom n 2 where the hessian of n is nondegenerate for one resonance condition 0 i n there is an immersed n 1 dimensional submanifold in action variable space where almost every point corresponds to a resonant torus for the unperturbed system which is foliated by n 1 dimensional ergodic components it is shown in this paper that there is a subset of with positive n 1 dim lebesgue measure such that for each resonant torus corresponding to a point in this set at least two n 1 dimensional tori can survive perturbations generically one is hyperbolic and the other one is elliptic

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

a renormalization approach to lower dimensional tori with brjuno frequency vectors abstract we extend the renormalization scheme for vector fields on t d r m in order to construct lower dimensional invariant tori with brjuno frequency vectors for near integrable hamiltonian flows for every brjuno frequency vector r d and every vector r d satisfying a diophantine condition with respect to there exists an analytic manifold w of infinitely renormalizable hamiltonian vector fields each vector field on w is shown to have an analytic invariant torus with frequency vector

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

solving partial differential equations on manifolds from incomplete interpoint distance solutions of partial differential equations pdes on manifolds have provided important applications in different fields in science and engineering existing methods are mainly based on discretization of manifolds as implicit functions triangle meshes or point clouds where the manifold structure is approximated by either zero level set of an implicit function or a set of points in many applications manifolds might be only provided as an interpoint distance matrix with possible missing values this paper discusses a framework to discretize pdes on manifolds represented as incomplete interpoint distance information without conducting a time consuming global coordinates reconstruction we propose a more efficient strategy by discretizing differential operators only based on pointwisely local reconstruction our local reconstruction model is based on the recent advances of low rank matrix completion theory where only a very small random portion of distance information is required this method enables u

periodic solutions for completely resonant nonlinear wave equations with dirichlet boundary conditions we consider the nonlinear string equation with dirichlet boundary conditions utt uxx u with u u3 o u5 odd and analytic 0 and we construct small amplitude periodic solutions with frequency for a large lebesgue measure set of close to 1 this extends previous results where only a zero measure set of frequencies could be treated the ones for which no small divisors appear the proof is based on combining the lyapunov schmidt decomposition which leads to two separate sets of equations dealing with the resonant and non resonant fourier components respectively the q and the p equations with resummation techniques of divergent powers series allowing us to control the small divisors problem the main difficulty with respect to the nonlinear wave equations utt uxx mu u m 0 is that not only the p equation but also the q equation is infinite dimensional

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

conditions for unique graph realizations the graph realization problem is that of computing the relative locations of a set of vertices placed in euclidean space relying only upon some set of inter vertex distance measurements this paper is concerned with the closely related problem of determining whether or not a graph has a unique realization both these problems are np hard but the proofs rely upon special combinations of edge lengths if one assumes the vertex locations are unrelated then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths this paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension efficient sequential and nc algorithms are presented for each condition although these algorithms have very different flavors in different dimensions

fully automatic hp adaptivity in three dimensions the paper presents a three dimensional implementation of the fully automatic hp adaptive mesh refinement strategy proposed in l demkowicz w rachowicz ph devloo a fully automatic hp adaptivity j sci comput 17 1 3 2002 127 155 the methodology is based on a minimization of the projection based interpolation error with a fixed number of degrees of freedom a coarse grid is first uniformly refined in both h and p to yield a corresponding fine grid a two grid solver is used to solve the problem on the fine grid or iterate towards the fine grid solution the fine grid solution possibly partially converged serves then as a reference solution to be interpolated on the original coarse mesh and the next optimal coarse grid to be determined the difference between the coarse and fine grid solutions provides an excellent error estimate for the coarse grid error the procedure is repeated with the coarse and fine meshes generated during the adaptive process simultaneously until a prescribed error tolerance for the coarse grid error is met the solution obtained on the last fine grid is considered to be the final product delivered by the adaptive procedure

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

a quasi periodic poincare s theorem we study the persistence of invariant tori on resonant surfaces of a nearly integrable hamiltonian system under the usual kolmogorov non degenerate condition by introducing a quasi linear iterative scheme to deal with small divisors we generalize the poincare theorem on the maximal resonance case i e the periodic case to the general resonance case i e the quasi periodic case by showing the persistence of majority of invariant tori associated to non degenerate relative equilibria on any resonant surface

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

degenerate elliptic resonances quasi periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations the parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations here we prove that for a class of elliptic tori a resummation algorithm can be devised and proved to be convergent thus extending to such lower dimensional invariant tori the methods employed to prove convergence of the lindstedt series either for the maximal i e kam tori or for the hyperbolic lower dimensional invariant tori

fractional lindstedt series the parametric equations of the surfaces on which highly resonant quasiperiodic motions develop lower dimensional tori cannot be analytically continued in general in the perturbation parameter e i e they are not analytic functions of e however rather generally quasiperiodic motions whose frequencies satisfy only one rational relation resonances of order 1 admit formal perturbation expansions in terms of a fractional power of e depending on the degeneration of the resonance we find conditions for this to happen and in such a case we prove that the formal expansion is convergent after suitable resummation

gevrey smoothness of invariant tori for analytic nearly integrable hamiltonian systems under russmann s non degeneracy condition abstract in this paper we prove gevrey smoothness of the persisting invariant tori for small perturbations of an analytic integrable hamiltonian system with russmann s non degeneracy condition by an improved kam iteration method with parameters

frequency locking in an injection locked frequency divider equation we consider a model for a resonant injection locked frequency divider and study analytically the locking onto rational multiples of the driving frequency we provide explicit formulae for the widt

continuous mesh framework part i well posed continuous interpolation error in the context of mesh adaptation riemannian metric spaces have been used to prescribe orientation density and stretching of anisotropic meshes but such structures are only considered to compute lengths in adaptive mesh generators in this article a riemannian metric space is shown to be more than a way to compute a length it is proven to be a reliable continuous mesh model in particular we demonstrate that the linear interpolation error can be evaluated continuously on a riemannian metric space from one hand this new continuous framework proves that prescribing a riemannian metric field is equivalent to the local control in the mathbf l 1 norm of the interpolation error this proves the consistency of classical metric based mesh adaptation procedures on the other hand powerful mathematical tools are available and are well defined on riemannian metric spaces calculus of variations differentiation optimization dots whereas these tools are not defined on discrete meshes

hyperbolic low dimensional invariant tori and summations of divergent series we consider a class of a priori stable quasi integrable analytic hamiltonian systems and study the regularity of low dimensional hyperbolic invariant tori as functions of the perturbation parameter we show that under natural nonresonance conditions such tori exist and can be identified through the maxima or minima of a suitable potential they are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin the invariant tori admit an asymptotic series at the origin with taylor coefficients that grow at most as a power of a factorial and a remainder that to any order n is bounded by the n 1 st power of the argument times a power of n we show the existence of a summation criterion of the generically divergent series in powers of the perturbation size that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory

floquet solutions for the 1 dimensional quasi periodic schrodinger equation we show that the 1 dimensional schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a floquet representation for almost every energye in the upper part of the spectrum we prove that the upper part of the spectrum is purely absolutely continuous and that for a generic potential it is a cantor set we also show that for a small potential these results extend to the whole spectrum

euclidean distance geometry and applications euclidean distance geometry is the study of euclidean geometry based on the concept of distance this is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in euclidean space realizing those given distances we survey the theory of euclidean distance geometry and its most important applications with special emphasis on molecular conformation problems

borel summability and lindstedt series resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation such series may be sometimes summed via suitable sum rules defining c functions of the perturbation strength here we find sufficient conditions for the borel summability of their sums in the case of two dimensional rotation vectors with diophantine exponent 1 e g with ratio of the two independent frequencies equal to the golden mean

an optimization based framework for anisotropic simplex mesh adaptation we present a general framework for anisotropic h adaptation of simplex meshes given a discretization and any element wise localizable error estimate our adaptive method iterates toward a mesh that minimizes error for a given degrees of freedom utilizing mesh metric duality we consider a continuous optimization problem of the riemannian metric tensor field that provides an anisotropic description of element sizes first our method performs a series of local solves to survey the behavior of the local error function this information is then synthesized using an affine invariant tensor manipulation framework to reconstruct an approximate gradient of the error function with respect to the metric tensor field finally we perform gradient descent in the metric space to drive the mesh toward optimality the method is first demonstrated to produce optimal anisotropic meshes minimizing the l 2 projection error for a pair of canonical problems containing a singularity and a singular perturbation the effectiveness of the framework is then demonstrated in the context of output based adaptation for the advection diffusion equation using a high order discontinuous galerkin discretization and the dual weighted residual dwr error estimate the method presented provides a unified framework for optimizing both the element size and anisotropy distribution using an a posteriori error estimate and enables efficient adaptation of anisotropic simplex meshes for high order discretizations

summation of divergent series and borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms we consider a class of second order ordinary differential equations describing one dimensional systems with a quasiperiodic analytic forcing term and in the presence of damping as a physical application one can think of a resistor inductor varactor circuit with a periodic or quasiperiodic forcing function even if the range of applicability of the theory is much wider in the limit of large damping we look for quasiperiodic solutions which have the same frequency vector of the forcing term and we study their analyticity properties in the inverse of the damping coefficient we find that even the case of periodic forcing terms is nontrivial as the solution is not analytic in a neighborhood of the origin it turns out to be borel summable in the case of quasiperiodic forcing terms we need renormalization group techniques in order to control the small divisors arising in the perturbation series we show the existence of a summation criterion of the series in this case also however this cannot be interpreted as borel summability

lra local rigid averaging of stretchable non rigid shapes we present a novel algorithm for generating the mean structure of non rigid stretchable shapes following an alignment process which supports local affine deformations we translate the search of the mean shape into a diagonalization problem where the structure is hidden within the kernel of a matrix this is the first step required in many practical applications where one needs to model bendable and stretchable shapes from multiple observations

a kam theorem for hyperbolic type degenerate lower dimensional tori in hamiltonian systems a kam theorem for degenerate lower dimensional tori in nearly integrable hamiltonian systems is given in this paper for the non degenerate cases both hyperbolic and elliptic the kam theorem has been well established by many authors 8 9 11 13 14 17

on the absolutely continuous spectrum of the perturbed hill s equation on considere l equation de hill y q x y y avec x et complexe ou q x est reel continu par morceaux et periodique de periode 1 et l equation augmentee perturbee par une fonction l 1 y q x p x y y ou p x est reel et absolument integrable le long de l axe reel

resummation of perturbation series and reducibility for bryuno skew product flows we consider skew product systems on d l sl 2 for bryuno base flows close to constant coefficients depending on a parameter in any dimension d and we prove reducibility for a large measure set of values of the parameter the proof is based on a resummation procedure of the formal power series for the conjugation and uses techniques of renormalisation group in quantum field theory

periodic and quasi periodic orbits for the standard map we consider both periodic and quasi periodic solutions for the standard map and we study the corresponding conjugating functions i e the functions conjugating the motions to trivial rotations we compare the invariant curves with rotation numbers satisfying the bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents n p n q n we prove the following results for n 1 for rotation numbers n n we study the radius of convergence of the conjugating functions and we find lower bounds on them which tend to a limit which is a lower bound on the corresponding quantity for 2 the periodic orbits consist of points which are more and more close to the invariant curve with rotation number 3 such orbits lie on analytical curves which tend uniformly to the invariant curve

periodic solutions for a class of nonlinear partial differential equations in higher dimension we prove the existence of periodic solutions in a class of nonlinear partial differential equations including the nonlinear schrodinger equation the nonlinear wave equation and the nonlinear beam equation in higher dimension our result covers cases of completely resonant equations where the bifurcation equation is infinite dimensional such as the nonlinear schrodinger equation with zero mass for which solutions which at leading order are wave packets are shown to exist

oscillator synchronisation under arbitrary quasi periodic forcing we study the problem of existence of response solutions for a real analytic one dimensional system consisting of a rotator subject to a small quasi periodic forcing with bryuno frequency vector we prove that at least one response solution always exists without any assumption on the forcing besides smallness and analyticity this strengthens the results available in the literature where generic non degeneracy conditions are assumed the proof is based on a diagrammatic formalism and relies on renormalisation group techniques which exploit the formal analogy with problems of quantum field theory a crucial role is played by remarkable identities between classes of diagrams

on lower dimensional invariant tori in reversible systems abstract in this paper a result on the persistence of lower dimensional invariant tori in reversible systems is obtained under some weak non degenerate conditions such a result can be applied to the case where normal frequencies are not simple and some of them may be zero we also give an answer of a conjecture of m b sevryuk 1986 reversible systems lecture notes in math vol 1211 springer verlag new york berlin about the existence of lower dimensional invariant tori in resonant zones

absolutely continuous spectra of quasiperiodic schrodinger operators several aspects of the general and constructive spectral theory of quasiperiodic schrodinger operators in one dimension are discussed an explicit formula for the absolutely continuous a c spectral densities that yields an immediate proof of the fact that the kolmogorov arnold moser kam spectrum constructed by dinaburg sinai and russmann funkt anal prilozen 9 8 1975 ann acad sci 357 90 1980 is a subset of the a c spectrum is provided some quasiperiodicity properties of the deift simon a c eigenfunctions are proved namely that the normalized phase of such eigenfunctions is a quasiperiodic distribution in the constructive part the dinaburg sinai russmann theory is extended to quasiperiodic perturbations of periodic schrodinger operators using a kam hamiltonian formalism based on a new treatment of perturbations of harmonic oscillators particular attention is devoted to the dependence upon the eigenvalue parameter and a complete control of kam objects is achieved using the notion

global density of reducible quasi periodic cocycles on bf t 1 times roman su 2 we prove that given a in a set of total haar measure in t 1 r z the set of a c t 1 su 2 for which the skew product system a t 1 su 2 t 1 su 2 a y a y is reducible that is a b a 0 b 1 for some a 0 su 2 b c t 1 su 2 is dense for the c topology

minimal invariant tori in the resonant regions for nearly integrable hamiltonian systems consider a real analytical hamiltonian system of kam type h p q n p p p q that has n degrees of freedom n 2 and is positive definite in p let r n 0 k z n in this paper we show that for most rotation vectors in in the sense of n 1 dimensional lebesgue measure there is at least one n 1 dimensional invariant torus these tori are the support of corresponding minimal measures the lebesgue measure estimate on this set is uniformly valid for any perturbation

existence of kam tori in degenerate hamiltonian systems abstract this article is devoted to the proof that invariant tori still exist in perturbed integrable hamiltonian systems with n degrees of freedom h p q n p p p q if the hypothesis of nondegeneracy is replaced by a much weaker one that the rank of the mapping n p can be any positive integer smaller than n provided the image of any open set under the mapping includes a cn 2 curved one dimensional submanifold

birkhoff kolmogorov arnold moser tori in convex hamiltonian systems for the hamiltonian systems of kam type it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

perturbations of lower dimensional tori in the resonant zone for reversible systems consider a nearly integrable system which is reversible with respect to an involution of n n type for a given l resonant surface 1 l n we show that there is a subset of it with positive measure such that in each resonant torus corresponding to a point of this set at least one dimensional invariant submanifold can survive any small perturbations

globally and locally attractive solutions for quasi periodically forced systems abstract we consider a class of differential equations x x g x f t with r d describing one dimensional dissipative systems subject to a periodic or quasi periodic diophantine forcing we study existence and properties of trajectories with the same quasi periodicity as the forcing for g x x 2 p 1 p n we show that when the dissipation coefficient is large enough there is only one such trajectory and that it describes a global attractor in the case of more general nonlinearities including g x x 2 describing the varactor equation we find that there is at least one trajectory which describes a local attractor

compensations in small divisor problems a general direct method alternative to kam theory apt to deal with small divisor problems in the real analytic category is presented and tested on several small divisor problems including the construction of maximal quasi periodic solutions for nearly integrable non degenerate hamiltonian or lagrangian systems and the construction of lower dimensional resonant tori for nearly integrable hamiltonian systems the method is based on an explicit graph theoretical representation of the formal power series solutions which allows to prove compensations among the monomials forming such representation

quasiperiodic hamiltonian motions scale invariance harmonic oscillators the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point