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

invariant manifolds of partial functional differential equations this paper is concerned with the existence smoothness and attractivity of invariant manifolds for evolutionary processes on general banach spaces when the nonlinear perturbation has a small global lipschitz constant and locally ck smooth near the trivial solution such a nonlinear perturbation arises in many applications through the usual cut off procedure but the requirement in the existing literature that the nonlinear perturbation is globally ck smooth and has a globally small lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut off function our general results are illustrated by and applied to partial functional differential equations for which the phase space c r 0 x where r 0 and x being a banach space has no smooth inner product structure and for which the validity of variation of constants formula is still an interesting open problem

center manifolds for invariant sets abstract we derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector field in r n by applying the hadamard graph transform technique it is shown that associated to a natural dynamical characteristic of the linearized flow along the invariant set there exists an invariant manifold called a center manifold of the invariant set which contains every locally bounded solution in particular contains the invariant set and is persistent under small perturbations

existence of bounded solutions of a second order system with dissipation in this article we study the following second order system of ordinary differential equations with dissipationu cu dau khu pt u rn t r where c d and k are positive constants h rn rn is a locally lipschitz function and p r rn is a continuous and bounded function a is a n n matrix whose eigenvalues are positive under these conditions we prove that for some values of c d and k this system has a bounded solution which is exponentially asymptotically stable moreover if p t is almost periodic then this bounded solution is also almost periodic these results are applied to the spatial discretization of very well known second order partial differential equations

a center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the gurtin mccamy equation

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

tsallis conditional mutual information in investigating long range correlation in symbol sequences abstract the presence of long range correlation lrc as opposed to markov chain of a finite order is of primary interest in the study of symbol sequences such as dna among others methods using information theory have been developed for markov chain order estimation in this work we consider the tsallis entropy and define the tsallis conditional mutual information tcmi for markov chain order estimation which is more suitable to identify large orders suggesting effectively lrc the tcmi of order m for a tsallis parameter q is computed for increasing m and a significance test is performed for each m until tcmi is found non significant randomization and parametric significance tests for tcmi are developed for the latter the null distribution of tcmi is approximated with a gamma distribution deriving analytic expressions for its parameters we assess the accuracy of order estimation with the two tests for tcmi and compare them to the respective tests using the shannon entropy extended simulations on markov chains of different orders and structures of the transition probability matrix show that shannon and tsallis tests with parameter q 2 have similar performance for small orders when the problem becomes more demanding for higher markov chain orders and lrcs tsallis tests for q 2 have better approximation to the correct order with the sequence length n finally the tsallis tests are favorably compared to shannon tests to real dna sequences

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

on a convergent dsa preconditioned source iteration for a dgfem method for radiative transfer abstract we consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution the even parity equations are solved using a diffusion synthetic accelerated source iteration we provide a convergence analysis for the infinite dimensional iteration as well as for its discretized counterpart the diffusion correction is computed by a subspace correction which leads to a convergence behavior that is robust with respect to the discretization the proven theoretical contraction rate deteriorates for scattering dominated problems we show numerically that the preconditioned iteration is in practice robust in the diffusion limit moreover computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect the theoretical methodology is presented for plane parallel geometries with isotropic scattering but the approach and proofs generalize to multi dimensional problems and more general scattering operators verbatim

analysis of a full discretization scheme for 2d radiative conductive heat transfer systems abstract this paper deals with the convergence of numerical scheme for combined nonlinear radiation conduction heat transfer system in a gray absorbing and non scattering two dimensional medium the radiative transfer equation is solved using a discontinuous galerkin method with upwind fluxes the conductive equation is discretized using the finite element method moreover the crank nicolson scheme is applied for time discretization of the semi discrete nonlinear coupled system existence and uniqueness of the solution for the continuous and full discrete system are presented the convergence proof follows from the application of a discrete fixed point theorem involving only the temperature fields at each time step the order of approximation error stability and order of convergence are investigated finally the theoretical stability and convergence results are supported with numerical examples

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

center manifolds for periodic functional differential equations of mixed type abstract we study the behaviour of solutions to nonlinear functional differential equations of mixed type mfdes that remain sufficiently close to a prescribed periodic solution under a discreteness condition on the floquet spectrum we show that all such solutions can be captured on a finite dimensional invariant center manifold that inherits the smoothness of the nonlinearity this generalizes the results that were obtained previously in h j hupkes s m verduyn lunel center manifold theory for functional differential equations of mixed type j dynam differential equations 19 2007 497 560 for bifurcations around equilibrium solutions to mfdes

analysis of a fully discrete scheme for neutron transport in two dimensional geometry we derive error estimates for a fully discrete scheme for the numerical solution of the neutron transport equation in two dimensional cartesian geometry obtained by using a special quadrature rule for the angular variable and the discontinuous galerkin finite element method with piecewise linear trial function for the space variable

chebyshev spectral sn method for the neutron transport equation we study convergence of a combined spectral and sn discrete ordinates approximation for a multidimensional steady state linear transport problem with isotropic scattering the procedure is based on expansion of the angular flux in a truncated series of chebyshev polynomials in spatial variables that results in the transformation of the multidimensional problems into a set of one dimensional problems the convergence of this approach is studied in the context of the discrete ordinates equations based on a special quadrature rule for the scattering integral the discrete ordinates and quadrature errors are expanded in truncated series of chebyshev polynomials of degree l and the convergence is derived assuming l s t 4 p s s where s t and s s are total and scattering cross sections respectively

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

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

entropy estimates of small data sets estimating entropies from limited data series is known to be a non trivial task naive estimations are plagued with both systematic bias and statistical errors here we present a new balanced estimator for entropy functionals shannon renyi and tsallis specially devised to provide a compromise between low bias and small statistical errors for short data series this new estimator outperforms other currently available ones when the data sets are small and the probabilities of the possible outputs of the random variable are not close to zero otherwise other well known estimators remain a better choice the potential range of applicability of this estimator is quite broad specially for biological and digital data series

entropy and local uncertainty of data from sensory neurons we present an empirical comparison between neural interspike interval sequences obtained from two different kinds of sensory receptors both differ in their internal structure as well as in the strength of correlations and the degree of predictability found in the respective spike trains as a further tool in this context we suggest the local uncertainty assigning a well defined predictability to individual spikes the local uncertainty is demonstrated to reveal significant patterns within the interspike interval sequences even when its overall structure is almost random our approach is based on the concept of symbolic dynamics and information theory

smooth centre manifolds for impulsive delay differential equations abstract the existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations several intermediate results of theoretical interest are developed including a variation of constants formula for linear equations in the phase space of right continuous regulated functions linear variational equation and smoothness of the nonautonomous process and a floquet theorem for periodic systems three examples are provided to illustrate the results

guessing probability distributions from small samples we propose a new method for the calculation of the statistical properties e g the entropy of unknown generators of symbolic sequences the probability distributionp k of the elementsk of a population can be approximated by the frequenciesf k of a sample provided the sample is long enough so that each elementk occurs many times our method yields an approximation if this precondition does not hold for a givenf k we recalculate the zipf ordered probability distribution by optimization of the parameters of a guessed distribution we demonstrate that our method yields reliable results

smoothness of invariant manifolds in this paper we present some functional analytic tools that allow us to prove a theorem on ck smoothness of general discrete and continuous invariant manifolds for ck systems a crucial role in this proof will be played by a theorem on ck parameter dependence of a fixed point of a contraction defined on a scale of banach spaces this fixed point principle is stated independently and may be of interest in itself further functional analytic considerations concerning smoothness properties of the nemytski operator are presented in a rather general form throughout the paper we treat the two cases of continuous and discrete dynamical systems in a unified manner thereby we use some concepts and results from the so called calculus on measure chains which was developed by the author in res in math 18 1990 18 56 knowledge of this calculus is not necessary for understanding the paper

prediction and entropy of nonlinear dynamical systems and symbolic sequences with lro abstract following shannon we introduce higher order entropies and derive dynamic entropies the nth order dynamic entropy conditional entropy is a measure of the uncertainty of the next state which follows after the observation of n foregoing states the asymptotic behaviour of the dynamic entropies at large n is studied for several nonlinear model systems and for symbolic sequences with long range order lro for example we investigate 1d maps texts dna strings and time series it is shown that the existence of long correlations improves the possibility of predictions characteristic scaling laws for the higher order shannon entropies and the conditional entropies are derived and a new interpolation formula is tested finally instead of the dynamic entropies which yield mean values of the uncertainty predictability we investigate the local values of the uncertainty predictability

wong zakai approximations and center manifolds of stochastic differential equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of the stochastic differential equation driven by a l dimensional brownian motion we prove that the solutions of wong zakai approximations converge in the mean square to the solutions of the stratonovich stochastic differential equation we also show that for a simple multiplicative noise the center manifold of the wong zakai approximations converges to the center manifold of the stratonovich stochastic differential equation

a finite element method for the neutron transport equation in an infinite cylindrical domain we study the spatial discretization in a fully discrete scheme for the numerical solution of a model problem for the neutron transport equation in an infinite cylindrical domain based on using an interpolation technique in the discontinuous galerkin finite element procedure we derive an almost optimal error estimate for the scalar flux in the l2 norm combining a duality argument applied to the above result together with the previous semidiscrete error estimates for the velocity discretizations we also obtain globally optimal error bounds for the critical eigenvalues

statistical linguistic characterization of variability in observed and synthetic daily precipitation series abstract this paper deals with an application of zipf law in climatology this analysis allows the extraction of information not available by standard methods in particular rainfall temporal aggregation patterns associated with different climates are characterized by means of exponents derived from the resulting scaling laws the analogy with linguistic analysis is obtained using a particular coding of precipitation as a discrete variable with four states corresponding to four standard precipitation thresholds each weekly symbolic sequence of observed precipitation is considered as a word and each local station defines a language characterized by the observed words in a period representative of the climatology to characterize these precipitation languages we obtained characteristic exponents derived from the zipf law for a set of representative stations of the main koppen s climates and subclimates we found different scaling behaviors for different subclimates given by a single exponent in the range 0 6 humid tropical climates to 1 4 polar climates some humid middle latitude subclimates exhibit a crossover with two different characteristic exponents corresponding to high and low frequency aggregation patterns no explanation for this behavior is provided as an application of the proposed methodology the same analysis was applied to stochastic time series obtained from standard weather generators we found that the language defined by a first order markov process agrees with the observed language of the observed series

center manifolds under nonuniform hyperbolicity maximal regularity abstract we establish the existence of invariant center manifolds with maximal c r regularity for a nonautonomous dynamics with discrete time we consider the general case of perturbations of a nonuniform exponential trichotomy our proof uses the fiber contraction principle and allows linear perturbations without any further effort

invariant manifolds for volterra integral equations of convolution type abstract in this paper we develop some elements of a qualitative theory for nonlinear volterra integral equations of convolution type our starting point is a local semiflow associated with the equation and acting on a space of compactly supported forcing functions within that framework we discuss the variation of constants formula the saddle point property the center manifold and hopf bifurcation some equations from population biology get special attention

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

viscous singular shock profiles for a system of conservation laws modeling two phase flow abstract this paper is concerned with singular shocks for a system of conservation laws via the dafermos regularization u t f u x t u x x for a system modeling incompressible two phase fluid flow the existence of viscous profiles is proved using geometric singular perturbation theory the weak convergence and the growth rate of the viscous solution are also derived the weak limit is the sum of a piecewise constant function and a measure supported on a shock line and the maximum value of the viscous solution is of order exp 1

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

bounded solutions to singularly perturbed systems of o d e abstract we consider existence and stability properties of bounded solutions to singular systems of o d e the method we suggest considers the concept of singular maps between scales of banach spaces and their properties this theory is then applied to obtain existence and transversality of bounded solutions to singular systems of o d e

smooth convergence of random center manifolds for spdes in varying phase spaces abstract in this paper we study the limiting behavior of center manifolds for a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces we first prove the existence and smoothness of random center manifolds for these equations then we establish the smooth convergence of these random center manifolds as the phase spaces approach to their singular limit

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

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

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

stability under galerkin truncation of a stable runge kutta discretizations in time we consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces we approximate their semiflow by an implicit a stable runge kutta discretization in time and a spectral galerkin truncation in space we show regularity of the galerkin truncated semiflow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value we also prove convergence of the space time discretization without any condition that couples the time step to the spatial resolution then we estimate the galerkin truncation error for the semiflow of the evolution equation its runge kutta discretization and their respective derivatives showing how the order of the galerkin truncation error depends on the smoothness of the initial data our results apply in particular to the semilinear wave equation and to the nonlinear schr odinger equation

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

error estimates for the combined spatial and angular approximations of the transport equation for slab geometry we study the convergence of the discrete ordinates method in the numerical solution of the transport equation for slab geometry in particular the combined effect of spatial and angular approximations is analyzed general conditions are given which guarantee the stability of the combined approximations error estimates are derived for both the scalar flux in the l p norm 1 leqq p leqq infty and for the approximate critical eigenvalues a comparison with numerical results indicates that the eigenvalue error estimates are optimal

multiple homoclinic orbits in conservative and reversible systems we study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows we consider the existence of two homoclinic orbits in the bellows configuration where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times in conservative systems one finds one parameter families of suspended horseshoes parameterized by the level of the first integral a somewhat similar picture occurs in reversible systems with two homoclinic orbits that are both symmetric the lack of a first integral implies that complete horseshoes do not exist we provide a description of orbits that necessarily do exist a second possible configuration in reversible systems occurs if a non symmetric homoclinic orbit exists and forms a bellows together with its symmetric image we describe the nonwandering set in an unfolding the nonwandering set is shown to simultaneously contain one parameter families of periodic orbits hyperbolic periodic orbits of different index and heteroclinic cycles between these periodic orbits

l p and eigenvalue error estimates for the discrete ordinates methods for two dimensional neutron transport the convergence of the discrete ordinates method is studied for angular discretization of the neutron transport equation for a two dimensional model problem with the constant total cross section and isotropic scattering considering a symmetric set of quadrature points on the unit circle error estimates are derived for the scalar flux in l p norms for 1 leqq p leqq infty a postprocessing procedure giving improved l infty estimates is also analyzed finally error estimates are given for simple isolated eigenvalues of the solution operator

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

the wong zakai approximations of invariant manifolds and foliations for stochastic evolution equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise we prove that the solutions of wong zakai approximations almost surely converge to the solutions of the stratonovich stochastic evolution equation we also show that the invariant manifolds and stable foliations of the wong zakai approximations converge to the invariant manifolds and stable foliations of the stratonovich stochastic evolution equation respectively

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given

center manifolds for infinite delay we obtain a c1 center manifold theorem for perturbations of delay difference equations in banach spaces with infinite delay our results extend in several directions the existing center manifold theorems besides considering infinite delay equations we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable unstable and central behaviors with respect to arbitrary asymptotic rates ec n for some diverging sequence n this includes as a very special case the usual exponential behavior with n n

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