method and apparatus for encoding envelopes of harmonic signals and method and apparatus for decoding envelopes of harmonic signals provided are a methods and apparatuses for encoding decoding an audio signal to efficiently encode decode a harmonic envelope the method of encoding an audio signal includes performing harmonic analysis with respect to an input signal to determine harmonic parameters with respect to harmonic signals correlating the amplitudes of the harmonic signals to signals in a time domain instead of signals in a frequency domain applying a time frequency transformation operation to the amplitudes in the time domain to generate time frequency transformed values in the frequency domain and encoding the time frequency transformed values when expressing a harmonic envelope the amplitudes of the harmonic signals are regarded as signals in the time domain so as to perform a time frequency transformation and only a part from among the transformed values is selected to be encoded therefore sound quality is not affected and coding efficiency greatly improves

dichotomies and asymptotic solutions of nonlinear differential systems the problem of asymptotic equivalence between two systems of ordinary differential equations has been studied by many authors l 5 it was first used by levinson in his theorem on asymptotic integration 6 71 in l 51 this problem is studied when system 1 2 is stable a t is constant and g t y b t is integrable in 3 theorems 5 111 it is assumed that the linear system 1 2 has an ordinary dichotomy and the nonlinear term g t y is lipschitz continuous in 2 the previous results are extended to nonlinear perturbations g t y which are not lipschitz continuous such that g t y l i r t y where r t y is monotone in y we emphasize that in all these works only the asymptotic equivalence between bounded solutions is established nothing is said about a possible correspondence between unbounded solutions of 1 1 and 1 2 in this paper we suppose that the nonlinear term g t y satisfies some integrability conditions that we summarize in hypothesis h2 these conditions permit one to consider systems with large nonlinearities for example equations where g t y is an oscillatory function of the variable y no lipschitz conditions or monotonicity properties are required of g t y however we will use dichotomic properties of system 1 2

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

quantization of harmonic amplitudes representing speech in a speech coding and decoding system in which a timewise segment of an acoustic speech signal is represented by a frame of a data signal characterized by a fundamental frequency and spectral harmonics a current frame is reconstructed using a set of prediction signals based on the number of spectral harmonics for the current frame and a preceding frame and reconstructed signal parameters characterizing the preceding frame the number of spectral harmonics for the current and preceding frames are reconstructed from at least a pair of digitally encoded signals that are generated using error protection codes for all of their bits

gradient schrodinger operators manifolds with density and applications abstract the aim of this paper is twofold on the one hand the study of gradient schrodinger operators on manifolds with density we classify the space of solutions when the underlying manifold is parabolic as an application we extend the naber yau liouville theorem and we prove that a complete manifold with density is parabolic if and only if it has finite capacity moreover we show that the linear space given by the kernel of a nonnegative gradient schrodinger operator is one dimensional provided there exists a bounded function in the kernel and the underlying manifold is parabolic on the other hand the topological and geometrical classification of complete weighted h stable hypersurfaces immersed in a manifold with density n g satisfying a lower bound on its bakry emery ricci tensor also we classify weighted stable surfaces in a three manifold with density whose perelman scalar curvature in short p scalar curvature together with the squared norm of the gradient of the density satisfy a lower bound finally we discuss the relationship of manifolds with density mean curvature flow mcf ricci flow and optimal transportation theory we obtain classification results for stable self similar solutions to the mcf and also for stable translating solitons to the mcf as far as we know the first classification result moreover for gradient ricci solitons we recover the hamilton ivey perelman classification assuming only a l 2 type bound on the scalar curvature and a inequality between the scalar curvature and the ricci tensor we also classify gradient ricci solitons when the scalar curvature does not change sign we finish by classifying critical transportation plans for the boltzman entropy on parabolic manifolds in particular we recover the case of the gaussian measure

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

robustness of discrete dynamics via lyapunov sequences for a nonautonomous dynamics with discrete time defined by a sequence of matrices we give a complete characterization of nonuniform exponential contractions and nonuniform exponential dichotomies in terms of lyapunov sequences we note that these include as very special cases uniform exponential contractions and uniform exponential dichotomies due to the central role played by these properties in a substantial part of the theory of dynamical systems in particular in connection with the study of stable and unstable invariant manifolds it is important to have available optimal characterizations that are more amenable to check whether a given dynamics has such a property we also obtain inverse theorems that give explicitly lyapunov sequences for a given contraction or dichotomy as a nontrivial application we establish the robustness under sufficiently small linear perturbations both of nonuniform exponential contractions and nonuniform exponential dichotomies we emphasize that when compared to former work our proof of the robustness property is much simpler

lyapunov functions for strong exponential contractions abstract our main aim is to give a complete characterization of an exponential contraction in terms of lyapunov sequences for discrete time or lyapunov functions for continuous time in particular we obtain inverse theorems giving explicitly a quadratic lyapunov function for each exponential contraction we consider the general cases of 1 nonautonomous dynamics either obtained from a product of linear operators or from a nonautonomous differential equation respectively for discrete and continuous time 2 nonuniform exponential contractions in which the uniform stability is replaced by a nonuniform stability 3 strong exponential behavior in the sense that we have simultaneously lower and upper contraction bounds we emphasize that the two last properties are ubiquitous in the context of ergodic theory and in particular for almost all trajectories of any measure preserving flow which justifies completely the study of this general type of exponential behavior as a nontrivial application of our work we establish the robustness of any strong nonuniform exponential contraction that is the persistence of the asymptotic stability of a strong nonuniform exponential contraction under sufficiently small linear perturbations

a grobman hartman theorem for nonuniformly hyperbolic dynamics abstract we establish a version of the grobman hartman theorem in banach spaces for nonuniformly hyperbolic dynamics we also consider the case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time more precisely we consider sequences of lipschitz maps a m f m such that the linear parts a m admit a nonuniform exponential dichotomy and we establish the existence of a unique sequence of topological conjugacies between the maps a m f m and a m furthermore we show that the conjugacies are holder continuous with holder exponent determined by the ratios of lyapunov exponents with the same sign to the best of our knowledge this statement appeared nowhere before in the published literature even in the particular case of uniform exponential dichotomies although some experts claim that it is well known in this case we are also interested in the dependence of the conjugacies on the perturbations f m we show that it is holder continuous with the same holder exponent as the one for the conjugacies we emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial in particular we need to consider several additional lyapunov norms

apparatus for linear predictive coding and decoding of speech using residual wave form time access compression a speech coding and decoding apparatus for use in linear predictive coding of a speech signal includes coding apparatus having a pitch analyzing element for separating an analysis frame into one or more blocks and calculating the strength of correlativity between pitch periods of a residual waveform in each block a residual partially compresing element compresses the time axis of the residual waveform in a block having a high correlativity strength a residual quantizing element quantizes the compressed residual waveform while preferentially allotting quantization allotment bits to the compressed portion of the residual waveform decoding apparatus includes a residual inverse quantizing element for inversely quantizing the residual waveform by the same bit allotment a residual partially expanding element expands the compressed portion of the residual waveform to its original length and supplies it to a linear predictive synthesis filter

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

vector quantization in a harmonic speech coding arrangement a harmonic speech coding arrangement where vector quantization is used to improve speech quality parameters are determined at the analyzer 120 of an illustrative coding arrangement to model the magnitude and phase spectra of the input speech a first codebook of vectors is searched for a vector that closely approximates the difference between the true and estimated magnitude spectra a second codebook of vectors is searched for a vector that closely approximates the difference between the true and the estimated phase spectra indices and scaling factors for the vectors are communicated to the synthesizer 160 such that scaled vectors can be added into the magnitude and phase spectra for use at the synthesizer in generating speech as a sum of sinusoids

characterizations of dichotomies of evolution equations on the half line abstract in this paper we investigate the characterization of dichotomies of an evolution family u u t s t s 0 of bounded linear operators on banach space x we introduce operators i0 and ix on subspaces of lp r x using the integral equation u t u t s u s ts u t f d the exponential and ordinary dichotomies of u are characterized by properties of i0 ix

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

transverse homoclinic orbits in periodically perturbed parabolic equations on considere le systeme autonome x ax g x ou a est un operateur sectoriel sur un espace de banach x on suppose que l equation differentielle a une solution bornee t telle que son equation variationnelle a une dichotomie exponentielle sur des demi droites on utilise la methode de lyapunov schmidt pour obtenir la fonction de bifurcation pour des solutions bornees de l equation perturbee non autonome

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

properties of bounded solutions of linear and nonlinear evolution equations homoclinics of a beam equation abstract the objective of this work is to discuss the existence bifurcation and regularity with respect to time and parameters of bounded solutions of infinite dimensional equations the authors present an application of their results to the study of homoclinic solutions of a nonlinear forced beam equation they use the approach of alternative method and semigroup theory

robustness via lyapunov functions for nonautonomous linear equations x a t x we give a complete characterization of nonuniform exponential dichotomies in terms of strict quadratic lyapunov functions nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies in particular we construct explicitly strict lyapunov functions for each exponential dichotomy as a nontrivial application we establish in a simple and direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations this represents a considerable simplification of former work

methods for speech quantization and error correction the redundancy contained within the spectral amplitudes is reduced and as a result the quantization of the spectral amplitudes is improved the prediction of the spectral amplitudes of the current segment from the spectral amplitudes of the previous is adjusted to account for any change in the fundamental frequency between the two segments the spectral amplitudes prediction residuals are divided into a fixed number of blocks each containing approximately the same number of elements a prediction residual block average prba vector is formed each element of the prba is equal to the average of the prediction residuals within one of the blocks the prba vector is vector quantized or it is transformed with a discrete cosine transform dct and scalar quantized the perceived effect of bit errors is reduced by smoothing the voiced unvoiced decisions an estimate of the error rate is made by locally averaging the number of correctable bit errors within each segment if the estimate of the error rate is greater than a threshold then high energy spectral amplitudes are declared voiced

exponential dichotomy of evolution equations and admissibility of function spaces on a half line consider an evolution family u u t s ts0 on a half line r and an integral equation u t u t s u s t s u t f d we characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of lp type the lorentz spaces lp q and many other function spaces occurring in interpolation theory we then apply our results to study the robustness of the exponential dichotomy of evolution families on a half line under small perturbations

robustness of nonuniform μ ν dichotomies in banach spaces abstract this paper focuses on the linear perturbation of dynamical systems in banach spaces which corresponds to a nonautonomous dynamical system admitting a nonuniform dichotomy we establish the robustness or roughness of nonuniform dichotomies in the sense that the linear dynamical system persists under a sufficiently small linear perturbation

the fredholm alternative and exponential dichotomies for parabolic equations abstract it is proved in this paper for parabolic equations that the fredholm alternative holds for bounded solutions if the homogeneous equation has exponential dichotomies on r and on r and moreover that the converse is also valid under the condition of bounded growth

parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights abstract let p be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight e h the aim of this paper is twofold first by assuming certain control on the h mean curvature of p we establish comparisons for the h capacity of extrinsic balls in p from which we deduce criteria ensuring the h parabolicity or h hyperbolicity of p second we employ functions with geometric meaning to describe submanifolds of bounded h mean curvature which are confined into some regions of the ambient manifold as a consequence we derive half space and bernstein type theorems generalizing previous ones our results apply for some relevant h minimal submanifolds appearing in the singularity theory of the mean curvature flow

smooth invariant manifolds in banach spaces with nonuniform exponential dichotomy we establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v a t v f t v in banach spaces assuming that the linear equation v a t v admits a nonuniform exponential dichotomy our proof of the ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of ck functions with holder continuous kth derivative this is a closed subset of the space of continuous functions with the supremum norm by an apparently not so well known lemma of henry see proposition 3 the estimates showing that the functions maintain the original bounds when transformed under the fixed point operator are obtained through a careful application of the faa di bruno formula for the higher derivatives of the compositions see 31 and 35 as a consequence we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class ck

transversality of homoclinic orbits and exponential dichotomies for parabolic equations abstract in this paper we discuss the existence of exponential dichotomies on r of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations and by making use of the results on exponential dichotomies of this paper we investigate the transversality of homoclinic orbits for parabolic equations

λ hypersurfaces on shrinking gradient ricci solitons abstract we consider an orientable smooth non totally geodesic hypersurface properly immersed on the cylinder shrinking soliton s 2 k 1 k r l g f where g is the product metric and the weight function f is defined by f p x x 2 4 if has weighted weak index less than or equal to l 1 we show that splits off a linear space also we classify the f stable orientable complete hypersurfaces properly immersed on s 2 k 1 k r g f with f p t t 2 4 obtaining as result s 2 k 1 k t or s 2 k 1 k 1 r

short time fourier analysis of sampled speech the theoretical basis for the representation of a speech signal by its short time fourier transform is developed a time frequency representation for linear time varying systems is applied to the speech production model to formulate a quasi stationary representation for the speech waveform short time fourier analysis of the resulting representation yields the relationship between the short time fourier transform of the speech and the speech production model

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

phase quantization method and apparatus a phase quantization method and apparatus in which the phase information of the input signal such as at the time of the sinusoidal synthesis encoding can be quantized efficiently the phase of the input signal derived from speech signals from an input terminal 11 is found by a phase detection unit 12 and scalar quantized by a scalar quantizer 13 the spectral amplitude weighting k of each harmonics is calculated by a weighting calculation unit 18 based on the lpc coefficients from a terminal 17 using the weighting k a bit allocation calculation unit 19 calculates an optimum number of quantization bits of respective harmonics to send the calculated optimum number to the scalar quantizer 13

conjugacies for linear and nonlinear perturbations of nonuniform behavior we construct topological conjugacies between linear and nonlinear evolution operators that admit either a nonuniform exponential contraction or a nonuniform exponential dichotomy we consider evolution operators defined by nonautonomous differential equations x a t x f t x in a banach space the conjugacies are obtained by first considering sufficiently small linear and nonlinear perturbations of linear equations x a t x in the case of linear perturbations we construct in a more or less explicit manner topological conjugacies between the two linear flows in the case of nonlinear perturbations we obtain a version of the grobman hartman theorem for nonuniformly hyperbolic dynamics furthermore all the conjugacies that we construct are locally holder continuous provided that the vectors fields are of class c1 as a byproduct of our approach we give conditions for the robustness of strong nonuniform exponential behavior in the sense that under sufficiently small perturbations the structure determined by the stable and unstable bundles persists up to small variations we also show that the constants determining the nonuniform exponential contraction or nonuniform exponential dichotomy vary continuously with the perturbation all the results are obtained in banach spaces

robustness of general dichotomies abstract for a linear equation v a t v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates e c t for some function t this includes as a special case the usual exponential behavior when t t we also consider the general case of nonuniform exponential dichotomies we establish the robustness of the exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy

exponential dichotomy roughness on banach spaces in the present paper we extend existing results on exponential dichotomy roughness for linear ode systems to infinite dimensional banach space we give new conditions for the existence of exponential dichotomy roughness in infinite dimensional space and in the finite interval case we also improve previous results by indicating the exact values of the dichotomic constants of the perturbed equation

roughness of tempered exponential dichotomies for infinite dimensional random difference equations abstract in this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in banach spaces such a dichotomy has a tempered bound and describes nonuniform hyperbolicity we prove the roughness without assuming their invertibility and the integrability condition of the multiplicative ergodic theorem we give an explicit bound for the linear perturbation such that the dichotomy is persistent we also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

processing of acoustic waveforms a sinusoidal model for acoustic waveforms is applied to develop a new analysis synthesis technique which characterizes a waveform by the amplitudes frequencies and phases of component sine waves these parameters are estimated from a short time fourier transform rapid changes in the highly resolved spectral components are tracked using the concept of birth and death of the underlying sine waves the component values are interpolated from one frame to the next to yield a representation that is applied to a sine wave generator the resulting synthetic waveform preserves the general waveform shape and is perceptually indistinguishable from the original furthermore in the presence of noise the perceptual characteristics of the waveform as well as the noise are maintained the method and devices disclosed herein are particularly useful in speech coding time scale modification frequency scale modification and pitch modification

strong nonuniform behaviour a datko type characterization abstract we obtain a datko type characterization of several classes of a strong nonuniform exponential behaviour we consider separately the case of exponential contractions expansions and dichotomies in addition we consider both discrete and continuous time as a nontrivial application of our work we study the robustness property of strong nonuniform exponential contractions expansions and dichotomies

stability of dichotomies in difference equations with infinite delay abstract for delay difference equations with infinite delay we consider the notion of nonuniform exponential dichotomy this includes the notion of uniform exponential dichotomy as a very special case our main aim is to establish a stable manifold theorem under sufficiently small nonlinear perturbations we also establish the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations finally we characterize the nonuniform exponential dichotomies in terms of strict lyapunov sequences in particular we construct explicitly a strict lyapunov sequence for each exponential dichotomy

smoothness of invariant manifolds for nonautonomous equations for semiflows generated by ordinary differential equations v a t v admitting a nonuniform exponential dichotomy we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v a t v f t v as an application we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow in particular we obtain smooth invariant manifolds for a class of vector fields that need not be c1 for any 0 1 to the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting we emphasize that we do not need to assume the existence of an exponential dichotomy but only the existence of a nonuniform exponential dichotomy with sufficiently small nonuniformity when compared to the lyapunov exponents of the original linear equation furthermore for example in the case of stable manifolds we only need to assume that there exist negative lyapunov exponents while we also allow zero exponents our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones

method for modeling speech harmonic magnitudes a system or method for modeling a signal such as a speech signal wherein harmonic frequencies and amplitudes are identified 106 and the harmonic magnitudes are interpolated 110 to obtain spectral magnitudes at a set of fixed frequencies an inverse transform is applied 112 to the spectral magnitudes to obtain a pseudo auto correlation sequence from which linear prediction coefficients are calculated 114 from the linear prediction coefficients model harmonic magnitudes are generated by sampling the spectral envelope 118 defined by the linear prediction coefficients a set of scale factors are then calculated 120 as the ratio of the harmonic magnitudes to the model harmonic magnitudes and interpolated to obtain a second set of scale factors 122 at the set of fixed frequencies the spectral envelope magnitudes at the set of fixed frequencies 124 are multiplied by the second set of scale factors 126 to obtain new spectral magnitudes and the process is iterated to obtain final linear prediction coefficients

the structure of phi stable minimal hypersurfaces in manifolds of nonnegative p scalar curvature suppose n n g is an n dimensional riemannian manifold with a given smooth measure m the p scalar curvature is defined as p g r m infty g r g 2 delta g rm log phi nabla g rm log phi g 2 where dm phi dvol g and r g is the scalar curvature of n n g in this paper under a technical assumption on phi we prove that phi stable minimal oriented hypersurface in the three dimensional manifold with nonnegative p scalar curvature must be conformally equivalent to either the complex plane mathbb c or the cylinder mathbb r times mathbb s 1

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

stable manifolds for nonautonomous equations without exponential dichotomy abstract we consider nonautonomous ordinary differential equations v a t v in banach spaces and under fairly general assumptions we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v a t v f t v which corresponds to the set of negative lyapunov exponents of the original linear equation the main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity i e a small deviation from the classical notion of uniform exponential dichotomy in fact we showed that essentially any linear equation v a t v admits a nonuniform exponential dichotomy and thus the above assumption only concerns the smallness of the nonuniformity of the dichotomy this smallness is a rather common phenomenon at least from the point of view of ergodic theory almost all linear variational equations obtained from a measure preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity we emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small only sufficiently small our approach is related to the notion of lyapunov regularity which goes back to lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations

nonuniform behavior and robustness we establish the robustness of linear cocycles in banach spaces admitting a nonuniform exponential dichotomy we first obtain robustness results for positive and negative time by establishing exponential behavior along certain subspaces and showing that the associated sequences of projections have bounded exponential growth we then establish a robustness result in z by constructing explicitly appropriate projections on the stable and unstable subspaces we emphasize that in general these projections may be different from those obtained separately from the robustness for positive and negative time we also consider the case of strong nonuniform exponential dichotomies

stability of delay equations via lyapunov functions abstract the importance of lyapunov functions is well known in the general setting of nonautonomous linear delay equations v l t v t we show how to characterize completely the existence of a nonuniform exponential contraction or of a nonuniform exponential dichotomy in terms of lyapunov functions this includes uniform exponential behavior as a very special case and it provides an alternative usually simpler and particularly more direct approach to verify the existence of exponential behavior or to obtain the robustness of the dynamics under sufficiently small perturbations

exponential separation exponential dichotomy and spectral theory for linear systems of ordinary differential equations abstract this paper is concerned with linear nonautonomous systems of ordinary differential equations a criterion for exponential separation in terms of exponential dichotomy is given as corollaries we obtain the roughness theorem for exponential separation and the new result that an upper triangular system on a half line is exponentially separated if and only if the system corresponding to its diagonal is a minimal decomposition into exponentially separated subspaces is defined it turns out that it is in general finer than the sacker sell spectral decomposition but that the two decompositions coincide for almost periodic systems

existence of nonuniform exponential dichotomies and a fredholm alternative abstract we consider linear equations x a t x in a banach space with nonuniform exponential dichotomies in r and r and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations we also obtain a fredholm alternative for the existence of bounded solutions under nonlinear perturbations

robustness for impulsive equations for nonautonomous linear impulsive differential equations in banach spaces we establish the robustness of exponential contractions and exponential dichotomies in the sense that the exponential behavior persists under sufficiently small linear perturbations we also consider the more general case of nonuniform exponential behavior