countability and sequentiality in fuzzy topological spaces abstract lowen s good extension criterion is used to test countability and sequentiality definitions in fuzzy topology quasi coincidence is found to yield better extensions that classical set membership

an algorithm to generate canonical transformations application to normal forms we describe a new method to generate symplectic diffeomorphisms this method is based on primitive functions and it was introduced in nonlinearity 13 2000 1483 we apply the algorithm to the construction of normal forms of kam tori a concrete application developed in detail and implemented numerically is normal forms near invariant circles

erratic behavior of invariant circles in standard like mappings abstract we study mappings like the standard map but with the sine replaced by another function by varying a parameter in this function it changes smoothly from the sine to a piecewise linear pwl function the function is always analytic except when it is pwl taking the function close enough to the pwl function it is found that kam tori can reappear after they disappeared when the nonlinearity is increased when plotting the boundary around the area where a kam torus with noble winding number exists in the plane of the non linearity versus the distance of the function to the pwl function self similarity is observed

on the master symmetries related to certain classes of integrable hamiltonian systems we present the complete classification of master symmetries related to a non degenerate hamiltonian system that is integrable in the arnol d liouville s sense it is shown that a integrable hamiltonian system cannot have generators of degree greater than 2 specific properties of the master symmetries classified in terms of the action angle variables are investigated

nature of singularities for analyticity domains of invariant curves we present theoretical arguments based on infinite dimensional bifurcation theory and numerical evidence based on nonperturbative methods that the boundaries of analyticity of invariant curves can be described as an accumulation of branch points which are typically of order 2 we show how this fact would explain previous numerical results of several authors and how it suggests more efficient numerical algorithms which we implement

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

a criterion for non existence of invariant tori for hamiltonian systems abstract a particularly interesting class of invariant tori for a hamiltonian system satisfying the legendre condition is the class of invariant lagrangian graphs a classical result of the calculus of variations is that for such systems every orbit on an invariant lagrangian graph has minimal action this provides a simple criterion for non existence of invariant lagrangian graphs namely if an orbit has a variation which decreases the action or it possesses conjugate points then there is no invariant lagrangian graph containing it as an example the criterion is applied to the motion of a particle in the field of two waves to prove that there are no invariant tori with initial velocity between those of the two waves if the amplitudes of the waves exceed certain values

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

fractal boundary for the existence of invariant circles for area preserving maps observations and renormalisation explanation abstract breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters the critical line in parameter space turns out to have a cantor set of cusps which can be organized in a self similar tree a theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator the results of some other researchers on similar systems are shown to fit this explanation as well

singularities of periodic orbits near invariant curves abstract we show numerically for standard like maps how the singularities in the complex parameter of the function which conjugates the map to a rotation of rational period behave when the period goes to an irrational number furthermore we propose a numerical method to extrapolate the radius of convergence of the series parametrizing the solution of periodic orbits the results are compared with analyses performed by pade approximants greene s method root criterion and the prediction by renormalization theory

extended integrability and bi hamiltonian systems the current notion of integrability of hamiltonian systems was fixed by liouville in a famous 1855 paper it describes systems in a 2k dimensional phase space whose trajectories are dense on tori q or wind on toroidal cylinders m q m within liouville s construction the dimension q cannot exceed k and is the main invariant of the system in this paper we generalize liouville integrability so that trajectories can be dense on tori k of arbitrary dimensions q 1 2k 1 2k and an additional invariant v 2 q k v 2 q 2 can be recovered the main theorem classifies all k k 1 2 canonical forms of hamiltonian systems that are integrable in a newly defined broad sense an integrable physical problem having engineering origin is presented the notion of extended compatibility of two poisson structures is introduced the corresponding bi hamiltonian systems are shown to be integrable in the broad sense

theory of tensor invariants of integrable hamiltonian systems i incompatible poisson structures this paper develops a new theory of tensor invariants of a completely integrable non degenerate hamiltonian system on a smooth manifoldmn the central objects in this theory are supplementary invariant poisson structurespc which are incompatable with the original poisson structurep1 for this hamiltonian system a complete classification of invariant poisson structures is derived in a neighbourhood of an invariant toroidal domain this classification resolves the well known inverse problem that was brought into prominence by magri s 1978 paper deveoted to the theory of compatible poisson structures applications connected with the kam theory with the kepler problem with the basic integrable problem of celestial mechanics and with the harmonic oscillator are pointed out a cohomology is defined for dynamical systems on smooth manifolds the physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of poisson structures are introduced to study the diversity of pairs of poisson structures incompatible in magri s sense it is proved that if a dynamical systemv preserves two strongly dynamically compatible poisson structuresp1 andp2 in a general position then this system is completely integrable such a systemv generates a hierarchy of integrable dynamical systems which in general are not hamiltonian neither with respect top1 nor with respect top2 necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible poisson structures

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

determination of the threshold of the break up of invariant tori in a class of three frequency hamiltonian systems we consider a class of hamiltonians with three degrees of freedom that can be mapped into quasi periodically driven pendulums the purpose of this paper is to determine the threshold of the break up of invariant tori with a specific frequency vector we apply two techniques the frequency map analysis and renormalization group methods the renormalization transformation acting on a hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the hamiltonian and a rescaling of phase space around the considered torus we give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break up of invariant tori furthermore we obtain by these methods numerical values of the threshold of the break up of the last invariant torus

a remark on the kam theorem applied to a four vortex system we consider a planar four vortex system with unit intensities and apply the kam theorem for two dimensional tori with fixed frequency we obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation number 5 1 2 and compare our result with numerical experiments

bounds on the trajectories of a system of weakly coupled rotators in this paper we study a classical mechanical system of weakly coupled rotators on a one dimensional lattice such systems are of interest in statistical mechanics we prove that for any site in the system there is a large set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site in the transformed system is essentially indistinguishable from that of an integrable system for a long but finite time alternatively the trajectory of this site lies very close to a torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators all the estimates in this theory areindependent of the number of degrees of freedom in the system we propose this mechanism as an explanation of certain numerical experiments

critical kam circles and the brjuno function the authors consider a complex hamiltonian map and show how to approximate the critical value of the perturbation parameter at which a given kam circle disappears by means of a function the brjuno function which only depends on the continued fraction expansion of the rotation number

a posteriori kam theory with optimal estimates for partially integrable systems abstract in this paper we present a posteriori kam results for existence of d dimensional isotropic invariant tori for n dof hamiltonian systems with additional n d independent first integrals in involution we carry out a covariant formulation that does not require the use of action angle variables nor symplectic reduction techniques the main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates which may cause difficulties and underperformance when quantifying the hypotheses of the kam theorem in such reduced coordinates the results include ordinary and generalized iso energetic kam theorems the approach is suitable to perform numerical computations and computer assisted proofs

bryuno function and the standard map for the standard map the homotopically non trivial invariant curves of rotation number satisfying the bryuno condition are shown to be analytic in the perturbative parameter e provided e is small enough the radius of convergence of the lindstedt series sometimes called critical function of the standard map is studied and the relation with the bryuno function b is derived the quantity log 2 b is proved to be bounded uniformly in

singularity analysis by pade approximants of some holomorphic maps the analysis of the singularity structure of some series arising in the conjugation theory of holomorphic maps is performed using the pade approximants the validity of pa is checked for functions whose analytic structure is known one observes that the numerical results are extremely sensitive to errors due to the use of a finite precision normal or double precision used in fortran codes is in most cases not sufficient to perform a numerical analysis of the singularity structure in order to have significant results at high orders all the computations were carried out using codes which allow us to operate with a sufficiently large number of digits the case of linearizable diffeomorphisms and of mappings tangent to the identity of c o was considered pa of both the direct and the inverse conjugating functions of a quadratic map to its normal form were computed and a comparison was performed with analytical results when available when the singularity pattern was unknown the poles and the zeros of the pa provided a coherent picture which in some cases allowed rigorous results to be established

a theorem on sequentiality and compactness with its fuzzy extension abstract an amended proof of a theorem of franklin s on sequentiality and sequential compactness is presented and a corresponding fuzzy version formulated using the theory of quasi coincidence

natural boundaries for area preserving twist maps we consider kam invariant curves for generalizations of the standard map of the form x y x y y f x wheref x is an odd trigonometric polynomial we study numerically their analytic properties by a pade approximant method applied to the function which conjugates the dynamics to a rotation in the complex plane natural boundaries of different shapes are found in the complex plane the analyticity region appears to be a strip bounded by a natural boundary whose width tends linearly to 0 as tends to the critical value

computation of domains of analyticity for some perturbative expansions of mechanics abstract we compute the domain of analyticity of some perturbative expansions for invariant circles appearing in mechanics we use pade approximants for the perturbative expansions and introduce methods to ascertain their domain of convergence we also use non perturbative methods based on direct computation of the invariant circles and in analogy with greene s criterion approximation by circles with rational rotation we find that the domains computed by all the methods agree within the limits of accuracy we also study rigorously the nature of the singularities when the frequency is rational

a renormalization group explanation of numerical observations of analyticity domains a recent paper considers the dependence of the size of analyticity domains of some functions appearing in kam theory as a function of the distance to breakdown they tentatively conclude that the relation is linear in this note we argue that mckay s renormalization group picture predicts a power law dependence with an exponent close to 1 but not equal to 1

variational method for finding periodic orbits in a general flow a variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems an initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop the newton descent partial differential equation that governs this evolution is an infinitesimal step version of the damped newton raphson iteration the feasibility of the method is demonstrated by its application to the henon heiles system the circular restricted three body problem and the kuramoto sivashinsky system in a weakly turbulent regime

renormalization method for computing the threshold of the large scale stochastic instability in two degrees of freedom hamiltonian systems an approximate renormalization procedure is derived for the hamiltonianh v x t v2 2 m cosx p cosk x t it gives an estimate of the large scale stochastic instability threshold which agrees within 5 10 with the results obtained from direct numerical integration of the canonical equations it shows that this instability is related to the destruction of kam tori between the two resonances and makes the connection with kam theory possible improvements of the method are proposed the results obtained forh allow us to estimate the threshold for a large class of hamiltonian systems with two degrees of freedom

about the existence of recursion operators for completely integrable hamiltonian systems near a liouville torus it is shown that the existence of a recursion operator near a liouville torus of a completely integrable hamiltonian system is not always satisfied and that such an existence implies very strong conditions on the hamiltonian and on the actions

accurate strategies for small divisor problems many interesting problems in mechanics are close to systems that can be solved exactly in such systems it is natural to consider perturbative expansions about the known system frequently it is possible to compute all of the terms in the expansion recursively which can lead to expressions involving denominators that become arbitrarily small

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

hamiltonian time dependent mechanics the usual formulation of time dependent mechanics implies a given splitting y r m of an event space y this splitting however is broken by any time dependent transformation including transformations between inertial frames the goal is the frame covariant formulation of time dependent mechanics on a bundle y r whose fibration y m is not fixed its phase space is the vertical cotangent bundle v y provided with the canonical 3 form and the corresponding canonical poisson structure an event space of relativistic mechanics is a manifold z whose fibration z r is not fixed

complete involutive algebras of functions on cotangent bundles of homogeneous spaces homogeneous spaces of all compact lie groups admit riemannian metrics with completely integrable geodesic flows by means of c smooth integrals 9 10 the purpose of this paper is to give some constructions of complete involutive algebras of analytic functions polynomial in velocities on the co tangent bundles of homogeneous spaces of compact lie groups this allows us to obtain new integrable riemannian and sub riemannian geodesic flows on various homogeneous spaces such as stiefel manifolds flag manifolds and orbits of the adjoint actions of compact lie groups

computer assisted proofs in analysis and programming in logic a case study in this paper we present a computer assisted proof of the existence of a solution for the feigenbaum equation varphi x frac 1 lambda varphi varphi lambda x there exist by now various such proofs in the literature although the one presented here is new the main purpose of this paper is not to provide yet another version but to give an easy to read and self contained introduction to the technique of computer assisted proofs in analysis our proof is written in prolog programming in logic a programming language which we found to be well suited for this purpose in this paper we also give an introduction to prolog so that even a reader without prior exposure to programming should be able to verify the correctness of the proof

on the numerical treatment of a small divisor problem quasiperiodic solutions of perturbed integrable hamiltonian equations such as weakly coupled harmonic oscillators can be found by constructing an appropriate coordinate transformation which leads to a small divisor problem however the numerical difficulties are not merely caused by the small divisors but rather by the appearence of ghost solutions which appear in any reasonable discretization of the problem our numerical treatment based on a newton type iteration guarantees an approximation of the relevant solution of the nonlinear problem numerical solutions are found up to a critical value of the coupling constant which is much larger than the coupling constants allowed by the existence theory available so far

submersions fibrations and bundles when does a submersion have the homotopy lifting property when is it a locally trivial fibre bundle we establish characterizations in terms of consistency in the topology of the neighbouring fibres

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

high order control for symplectic maps abstract we revisit the problem of introducing an a priori control for devices that can be modeled via a symplectic map in a neighborhood of an elliptic equilibrium using a technique based on lie transform methods we produce a normal form algorithm that avoids the usual step of interpolating the map with a flow the formal algorithm is completed with quantitative estimates that bring into evidence the asymptotic character of the normal form transformation then we perform an heuristic analysis of the dynamical behavior of the map using the invariant function for the normalized map finally we discuss how control terms of different orders may be introduced so as to increase the size of the stable domain of the map the numerical examples are worked out on a two dimensional map of henon type

a method for accurate stability bounds in a small denominator problem the author considers the problem of obtaining realistic lower bounds for the siegel radius recent advances of the analysis of siegel disks allows one to give a very accurate numerical algorithm based on rigorous results he finds that for non quadratic polynomial maps the maximal siegel radius might correspond to rotation numbers different from the golden mean

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

realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory part ii the so called problem of the realization of the holonomic constraints of classical mechanics is here revisited in the light of nekhoroshev like classical perturbation theory precisely if constraints are physically represented by very steep potential wells with associated high frequency transversal vibrations then one shows that within suitable assumptions the vibrational energy and the energy associated to the constrained motion are separately almost constant for a very long time scale growing exponentially with the frequency i e with the rigidity of the constraint one aims to realize this result can also be applied to microscopic physics providing a possible entirely classical mechanism for the freezing of the high frequency degrees of freedom in terms of non equilibrium statistical mechanics according to some ideas expressed by boltzmann and jeans at the turn of the century in this part i we introduce the problem and prove a first theorem concerning the realization of a single constraint within a system of any number of degrees of freedom the problem of the realization of many constraints will be considered in a forthcoming part ii

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

on the symmetry lines of the standard mapping abstract the taylor and chirikov standard mapping is used as a model to represent the dynamics of non integrable systems by means of the poincare section of the trajectories intersecting a fixed plane of the phase space this study is based on a systematic use of the decomposition of the standard mapping in a product of two involutions as was birkhoff s in the restricted three body problem an infinite family of involutions is generated by these two involutions forming a group including the iterates of the mapping and their inverses the symmetry lines of the family of involutions determine the position of the periodic points and explain the onset of new chains of periodic points produced by the winding of the symmetry lines around the stable periodic points

noncommutative integrability on noncompact invariant manifolds the mishchenko fomenko theorem on noncommutative completely integrable hamiltonian systems on a symplectic manifold is extended to the case of noncompact invariant submanifolds

invariant curves for area preserving twist maps far from integrable one parameter families of area preserving twist maps of the formf e x y x y ef x y ef x are considered various invariant curves for the maps corresponding tof x sin andf x sinx 1 50 sin 5x are rigorously constructed forlarge values of the nonlinearity parameter for larger values of close to critical some numerical experiments are briefly discussed

breakdown of lindstedt expansion for chaotic maps in a previous paper of one of us europhys lett 59 330 336 2002 the validity of greene s method for determining the critical constant of the standard map sm was questioned on the basis of some numerical findings here we come back to that analysis and we provide an interpretation of the numerical results by showing that the conclusions of that paper were wrong as they relied on a plausible but untrue assumption hence no contradiction exists with respect to greene s method we show that the previous results based on the expansion in lindstedt series do correspond to the critical constant but for a different map the semi standard map ssm for such a map no greene s method analog is at disposal so that methods based on lindstedt series are essentially the only possible ones moreover we study the expansion for two simplified models obtained from the sm and ssm by suppressing the small divisors we call them the simplified sm and simplified ssm respectively the first case turns out to be re

the kam theory of systems with short range interactions i the existence of quasiperiodic trajectories for hamiltonian systems consisting of long chains of nearly identical subsystems with interactions which decay rapidly with increasing distance between the interacting components is studied such models are of interest in statistical mechanics it is shown that nonergodic motions persist for much larger perturbations than prior work indicated if the number of degrees of freedom of the system isn the allowed perturbation decreases only as an inverse power ofn as the number of degrees of freedom increases rather than the inverse power ofn which previous estimates yielded

a rigorous partial justification of greene s criterion we prove several theorems that lend support to greene s criterion for the existence or not of invariant circles in twist maps in particular we show that some of the implications of the criterion are correct when the aubry mather sets are smooth invariant circles or uniformly hyperbolic we also suggest a simple modification that can work in the case that the aubry mather sets have nonzero lyapunov exponents the latter is based on a closing lemma for sets with nonzero lyapunov exponents which may have several other applications

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

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

numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors we study numerically the complex domains of validity for kam theory in generalized standard mappings we compare methods based on pade approximants and methods based on the study of periodic orbits

recent developments in the theory of hamiltonian systems area preserving mappings of an annulus occur as poincare mappings of hamiltonian systems they were studied extensively by g d birkhoff recently aubry and mather investigated the subclass of so called monotone twist mappings for which they constructed independently closed invariant cantor sets their work led to important new results their theory is discussed and related to hamiltonian systems satisfying a legendre condition the connection of this theory with the stability problem kam theory and in particular the disintegration of invariant tori is discussed various constructions of mather sets are explained in which minimal solutions of variational problems play a central role this theory has a close relation to the differential geometric investigations by morse and hedlund on geodesics on two dimensional surfaces

rigorous estimates for a computer assisted kam theory nonautonomous hamiltonian systems of one degree of freedom close to integrable ones are considered let e be a positive parameter measuring the strength of the perturbation and denote by e c the critical value at which a given kam kolmogorov arnold moser torus breaks down a computer assisted method that allows one to give rigorous lower bounds for e c is presented this method has been applied in celletti falcolini porzio to be published in ann inst h poincare to the escande and doveil pendulum yielding a bound which is within a factor 40 2 of the value indicated by numerical experiments

global action angle coordinates for completely integrable systems with noncompact invariant submanifolds the obstruction to the existence of global action angle coordinates of abelian and noncommutative non abelian completely integrable systems with compact invariant submanifolds has been studied we extend this analysis to the case of noncompact invariant submanifolds

le probleme general des variables actions angles feuilletages symplectiquement complets monodromie des fibrations isotropes symplectiquement completes structure locale et theoreme des variables action angles classe de chern realisation d une variete de poisson au voisinage d une feuille compacte