center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

multi stability in monotone input output systems this paper studies the emergence of multi stability and hysteresis in those systems that arise under positive feedback starting from monotone systems with well defined steady state responses such feedback configurations appear routinely in several fields of application and especially in biology characterizations of global stability behavior are stated in terms of easily checkable graphical conditions

on a class of nonlinear dispersive dissipative interactions abstract we study a prototype dissipative dispersive equation ut a um x un xxx uk xx a consts which represents a wide variety of interactions at the critical value k m n 2 which separates dispersive and dissipation dominated phenomena these effects are in a detailed balance and the patterns formed do not depend on the amplitude in particular when m n 2 k 1 the equation can be transformed into a form free of convection and dissipation making it accessible to analysis both bounded and unbounded oscillations as well as solitary waves are found a variety of exact solutions are presented with a notable example being a solitary doublet for n 1 and a 2 3 2 the problem may be mapped into a linear equation leading to rational periodic or aperiodic solutions among others

ck invariant manifolds for maps of banach spaces abstract in this work we give a unified proof for the existence under appropriate gap conditions of the standard invariant manifolds for a ck map of a banach space near a fixed point where k 1 is an integer

fer s factorization as a symplectic integrator in this paper we analyze the main features of the so called fer expansion as a new method for integrating the ordinary differential equations derived from explicitly time dependent hamiltonian dynamical systems this method is based on a factorization of the evolution operator as an infinite product of exponentials of lie operators and thus exactly preserves the poincare integral invariants third and fourth order expansions are considered and numerical results are presented for a quadratic hamiltonian with various time dependent frequencies comparison is done with other numerical integration schemes

smooth conjugacy and s r b measures for uniformly and non uniformly hyperbolic systems we give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional anosov systems we also show that if a homeomorphism conjugating two smooth low dimensional anosov systems is absolutely continuous then it is as smooth as the maps we furthermore prove generalizations of these facts for non uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions

sharp regularity of linearization for c 1 1 hyperbolic diffeomorphisms c 1 linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions however c 1 smoothness is not enough to guarantee c 1 linearization for c 1 1 hyperbolic diffeomorphisms on banach spaces c 1 linearization was proved under a gap condition together with a band condition of the spectrum in this paper the result of c 1 linearization in banach spaces is strengthened to c 1 beta linearization with a constant beta 0 under a weaker band condition by a decomposition with invariant foliations the weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum furthermore we give an estimate for the exponent beta and prove that the estimate is sharp in the planar case

monotone systems under positive feedback multistability and a reduction theorem for feedback loops involving single input single output monotone systems with well defined i o characteristics a recent paper by angeli and sontag provided an approach to determining the location and stability of steady states a result on global convergence for multistable systems followed as a consequence of the technique the present paper extends the approach to multiple inputs and outputs a key idea is the introduction of a reduced system which preserves local stability properties

local contractions of banach spaces and spectral gap conditions abstract we study the linearization problem for a c k 1 k 1 contraction of a banach space e near a fixed point which satisfies a spectral gap condition and a narrow band condition both of order k we also assume that the part of the spectrum in each band satisfies a finite non resonant condition of order k relative to itself together with the part that lies in the larger bands we show that there is a c k linearization for sufficiently small 0 we give a precise estimate on in terms of the gap and band conditions

weierstrass s criterion and compact solitary waves weierstrass s theory is a standard qualitative tool for single degree of freedom equations used in classical mechanics and in many textbooks in this brief report we show how a simple generalization of this tool makes it possible to identify some differential equations for which compact and even semicompact traveling solitary waves exist in the framework of continuum mechanics these differential equations correspond to bulk shear waves for a special class of constitutive laws

a geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of 2 we give a proof based in geometric perturbation theory of a result proved by j n mather using variational methods namely the existence of orbits with unbounded energy in perturbations of a generic geodesic flow in 2 by a generic periodic potential

invariant pre foliations for non resonant non uniformly hyperbolic systems given an orbit whose linearization has invariant subspaces satisfying some non resonance conditions in the exponential rates of growth we prove existence of invariant manifolds tangent to these subspaces the exponential rates of growth can be understood either in the sense of lyapunov exponents or in the sense of exponential dichotomies these manifolds can correspond to slow manifolds which characterize the asymptotic convergence let x i i n be a regular orbit of a c 2 dynamical system f let s be a subset of its lyapunov exponents assume that all the lyapunov exponents in s are negative and that the sums of lyapunov exponents in s do not agree with any lyapunov exponent in the complement of s denote by e s xi the linear spaces spanned by the spaces associated to the lyapunov exponents in s we show that there are smooth manifolds w s xi such that f w s xi w s xi 1 and t xi w s xi e s xi we establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non resonance conditions these systems of invariant manifolds are not in general a foliation

differentiability and analyticity of topological entropy for anosov and geodesic flows in this paper we investigate the regularity of the topological entropyh top forc k perturbations of anosov flows we show that the topological entropy varies almost as smoothly as the perturbation the results in this paper along with several related results have been announced in kkpw

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

exponential infinite product representations of the time displacement operator fer s infinite product expansion 1958 is reconsidered and applied to the specific case of the time displacement operator in quantum mechanics an alternative version of this expansion due to wilcox is also discussed and found to be quite different from the original one in general the latter is expected to possess better convergence properties

stable manifolds to bounded solutions in possibly ill posed pdes abstract we prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations these equations may be ill posed a single specific solution an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions in the later cases which include several orbits we also establish the invariant manifolds of an orbit depend smoothly on the orbit analytically in the case of quasi periodic orbits and finitely differentiably in the case of more general families we first establish a general abstract theorem which under suitable spectral non degeneracy analyticity assumptions on the linearized equation establishes the existence of the desired manifold related results appear in the literature but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds this makes the results in this paper applicable to some several models of current interest that could not be treated otherwise we discuss in detail the boussinesq equation of water waves similar phenomena happen in other long wave approximations and complex ginzburg landau equation more recently we observed 11 that our results also apply to mean field games since the equations we consider may be ill posed part of the requirements for the stable manifold is that one can define the forward dynamics on them note also that the methods that are based in the existence of dynamics such as graph transform do not apply to ill posed equation we use the methods based on integral equations perron method associated with the partial dynamics but we need to take advantage of smoothing properties of the partial dynamics note that even if the families of solutions we started with are finite dimensional the stable manifolds may be infinite dimensional

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

compact solitary waves in linearly elastic chains with non smooth on site potential it was recently observed by saccomandi and sgura that one dimensional chains with nonlinear elastic interaction and regular on site potential can support compact solitary waves i e travelling solitary waves with strictly compact support in this paper we show that the same applies to chains with linear elastic interaction and an on site potential which is continuous but non smooth at minima some different features arise in particular the speed of compact solitary waves is not uniquely fixed by the equation we also discuss several generalizations of our findings

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

sternberg theorems for random dynamical systems in this paper we prove the smooth conjugacy theorems of sternberg type for random dynamical systems based on their lyapunov exponents we also present a stable and unstable manifold theorem with tempered estimates that are used to construct conjugacy 2005 wiley periodicals inc

continuous numerical solutions of coupled mixed partial differential systems using fer s factorization in this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate time dependent systems of the type ut a t uxx b t u 0 0 0 u 0 t u p t 0 u x 0 f x 06 x6p after truncation of an exact series solution the numerical solution is constructed using fer s factorization given 0 and t0 t1 with 0 t0 t1 and d t0 t1 f x t 06x6p t06t6t1g the error of the approximated solution with respect to the exact series solution is less than uniformly in d t0 t1 an algorithm is also included c 1999 elsevier science b v all rights reserved ams classication 65m15 34a50 35c10 35a50

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

differentiability at the tip of arnold tongues for diophantine rotations numerical studies and renormalization group explanations we study numerically the regularity of arnold tongues corresponding to diophantine rotation numbers of circle maps at the edge of validity of kam theorem this serves as a good test for the numerical stability of two different algorithms we find empirically that arnold tongues are only finitely differentiable at the tip we also find several scaling properties of the sobolev norms of the conjugacy near the breakdown we also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the sobolev regularity we also uncover empirically some other patterns which require explanation

compactlike discrete breathers in systems with nonlinear and nonlocal dispersive terms discrete breathers with purely anharmonic short range interaction potentials localize superexponentially becoming compactlike we analyze their spatial localization properties and their dynamical stability several branches of solutions are identified one of them connects to the well known page and sievers takeno lattice modes another one connects with the compacton solutions of rosenau the absence of linear dispersion allows for extremely long lived time quasiperiodic localized excitations adding long range anharmonic interactions leads to an extreme case of competition between length scales defining the spatial breather localization we show that short and long range interaction terms competition results in the appearance of several characteristic crossover lengths and essentially breaks the concept of compactness of the corresponding discrete breathers

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

discrete compactons some exact results in this letter we use the method of constructing exact solutions on lattices proposed by kinnersley and described in schmidt 1979 phys rev b 20 4397 to obtain exact compacton solutions in discrete models we examine the linear stability of such solutions both for the bright compacton and for the dark compacton cases we focus on a quantization condition that the width of the profile should satisfy we also use this quantization condition to examine the possibility of compact coherent structures travelling in discrete settings our results are obtained for sinusoidal profiles and then generalized to elliptic functions of arbitrary modulus the possibility of multi compacton solutions is considered

traveling wave solutions of the bbm like equations in this work we apply the factorization technique to the benjamin bona mahony like equations b m n in order to get traveling wave solutions we will focus on some special cases for which m n and we will obtain these solutions in terms of the special forms of weierstrass functions

livsic theorem for matrix cocycles we prove the liv sic theorem for arbitrary gl m r cocycles we con sider a hyperbolic dynamical system f x x and a holder continuous function a x gl m r we show that if a has trivial periodic data i e a f n 1 p a fp a p id for each periodic point p f n p then there exists a holder continuous function c x gl m r satisfying a x c fx c x 1 for all x2 x the main new ingredients in the proof are results of independent interest on relations between the periodic data lyapunov exponents and uniform estimates on growth of products along orbits for an arbitrary holder function a

ck centre unstable manifolds we consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps we show that every global centre unstable manifold can be expressed as a graph of a c k map provided that the nonlinearities are c k smooth the proofs are based on a lemma by d henry on a necessary and sufficient condition for a lipschitz map to be continuously differentiable

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

c1 linearization for planar contractions c1 linearization is of special interests because it can distinguish characteristic directions of dynamical systems it is known that planar c1 contractions with a fixed point at the origin o admit c1 linearization with sufficiently small 0 if 1 and admit c1 linearization if log 1 log 2 1 1 where 1 and 2 are eigenvalues of the linear parts of the contractions at o with 0 1 2 1 in this paper we improve the lower bound of to lower the condition of c1 linearization for planar contractions furthermore we prove that the derivatives of transformations in our c1 linearization are holder continuous and give estimates for the holder exponent finally we give a counter example to show that those estimates cannot be improved anymore

invariants for smooth conjugacy of hyperbolic dynamical systems ii we show that the eigenvalues of the derivatives at periodic points form a complete set of invariants for smooth local conjugacy of anosov diffeomorphisms oft2

sub stable and weak stable manifolds associated with finitely non resonant spectral subspaces in this work we study c k delta 2 leq k leq infty 0 leq delta leq 1 maps of a banach space near a fixed point we show the existence and uniqueness of a class of c k delta local invariant sub manifolds of the stable manifold which correspond to a spectral subspace satisfying a finite non resonance condition of order l leq k and an overriding condition of order l leq k condition 3 of theorem 1 we study the dependence of these invariant manifolds on a parameter that lies in a banach space we also show that a c k delta local weak stable manifold that satisfies these two conditions is unique in the class of c k delta maps the uniqueness is due to the fact that our method does not require a cut off function an infinite dimensional banach space does not always admit smooth cut off functions

magnus and fer expansions for matrix differential equations the convergence problem approximate solutions of matrix linear differential equations by matrix exponentials are considered in particular the convergence issue of magnus and fer expansions is treated upper bounds for the convergence radius in terms of the norm of the defining matrix of the system are obtained the very few previously published bounds are improved bounds to the error of approximate solutions are also reported all results are based just on algebraic manipulations of the recursive relation of the expansion generators

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

on invariant manifolds and invariant foliations without a spectral gap abstract it is known that a spectral gap condition is required for the existence and smoothness of invariant manifolds and invariant foliations near a fixed point however any planar mapping with a unipotent linear part having a center manifold on the whole phase plane does not have such a gap in this paper we give the existence and smoothness of invariant manifolds which are invariant submanifolds of the center manifold for the aforementioned planar mapping corresponding to the invariant submanifolds we further prove the existence and smoothness of invariant foliations on the center manifold

computation of nonlinear behavior of hamiltonian systems using lie algebraic methods lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general hamiltonian systems a procedure is presented for the computation of nonlinear effects of arbitrarily high degree and explicit formulas are given through effects of degree 5 expected applications include accelerator design charged particle beam and light optics other problems in the general area of nonlinear dynamics and perhaps with suitable modification the area of s matrix expansions in quantum field theory

some remarks on pathologies of renormalization group transformations for the ising model the results recently obtained by van enter fernandez and sokal on non gibbsianness of the measurev t b h arising from the application of a single decimation transformationt b with spacingb to the gibbs measure h of the ising model for suitably chosen large inverse temperature and nonzero external fieldh are critically analyzed in particular we show that if keeping fixed the same values of h andb one iterates a sufficiently large number of timesn the transformationt b one obtains a new measurev t b n h which is gibbsian and moreover very weakly coupled

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

bootstrap for local rigidity of anosov automorphisms on the 3 torus we establish a strong form of local rigidity for hyperbolic automorphisms of the 3 torus with real spectrum namely let l mathbb t 3 to mathbb t 3 be a hyperbolic automorphism of the 3 torus with real spectrum and let f be a c 1 small perturbation of l then f is smoothly c infty conjugate to l if and only if obstructions to c 1 conjugacy given by the eigenvalues at periodic points of f vanish by combining our result and a local rigidity result of kalinin and sadovskaya ergod theory dyn syst 29 117 136 2009 for conformal automorphisms this completes the local rigidity program for hyperbolic automorphisms in dimension 3 our work extends de la llave marco moriyon 2 dimensional local rigidity theory commun math phys 109 368 378 1987 ergod theory dyn syst 17 3 649 662 1997 commun math phys 109 4 681 689 1987

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

invariant manifolds associated to nonresonant spectral subspaces we show that if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain nonresonance conditions the map leaves invariant a smooth manifold tangent to this subspace this manifold is as smooth as the map when the smoothness is measured in appropriate scales but is unique amongcl invariant manifolds wherel depends only on the spectrum of the linearization or on some more general smoothness classes that we detail we show that if the nonresonance conditions are not satisfied a smooth invariant manifold need not exist and we also establish smooth dependence on parameters we also discuss some applications of these invariant manifolds and briefly survey related work

period doubling bifurcations for families of maps on ℝ n infinite sequences of period doubling bifurcations in one parameter families 1 pf of maps enjoy very strong universality properties this is known numerically in a multitude of cases and has been shown rigorously for certain 1 pf of maps on the interval these bifurcations occur in 1 pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 4 6692 in this paper we indicate the main steps of a proof that the same is true for 1 pf of analytic maps from n to n whose restriction to n is real

super exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients abstract we show that all eigenfunctions of linear partial differential operators in r n with polynomial coefficients of shubin type are extended to entire functions in c n of finite exponential type 2 and decay like exp z 2 for z in conic neighbourhoods of the form im z re z we also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super exponential decay of the above type whereas a loss of the holomorphicity occurs namely we show holomorphic extension into a strip z c n im z t for some t 0 the proofs are based on geometrical and perturbative methods in gelfand shilov spaces the results apply in particular to semilinear schrodinger equations of the form u x 2 u u f x u u our estimates on homoclinics are sharp in fact we exhibit examples of solutions of with super exponential decay which are meromorphic functions the key point of our argument being the celebrated great picard theorem in complex analysis

a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps rigorous results abstract in this paper we prove rigorous results on persistence of invariant tori and their whiskers the proofs are based on the parameterization method of x cabre e fontich r de la llave the parameterization method for invariant manifolds i manifolds associated to non resonant subspaces indiana univ math j 52 2 2003 283 328 x cabre e fontich r de la llave the parameterization method for invariant manifolds ii regularity with respect to parameters indiana univ math j 52 2 2003 329 360 the invariant manifolds results proved here include as particular cases of the usual strong stable and strong unstable manifolds but also include other non resonant manifolds the method lends itself to numerical implementations whose analysis and implementation is studied in a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical algorithms preprint 2005 a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical implementation and examples preprint 2005 the results are stated as a posteriori results namely that if one has an approximate solution which is not degenerate then one has a true solution not too far from the approximate one this can be used to validate the results of numerical computations

lie algebraic approach to fer s expansion for classical hamiltonian systems the so called fer s expansion is proposed as a solution for the time evolution operator of classical time dependent hamiltonian systems the quadratic hamiltonians treated as examples show that under very different regimes the second order approximation already gives extremely good results

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

on a regularity problem occurring in connection with anosov diffeomorphisms let be ac manifold and s and u be two holder foliations transverse and with uniformlyc leaves if a functionf is uniformlyc along the leaves of the two foliations then it isc on the proof is elementary

c1 hartman theorem for random dynamical systems abstract the c 1 hartman theorem states that every c 1 1 contraction in r d admits c 1 linearization in this paper we prove c 1 linearizability for c 1 contractive or expansive random diffeomorphisms with appropriate constants 0 1 and 0 1 which implies a random version of the c 1 hartman theorem moreover the conjugacy is proved to be tempered this result strengthens related results even in the deterministic case lowering the c 1 1 condition to c 1 and enhancing the c 1 smoothness of the conjugacy to c 1 in r d in the proof we use a smooth weak stable invariant manifold in the random sense to construct the conjugacy and to overcome difficulties from nonuniformity and measurability