homoclinic orbits to invariant tori near a homoclinic orbit to center center saddle equilibrium we consider a perturbation of an integrable hamiltonian vector field with three degrees of freedom with a center center saddle equilibrium having a homoclinic orbit or loop with the help of a poincare map chosen based on the unperturbed homoclinic loop we study the homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic kam tori on the center manifold near the equilibrium if the perturbation is such that the homoclinic loop is preserved i e the perturbation also has a homoclinic loop inherited from the unperturbed one we establish that in general the manifolds intersect along 8 12 or 16 transverse homoclinic orbits on the other hand in a more generic situation the loop is not preserved a condition for this fact is obtained by means of a melnikov like method the manifolds intersect along four transverse homoclinic orbits though a small neighborhood of the loop has to be excluded in a first approximation those homoclinic orbits can be detected as nondegenerate critical points of a melnikov potential defined on the 2 torus the number of homoclinic orbits is given by morse theory applied to the melnikov potential

instability of high dimensional hamiltonian systems multiple resonances do not impede diffusion abstract we consider models given by hamiltonians of the form h i p q t e h i j 1 n 1 2 p j 2 v j q j e q i p q t e where i i r d t d p q r n t t 1 these are higher dimensional analogues both in the center and hyperbolic directions of the models studied in 28 29 43 and are usually called a priori unstable hamiltonian systems all these models present the large gap problem we show that for 0 e 1 under regularity and explicit non degeneracy conditions on the model there are orbits whose action variables i perform rather arbitrary excursions in a domain of size o 1 this domain includes resonance lines and hence large gaps among d dimensional kam tori this phenomenon is known as arnold diffusion the method of proof follows closely the strategy of 28 29 the main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions i i r d we show that since these multiple resonances happen in sets of codimension greater than one in the space of actions i they can be contoured this corresponds to the mechanism called diffusion across resonances in the physics literature the present paper however differs substantially from 28 29 on the technical details of the proofs we have taken advantage of the theory of the scattering map developed in 31 notably the symplectic properties which were not available when the above papers were written we have analyzed the conditions imposed on the resonances in more detail more precisely we have found that there is a simple condition on the melnikov potential which allows us to conclude that the resonances are crossed in particular this condition does not depend on the resonances so that the results are new even when applied to the models in 28 29

ballistic transport in one dimensional quasi periodic continuous schrodinger equation abstract for the solution q t to the one dimensional continuous schrodinger equation i t q x t x 2 q x t v x q x t x r with r d satisfying a diophantine condition and v a real analytic function on t d we consider the growth rate of the diffusion norm q t d r x 2 q x t 2 d x 1 2 for any non zero initial condition q 0 h 1 r with q 0 d we prove that q t d grows linearly with t if v is sufficiently small

majorization and a schur horn theorem for positive compact operators the nonzero kernel case abstract schur horn theorems focus on determining the diagonal sequences obtainable for an operator under all possible basis changes formally described as the range of the canonical conditional expectation of its unitary orbit following a brief background survey we prove an infinite dimensional schur horn theorem for positive compact operators with infinite dimensional kernel one of the two open cases posed recently by kaftal weiss there they characterized the diagonals of operators in the unitary orbits for finite rank or zero kernel positive compact operators here we show how the characterization problem depends on the dimension of the kernel when it is finite or infinite dimensional we obtain exact majorization characterizations of the range of the canonical conditional expectation of the unitary orbits of positive compact operators with infinite dimensional kernel unlike the approximate characterizations of arveson kadison but extending the exact characterizations of gohberg markus and kaftal weiss recent advances in this subject and related subjects like traces on ideals show the relevance of new kinds of sequence majorization as in the work of kaftal weiss e g strong majorization and another majorization similar to what here we call p majorization and of kalton sukochev e g uniform hardy littlewood majorization and of bownik jasper e g riemann and lebesgue majorization likewise key tools here are new kinds of majorization which we call p and approximate p majorization 0 p

spectral monodromy of non self adjoint operators in the present paper we build a combinatorial invariant called the spectral monodromy from the spectrum of a single non self adjoint h pseudodifferential operator with two degrees of freedom in the semi classical limit our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self adjoint h pseudodifferential operators given by s vu ngoc quantum monodromy in integrable systems commun math phys 203 2 465 479 1999 the first simple case that we treat in this work is a normal operator in this case the discrete spectrum can be identified with the joint spectrum of an integrable quantum system the second more complex case we propose is a small perturbation of a self adjoint operator with a classical integrability property we show that the discrete spectrum in a small band around the real axis also has a combinatorial monodromy the main difficulty in this case is that we do not know the description of the spectrum everywher

quasiperiodic dynamics in hamiltonian 1 1 2 degree of freedom systems far from integrability abstract the subject of this paper is two quasiperiodicity in a large class of one and a half degree of freedom hamiltonian systems the main result is that such systems have invariant tori for any internal frequency that is of constant type and sufficiently large relative to the forcing frequency an explicit bound on the minimum value of the internal frequency is presented the systems under consideration are not required to be small perturbations of integrable ones

umbilical torus bifurcations in hamiltonian systems we consider perturbations of integrable hamiltonian systems in the neighbourhood of normally umbilic invariant tori these lower dimensional tori do not satisfy the usual non degeneracy conditions that would yield persistence by an adaption of kam theory and there are indeed regions in parameter space with no surviving torus we assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a generalised umbilical catastrophe combining techniques of kam theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large cantor sets applications to gyrostat dynamics are pointed out

existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems abstract in this paper we study a generalization of arnold s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori that has subsequently been referred to as arnold diffusion in case when the number of degrees of freedom n 3 namely we consider a widely studied model of a pendulum weakly coupled with n 1 rotors with the degeneracies in the unperturbed hamiltonian corresponding to different timescales existing in the problem using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation thus improving the estimate of gallavotti 1994 which he calls quasiflat and generalizing the analogous recent estimate of delshams et al 1996 for the rapidly quasiperiodically forced pendulum to a much larger class of hamiltonian systems in particular the exponentially small upper bound for the transversality measure of the splitting applies when the hamiltonian has extra degeneracies namely when the frequencies on a torus become near resonant in fact we show that in such a case the quantity in question becomes smaller which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger nekhoroshev exponent nekhoroshev exponent nevertheless we emphasize that getting uniform estimates for an arbitrary n 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector although recent developments show that the first order of canonical perturbation theory given by melnikov integrals generally cannot be accepted as the leading order answer for the splitting distance for the case of more than two degrees of freedom including the rapidly quasiperiodically forced pendulum problem we suggest an analytic perturbation the majority of whose fourier components are strictly nondashzero for which melnikov integrals can be vindicated as the leading order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions this allows us to bound the splitting distance from below furthermore having such a perturbation for the case of three degrees of freedom we use a simple number theoretical argument to find the asymptotics of the fourier series with exponentially small coefficients involved this enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure and by proving the domineering contribution of the first order of perturbation theory for the transversality measure to suggest a leading order answer for this quantity which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies we elucidate the numerous arithmetic issues that obstruct getting a compact leading order approximation for the splitting size most of which can be overcome in the case of three degrees of freedom as our example shows these obstacles can be also possibly avoided in the same fashion for an arbitrary n 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n 3 or n 2

the green function of an infinite fluid loaded membrane abstract in this paper the response of a fluid loaded plane structure a membrane to a concentrated line force excitation is considered in great detail the normalized velocity response here called the green function g depends upon a dimensionless range x0 km x where km is the free wavenumber on the membrane in a vacuum on the mach number m k 0 k m the ratio of wave phase speed km on the unloaded membrane to the sound speed k0 and on a parameter which can be regarded as a measure of fluid loading at the coincidence condition m 1 in the analogous problem involving a thin elastic plate the corresponding parameter is independent of frequency and plate thickness and may be regarded as an intrinsic measure of fluid loading moreover in cases of common interest steel in water aluminium in air that parameter is small in the present paper the asymptotic structure of g x0 m is therefore sought in the limit 0 naturally no single asymptotic expansion can be expected to be valid throughout the x0 m plane and the programme therefore involves the delineation of regions of that plane in which distinct asymptotic results apply the construction and discussion of those results and the asymptotic matching according to the procedures of the method of matched asymptotic expansions of results holding in adjoining regions the fourier integral for g is broken into surface wave and acoustic components and the asymptotic structure obtained for each previously obtained results for the behaviour at large distances are recovered with a demonstration that very large distances indeed x0 2 may be needed for their validity for some ranges of m and the drive point behaviour of g x0 0 m as 0 qua function of m is shown to correspond to that already discussed in the literature elsewhere in the covering of the whole x0 m plane by different asymptotic expressions a wide variety of analytical results is found reflecting the achievement in different regions of different balances among the five competing physical mechanisms represented in the model namely structural stiffness structural inertia fluid pressures fluid compressibility and fluid inertia these different balances give rise to a wide variety of expressions for the phase and amplitude of the surface wave and acoustic components which can now be used to isolate the dominant structural and acoustic mechanisms at any point in the x0 m plane

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

super exponential stability for generic real analytic elliptic equilibrium points we consider the dynamics in a neighborhood of an elliptic equilibrium point with a diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability generically both in a topological and measure theoretical sense any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point

kam theory and the 3d euler equation abstract we prove that the dynamical system defined by the hydrodynamical euler equation on any closed riemannian 3 manifold m is not mixing in the c k topology k 4 and non integer for any prescribed value of helicity and sufficiently large values of energy this can be regarded as a 3d version of nadirashvili s and shnirelman s theorems showing the existence of wandering solutions for the 2d euler equation moreover we obtain an obstruction for the mixing under the euler flow of c k neighborhoods of divergence free vectorfields on m on the way we construct a family of functionals on the space of divergence free c 1 vectorfields on the manifold which are integrals of motion of the 3d euler equation given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types we use the kam theory to establish some continuity properties of these functionals in the c k topology this allows one to get a lower bound for the c k distance between a divergence free vectorfield in particular a steady solution and a trajectory of the euler flow

degenerate lower dimensional tori in hamiltonian systems abstract we study the persistence of lower dimensional tori in hamiltonian systems of the form h x y z y 1 2 z m z e p x y z where x y z t n r n r 2 m e is a small parameter and m can be singular we show under a weak melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n tori can still survive from the small perturbation as an application we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable properly degenerate hamiltonian system for which neither the kolmogorov nor the g nondegenerate condition is satisfied

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

analytical numerical matching for fluid loaded structures with discontinuities the method of analytical numerical matching anm is applied to several example problems involving the radiation and vibration of a fluid loaded cylindrical shell with structural discontinuities these problems are used to verify the accuracy of the method compared to a purely numerical approach and to demonstrate the usefulness of anm for solving problems involving nonaxisymmetric forcing and constraints on structures that are otherwise axisymmetric the principal advantage of the anm method for this application is its ability to employ a spectral method modal decomposition in azimuth in the solution of a problem that is nonaxisymmetric anm divides the original problem into local and global subproblems that are solved separately the discontinuity is handled using a high resolution local finite element model smooth forces are derived from solving an analytical matching problem on the local domain and these smooth forces are then applied to a low resolution numerical global problem the superposition of solutions to the subproblems is equivalent to the solution of the original problem for fluid loading the local high resolution model is solved in vacuo and the effects of fluid loading are included completely by the low resolution global problem

invariant tori for the billiard ball map for an n dimensional domain q n 3 with a smooth bound ary which is strictly convex in a neighborhood of an elliptic closed geodesic the existence of a family of invariant tori for the billiard ball map with a positive measure is proved under the assumptions of nondegeneracy and n elementarity n 5 of the corresponding to poincare map moreover the conjugating diffeomorphism constructed is symplectic an analogous result is obtained in the case n 2 it is shown that the lengths of the periodic geodesics determine uniquely the invariant curves near the boundary and the billiard ball map on them up to a symplectic diffeomorphism let n be a strictly convex and compact domain in ir n n 2 2 with a boundary an of class coo in this paper we investigate the so called billiard ball map b near the boundary s an of the coball bundle l b an x e t an 1 1 i the billiard ball map is an exact symplectic map in the interior of l 6 which is singular on the boundary al the local behaviour of b near al was described by melrose using the equivalence theorem for glancing hypersurfaces proved by melrose 15 one can introduce local symplectic coordinates x in t an near any point p e s an so that b an gn i 2 o and the billiard ball map b assumes the form

persistence of lagrange invariant tori at tangent degeneracy abstract in the present paper without russmann s nondegenderate condition we show the persistence of kam invariant tori using this theorem we obtain the persistence of invariant tori for degenerate systems rising from restricted three body problems especially for lagrange equilibrium position l2 with certain degeneracy

weakly mixing invariant tori of hamiltonian systems we note that every finite or infinite dimensional real analytic hamiltonian system with a quasi periodic invariant kam torus of finite dimension d 2 can be perturbed in such a way that the new real analytic hamiltonian system has a weakly mixing invariant torus of the same dimension

diagonals of normal operators with finite spectrum let x lambda1 lambdan be a finite set of complex numbers and let a be a normal operator with spectrum x that acts on a separable hilbert space h relative to a fixed orthonormal basis e1 e2 for h a gives rise to a matrix whose diagonal is a sequence d d1 d2 with the property that each of its terms dn belongs to the convex hull of x not all sequences with that property can arise as the diagonal of a normal operator with spectrum x the case where x is a set of real numbers has received a great deal of attention over the years and is reasonably well though incompletely understood in this work we take up the case in which x is the set of vertices of a convex polygon in the critical sequences d turn out to be those that accumulate rapidly in x in the sense that sigmainfinityn 1 dist dn x s d e gammax with the following property if s d not equal 0 then d is not the diagonal of any such operator a we also show that while this is the only obstruction when n 2 there are other as yet unknown obstructions when n 3

small divisors with spatial structure in infinite dimensional hamiltonian systems a general perturbation theory of the kolmogorov arnold moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems the key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures this approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

the application of analytical numerical matching to structural discontinuities in structural acoustic problems analytical numerical matching anm is introduced to more efficiently model structural discontinuities in structural acoustic problems the general method is explained in detail for the simple example problem of normal incidence acoustic scattering from a periodically line supported elastic membrane a modal series solution exists for this problem the most slowly convergent part of this solution the dominant high resolution large wave number content is removed from the series representation and summed in closed form in what is called the anm local solution this local solution is represented by a piecewise polynomial that is zero outside of a small region the remaining part of the solution containing the overall low resolution content is called the anm global solution and converges very rapidly in the present example the series representing the global solution falls by three powers of the index faster than the classical modal solution the result is an anm composite local plus global solution that requires far less computational effort to resolve in addition to providing a clear abstraction of the method the example problem allows the equivalence of the anm solution to be verified and indicates the importance of resolving the fine scale effects of discontinuities in constrained scattering problems

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

numerical computation of normal forms around some periodic orbits of the restricted three body problem in this paper we introduce a general methodology for computing numerically the normal form around a periodic orbit of an autonomous analytic hamiltonian system the process follows two steps first we expand the hamiltonian in suitable coordinates around the orbit and second we perform a standard normal form scheme based on the lie series method this scheme is carried out up to some finite order and neglecting the remainder we obtain an accurate description of the dynamics in a small enough neighbourhood of the orbit in particular we obtain the invariant tori that generalize the elliptic directions of the periodic orbit on the other hand bounding the remainder one obtains lower estimates for the diffusion time around the orbit this procedure is applied to an elliptic periodic orbit of the spatial restricted three body problem the selected orbit belongs to the lyapunov family associated to the vertical oscillation of the equilibrium point l 5 the mass parameter mu has been chosen such that l 5 is unstable but the periodic orbit is still stable this allows to show the existence of regions of effective stability near l 5 for values of mu bigger that the routh critical value the computations have been done using formal expansions with numerical coefficients

free vortex calculations using analytical numerical matching with solution pyramiding the motion of a free vortex system is predicted using a novel method called analytical numerical matching anm analytical numerical matching is a hybrid mathematical and numerical technique that results in a new problem formulation with high accuracy and increased computational efficiency the approach capitalizer on the disparity of characteristic length scaler that occur in many fluid dynamics problems accordingly many problems can be decomposed into two distinct problems involving a local high resolution near field often solvable analytically end a global low resolution far field solved numerically the anm technique breaks the free vortex problem into these local and global problems each of which are easier end more efficient to solve than the traditional problem formulation in addition the anm decomposition can be repeated to form a hierarchy of nested problems of increasing resolution using a method called pyramiding the anm method with pyramiding called anm p is studied to assess its accuracy and computational efficiency for two dimensional and three dimensional vortex dynamics problems in calculations of vortex sheet rollup end vortex ring motion and interactions this method war found to be relatively easy to implement end exhibited reductions in computational time from a factor of three to more than an order of magnitude

an infinite dimensional schur horn theorem and majorization theory the main result of this paper is the extension of the schur horn theorem to infinite sequences for two nonincreasing nonsummable sequences and that converge to 0 there exists a positive compact operator a with eigenvalue list and diagonal sequence if and only if j 1n j j 1n j for every n if and only if q for some orthostochastic matrix q when and are summable requiring additionally equality of their infinite series obtains the same conclusion extending a theorem by arveson and kadison our proof depends on the construction and analysis of an infinite product of t transform matrices

canonical methods of constructing invariant tori by phase space sampling abstract invariant tori in phase space can be constructed via a nonperturbative canonical transformation applied to a known integrable hamiltonian h hitherto this process has been carried through with h corresponding to the isochrone potential and the harmonic oscillator in this paper we expand the applicability regime of the torus construction method by demonstrating that h can be based on a stackel potential the most general known form of an integrable potential also we present a simple scheme based on phase space sampling for recovering the angle variables on the constructed torus numerical examples involving axisymmetric galactic potentials are given

kam theory the legacy of kolmogorov s 1954 paper kolmogorov arnold moser or kam theory was developed for conservative dynamical systems that are nearly integrable integrable systems in their phase space usually contain lots of invariant tori and kam theory establishes persistence results for such tori which carry quasi periodic motions we sketch this theory which begins with kolmogorov s pioneering work

persistence of lower dimensional tori of general types in hamiltonian systems this work is a generalization to a result of j you 1999 we study the persistence of lower dimensional tori of general type in hamiltonian systems of general normal forms by introducing a modified linear kam iterative scheme to deal with small divisors we shall prove a persistence result under a melnikov type of non resonance condition which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori

an infinite dimensional version of the schur horn convexity theorem the schur horn convexity theorem states that forain rnp u diag a u u u n conv sna wherepdenotes the projection on the diagonal in this paper we generalize this result to the setting of arbitrary separable hilbert spaces it turns out that the theorem still holds if we take thel closure on both sides we will also give a description of the left hand side for nondiagonalizable hermitian operators in the last section we use this result to get an extension theorem for invariant closed convex subsets of the diagonal operators

separatrix splitting for systems with three time scales an exact expression for the determinant of the splitting matrix is derived for three degrees of freedom systems with three time scales it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in ann inst h poincare b 60 1 1994 the asymptotic validity of mel nikov s integrals is proved for the class of models considered which are polynomial perturbations the technique for exhibiting cancellations is inspired by renormalization theory in quantum electrodynamics and uses an analogue of dyson s equations to prove an infinite family of identities due to symmetries that remind us of ward s identities

geometry of arnold diffusion the goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of arnold diffusion we choose to speak of a specific model that of geometric rays in a periodic optical medium this model is equivalent to that of a particle in a periodic potential in mathbb r n with energy prescribed and to the geodesic flow in a riemannian metric on mathbb r n

the finite dimensional reversible kam theory abstract the main results of the theory of quasi periodic motions in finite dimensional reversible dynamical systems are surveyed lower dimensional invariant tori and excitation of elliptic normal modes are discussed in detail the case where the reversing involution is perturbed is considered several fundamental differences between the reversible and hamiltonian theories are listed

a reduction theorem for time dependent schrodinger operator with finite differentiable unbounded perturbation in this paper we consider the time dependent schrodinger operator with a temporal quasi periodic perturbation which is some function of class cl for any l 100 3n 2 1 by using the kam technique we establish a reduction theorem for the above operator and prove that the invariant tori of the nearly integrable hamiltonian system with unbounded small perturbation are linearly stable both for d d 1 and the limiting case d d 1

quasi periodic stability of subfamilies of an unfolded skew hopf bifurcation in the skew hopf bifurcation a quasi periodic attractor with nontrivial normal linear dynamics loses hyperbolicity the simplest setting concerns rotationally symmetric diffeomorphisms of s1 r2 their dynamics involve periodicity quasi periodicity and chaos including mixed spectrum the present paper deals with the persistence under symmetry breaking of quasi periodic invariant circles in this bifurcation it turns out that when adding sufficiently many unfolding parameters the invariant circle persists for a large hausdorff measure subset of a submanifold in parameter space

multiscale kam theorem for hamiltonian systems abstract in this paper we study the persistence of invariant tori in nearly integrable multiscale hamiltonian systems with highorder degeneracy in the integrable part such hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems we introduce multiscale nondegenerate condition and multiscale diophantine condition comparable to the usual diophantine condition using quasilinear kam method we prove a multiscale kam theorem

persistence of hyperbolic tori in hamiltonian systems abstract we generalize the well known result of graff and zehnder on the persistence of hyperbolic invariant tori in hamiltonian systems by considering non floquet frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies the preservation of part or full frequency components associated to the degree of non degeneracy is considered as applications we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non integrable hamiltonian system

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

an illustration of analytical numerical matching with finite element analysis for structural vibration problems analytical numerical matching anm is an accurate and efficient method for solving many types of problems with discontinuities the method separates local and global effects and solves separate subproblems using high resolution around the discontinuity and low resolution away from the discontinuity the work presented in this manuscript demonstrates a methodology for applying anm to a dynamic structure using finite element analysis fea for the solution of the high resolution local and the low resolution global subproblems the anm method is illustrated on a thick two dimensional beam having several displacement constraints attached to its lower surface ordinarily and here for verification purposes this problem would be solved using two dimensional plane elements due to the local discontinuities around the constraints and the thickness of the beam using anm these discontinuities and through thickness effects are modeled in the geometrically compact local problem using a high resolution mesh

the pythagorean theorem ii the infinite discrete case the study of the pythagorean theorem and variants of it as the basic result of noncommutative metric euclidean geometry is continued the emphasis in the present article is the case of infinite discrete dimensionality

analytical numerical matching for efficient calculation of scattering from cylindrical shells with lengthwise constraints structural discontinuities in highly coupled fluid structure systems are modeled by an approach called analytical numerical matching anm this method separates the low resolution global influence of a discontinuity from the relatively high resolution local effects a continuous smoothed replacement for a fundamental structural discontinuity is constructed so that the system is identically unchanged beyond a small smoothing region simultaneously the precise local effect of smoothing the discontinuity is retained in analytical form the smoothed problem is solved by numerical techniques with rapid convergence and reduced computational cost the original discontinuous character is restored using the analytical expression for the local difference between the smoothed and the original problems anm has been successfully applied to two dimensional cases of acoustic scattering from a thin infinitely long cylindrical shell with multiple structural discontinuities local solutions for longitudinal line discontinuities with radial tangential and rotational constraints have been formulated using anm line constrained scattering problems as well as line driven problems are investigated the ability of analytical numerical matching to replace a discontinuous physical problem by a well behaved continuous one for numerical evaluation while ultimately retaining the original geometry and physical behavior is illustrated

moser s theorem for lower dimensional tori moser s c version of kolmogorov s theorem on the persistence of maximal quasi periodic solutions for nearly integrable hamiltonian system is extended to the persistence of non maximal quasi periodic solutions corresponding to lower dimensional elliptic tori of any dimension n between one and the number of degrees of freedom the theorem is proved for hamiltonian

structure and f dependence of the a c i m for a unimodal map f of misiurewicz type by using a suitable banach space on which we let the transfer operator act we make a detailed study of the ergodic theory of a unimodal map f of the interval in the misiurewicz case we show in particular that the absolutely continuous invariant measure can be written as the sum of 1 square root spikes along the critical orbit plus a continuous background we conclude by a discussion of the sense in which the map f mapsto rho may be differentiable

invariant tori for nearly integrable hamiltonian systems with degeneracy does not fall into a hyperplanethrough the origin the condition 1 3 is the sharpest one we have not seenits proof yet in this paper we will give a nondegeneracy condition by meansof the derivatives of h p which is equivalent to the condition 1 3 in analyticcase and under this nondegeneracy condition we obtain the russmann s results for analytic case furthermore the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the nite orderderivatives of h p main resultstheorem a suppose that h h p f p q is analytic in t

total destruction of lagrangian tori for an integrable tonelli hamiltonian with d d 2 degrees of freedom we show that all of the lagrangian tori can be destroyed by analytic perturbations which are arbitrarily small in the cd topology

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

bi hamiltonian partially integrable systems given a first order dynamical system possessing a commutative algebra of dynamical symmetries we show that under certain conditions there exists a poisson structure on an open neighborhood of its regular not necessarily compact invariant manifold which makes this dynamical system into a partially integrable hamiltonian system this poisson structure is by no means unique bi hamiltonian partially integrable systems are described in some detail as an outcome we state the conditions of quasiperiodic stability the kam theorem for partially integrable hamiltonian systems

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

a proof of the isoenergetic kam theorem from the ordinary one a proof is given of the isoenergetic kam theorem for hamiltonian systems using the ordinary kam theorem and a transversality argument

diophantine tori and spectral asymptotics for nonselfadjoint operators we study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint h pseudodifferential operators in dimension 2 assuming that the classical flow of the unperturbed part possesses several invariant lagrangian tori enjoying a diophantine property we get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases in the first case we assume that the strength e of the perturbation is o hd for some d 0 and is bounded from below by a fixed positive power of h in the second case e is assumed to be sufficiently small but independent of h and we describe the eigenvalues completely in a fixed h independent domain in the complex spectral plane

frequency analysis for multi dimensional systems global dynamics and diffusion abstract frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system it was first devised in order to study the stability of the solar system j laskar icarus 88 1990 266 291 and then applied to the 2d standard mapping laskar et al physica d 56 1992 253 269 it is a powerful method for analyzing weakly chaotic motion in hamiltonian systems or symplectic maps for regular motions it yields an analytical representation of the solutions the analysis of the regularity of the frequency map with respect to the action space and of its variations with respect to time gives rise to two criteria for the regularity of the motion which are valid for multi dimensional systems for a 4d symplectic map plotting the frequency map in the frequency plane provides a clear representation of the global dynamics and reveals that high order resonances are of great importance in understanding the diffusion of non regular orbits through the invariant tori in particular it appears in several examples that diffusion along the resonance lines arnold diffusion is of less importance than diffusion across the resonances lines which can lead to large diffusion due to the phenomenon of overlap of higher order resonances chaotic layers many fine features of the dynamics are also revealed by frequency analysis which would require more theoretical study for a better understanding

differentiating the absolutely continuous invariant measure of an interval map f with respect to f let the map f 1 1 1 1 have a c i m absolutely continuous f invariant measure with respect to lebesgue let be the change of corresponding to a perturbation x f f 1 of f formally we have for differentiable a

explicit symplectic approximation of nonseparable hamiltonians algorithm and long time performance explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable hamiltonians this article proposes for arbitrary hamiltonians similar integrators which are explicit of any even order symplectic in an extended phase space and with pleasant long time properties they are based on a mechanical restraint that binds two copies of phase space together using backward error analysis kolmogorov arnold moser theory and additional multiscale analysis an error bound of o t l is established for integrable systems where t l and are respectively the long simulation time step size integrator order and some binding constant for nonintegrable systems with positive lyapunov exponents such an error bound is generally impossible but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear schrodinger equation

measuring the mixing efficiency in a simple model of stirring some analytical results and a quantitative study via frequency map analysis we prove the existence of invariant curves for a t periodic hamiltonian system which models a fluid stirring in a cylindrical tank when t is small and the assigned stirring protocol is piecewise constant furthermore using the numerical analysis of the fundamental frequency of laskar we investigate numerically the break down of invariant curves as t increases and we give a quantitative estimate of the efficiency of the mixing

persistence of lower dimensional hyperbolic invariant tori for generalized hamiltonian systems abstract in this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized hamiltonian systems here the generalized hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables in particular systems under consideration can be odd dimensional under russmann type non degenerate condition by introducing a modified linear kam iterative scheme we proved that the majority of the lower dimensional hyperbolic invariant tori persist under small perturbations for generalized hamiltonian systems

extreme points of the set of elements majorised by an integrable function resolution of a problem by luxemburg and of its noncommutative counterpart abstract let f be an arbitrary integrable function on a finite measure space x we characterise the extreme points of the set f of all measurable functions on x majorised by f providing a complete answer to a problem raised by w a j luxemburg in 1967 moreover we obtain a noncommutative version of this result

singularities of normal forms and topology of orbits in area preserving maps the existence of analytic normal forms for an area preserving map with an elliptic fixed point is considered if the linear frequency is diophantine the complexified map can be analytically conjugated with an integrable map in a disc of the complex radial coordinate r excluding a sequence of sets whose measure decreases exponentially fast as the origin is approached the analyticity in the angle theta is just a strip the non analyticity domains correspond to the regions where the topology of the orbits changes since nonlinear resonances are present and the modulus of the residue of the leading poles provided by perturbation theory is the square of the width of the corresponding chains of islands this suggests a relation between clusters of singularities of the normalizing transformation and changes in the topology of the orbits

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

existence and multiplicity of rotating periodic solutions for resonant hamiltonian systems abstract in the present paper we consider a class of resonant hamiltonian systems x j h x t x in r 2 n we will use saddle point reduction morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions i e x t t q x t for any t r with t 0 and q an symplectic orthogonal matrix in the case q k i 2 n for any positive integer k such a rotating periodic solution is just a quasi periodic solution moreover if h is even in x we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques

nekhoroshev estimates for finitely differentiable quasi convex hamiltonians abstract a major result about perturbations of integrable hamiltonian systems is the nekhoroshev theorem which gives exponential stability for all solutions provided the system is analytic and the integrable hamiltonian is generic in the particular but important case where the latter is quasi convex these exponential estimates have been generalized by marco and sauzin if the hamiltonian is gevrey regular using a method introduced by lochak in the analytic case in this paper using the same approach we investigate the situation where the hamiltonian is assumed to be only finitely differentiable for which it is known that exponential stability does not hold but nevertheless we prove estimates of polynomial stability

the schur horn theorem for operators with three point spectrum abstract we characterize the set of diagonals of the unitary orbit of a self adjoint operator with three points in the spectrum our result gives a schur horn theorem for operators with three point spectrum analogous to kadison s pythagorean theorem and carpenter s theorem which characterize the diagonals of orthogonal projections

frames of translates with prescribed fine structure in shift invariant spaces abstract for a given finitely generated shift invariant fsi subspace w l 2 r k we obtain a simple criterion for the existence of shift generated sg bessel sequences e f induced by finite sequences of vectors f w n that have a prescribed fine structure i e such that the norms of the vectors in f and the spectra of s e f are prescribed in each fiber of spec w t k we complement this result by developing an analogue of the so called sequences of eigensteps from finite frame theory in the context of sg bessel sequences that allows for a detailed description of all sequences with prescribed fine structure then given 1 n 0 we characterize the finite sequences f w n such that f i 2 i for 1 i n and such that the fine spectral structure of the shift generated bessel sequences e f has minimal spread i e we show the existence of optimal sg bessel sequences with prescribed norms in this context the spread of the spectra is measured in terms of the convex potential p w induced by w and an arbitrary convex function r r

a kam theorem for finitely differentiable hamiltonian systems abstract given l 2 2 d 4 we prove the persistence of a cantor family of kam tori of measure o e 1 2 l for any non degenerate nearly integrable hamiltonian system of class c l d t d where 1 is the diophantine power of the frequencies of the persistent kam tori and d r d is a bounded domain provided that the size e of the perturbation is sufficiently small this extends a result by d salamon in 25 according to which we do have the persistence of a single kam torus in the same framework moreover as for the persistence of a single torus the regularity assumption is essentially optimal

effective stability and kam theory abstract the two main stability results for nearly integrable hamiltonian systems are revisited nekhoroshev theorem concerning exponential lower bounds for the stability time effective stability and kam theorem concerning the preservation of a majority of the nonresonant invariant tori perpetual stability to stress the relationship between both theorems a common approach is given to their proof consisting of bringing the system to a normal form constructed through the lie series method the estimates obtained for the size of the remainder rely on bounds of the associated vectorfields allowing one to get the optimal stability exponent in nekhoroshev theorem for quasiconvex systems on the other hand a direct and complete proof of the isoenergetic kam theorem is obtained moreover a modification of the proof leads to the notion of nearly invariant torus which constitutes a bridge between kam and nekhoroshev theorems

a point wise criterion for quasi periodic motions in the kam theory abstract we consider initial value problems for nearly integrable hamiltonian systems we formulate a sufficient condition for each initial value to admit the quasi periodic solution with a diophantine frequency vector without any nondegeneracy of the integrable part we reconstruct the kam theorem under russmann s nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition our point wise version is of the form analogous to the corresponding problems for the integrable case we compare our framework with the standard kam theorem through a brief review of the kam theory

results on normal forms for fpu chains in this paper we prove among other results that near the equilibirum position any periodic fpu chain with an odd number n of particles admits a birkhoff normal form up to order 4 whereas any periodic fpu chain with n even admits a resonant normal form up to order 4 this resonant normal form of order 4 turns out to be completely integrable further for n odd we obtain an explicit formula of the hessian of its hamiltonian at the fixed point

normal linear stability of quasi periodic tori abstract we consider families of dynamical systems having invariant tori that carry quasi periodic motions our interest is the persistence of such tori under small nearly integrable perturbations this persistence problem is studied in the dissipative the hamiltonian and the reversible setting as part of a more general kam theory for classes of structure preserving dynamical systems this concerns the parametrized kam theory as initiated by moser j k moser on the theory of quasiperiodic motions siam rev 8 2 1966 145 172 j k moser convergent series expansions for quasi periodic motions math ann 169 1967 136 176 and further developed in g b huitema unfoldings of quasi periodic tori phd thesis university of groningen 1988 h w broer g b huitema f takens unfoldings of quasi periodic tori mem amer math soc 83 421 1990 1 82 h w broer g b huitema unfoldings of quasi periodic tori in reversible systems j dynam differential equations 7 1 1995 191 212 the corresponding nondegeneracy condition involves certain trans versality conditions on the normal linear leading part at the invariant tori we show that as a consequence a cantor family of diophantine tori with positive hausdorff measure is persistent under nearly integrable perturbations this result extends the above references since presently the case of multiple floquet exponents is included our leading example is the normal 1 1 resonance which occurs a lot in applications both hamiltonian and reversible as an illustration of this we briefly describe the lagrange top coupled to an oscillator