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 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

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

understanding the edge effect in tasep with mean field theoretic approaches we study a totally asymmetric simple exclusion process tasep with one defect site hopping rate q 1 near the system boundary regarding our system as a pair of uniform tasep s coupled through the defect we study various methods to match a finite tasep and an infinite one across a common boundary several approximation schemes are investigated utilizing the finite segment mean field fsmf method we set up a framework for computing the steady state current j as a function of the entry rate and q for the case where the defect is located at the entry site we obtain an analytical expression for j q which is in good agreement with monte carlo simulation results when the defect is located deeper in the bulk we refined the scheme of macdonald et al 1968 biopolymers 6 1 and find reasonably good fits to the density profiles before the defect site we discuss the strengths and limitations of each method as well as possible avenues for further studies

asymmetric simple exclusion model with local inhomogeneity we study a totally asymmetric simple exclusion model with open boundary conditions and a local inhomogeneity in the bulk it consists of a one dimensional lattice with particles hopping stochastically with equal rates to the right at all lattice sites except one where the jump rate is different approximate stationary state solutions and phase diagrams are obtained and compared with monte carlo simulation results

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

totally asymmetric exclusion processes with particles of arbitrary size the steady state currents and densities of a one dimensional totally asymmetric exclusion process tasep with particles that occlude an integer number d of lattice sites are computed using various mean field approximations and monte carlo simulations taseps featuring particles of arbitrary size are relevant for modelling systems such as mrna translation vesicle locomotion along microtubules and protein sliding along dna we conjecture that the nonequilibrium steady state properties separate into low density high density and maximal current phases similar to those of the standard d 1 tasep a simple mean field approximation for steady state particle currents and densities is found to be inaccurate however we find local equilibrium particle distributions derived from a discrete tonks gas partition function yield apparently exact currents within the maximal current phase for the boundary limited phases the equilibrium tonks gas distribution cannot be used to predict currents phase boundaries or the order of the phase transitions however we employ a refined mean field approach to find apparently exact expressions for the steady state currents boundary densities and phase diagrams of the d 1 tasep extensive monte carlo simulations are performed to support our analytic mean field results

centre manifolds for partial differential equations with delays a centre manifold theory for reaction diffusion equations with temporal delays is developed besides an existence proof we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order as an illustration this reduction procedure is applied to the hutchinson equation with diffusion

resolvent positive operators resolvent positive operators on an ordered banach space with generating and normal positive cone are by definition linear possibly unbounded operators whose resolvent exists and is positive on a right half line even though these operators are defined by a simple purely algebraic condition analogues of the basic results of the theory of q semigroups can be proved for them in fact if a is resolvent positive and has a dense domain then the cauchy problem associated with a has a unique solution for every initial value in the domain of a and the solution is positive if the initial value is positive also the converse is true if we assume that a has a non empty resolvent set and d a n e is dense in moreover every positive resolvent is a laplace stieltjes transform of a so called integrated semigroup and conversely every such increasing non degenerate integrated semigroup defines a unique resolvent positive operator

exact results for the asymmetric simple exclusion process with a blockage we present new results for the current as a function of transmission rate in the one dimensional totally asymmetric simple exclusion process tasep with a blockage that lowers the jump rate at one site from one tor 1 exact finitevolume results serve to bound the allowed values for the current in the infinite system this proves the existence of a nonequilibrium phase transition corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state a series expansion inr derived from the finite systems is proven to be asymptotic for all sufficiently large systems pade approximants based on this series which make specific assumptions about the nature of the singularity atr 1 match numerical data for the infinite system to 1 part in 104

local inhomogeneity in asymmetric simple exclusion processes with extended objects totally asymmetric simple exclusion processes tasep with particles which occupy more than one lattice site and with a local inhomogeneity far away from the boundaries are investigated these non equilibrium processes are relevant for the understanding of many biological and chemical phenomena the steady state phase diagrams currents and bulk densities are calculated using a simple approximate theory and extensive monte carlo computer simulations it is found that the phase diagram for tasep with a local inhomogeneity is qualitatively similar to homogeneous models although the phase boundaries are significantly shifted the complex dynamics is discussed in terms of domain wall theory for driven lattice systems

stable and unstable manifolds for hyperbolic bi semigroups we show the existence of local lipschitzian stable and unstable manifolds for the ill posed problem of perturbations of hyperbolic bi semigroups we do not assume backward nor forward uniqueness of solutions we do not use cut off functions because we do not assume global smallness conditions on the nonlinearities we introduce what we call dichotomous flows which recovers the symmetry between the past and the future thus we need to prove only a stable manifold theorem we modify the conley mcgehee moeckel approach to avoid appealing to wazewski principle and brouwer degree theory hence we allow both the stable and unstable directions to be infinite dimensional we apply our theorem to the elliptic system u u g u u in an infinite cylinder r

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

exclusion process for particles of arbitrary extension hydrodynamic limit and algebraic properties the behaviour of extended particles with exclusion interaction on a one dimensional lattice is investigated the basic model is called l asep as a generalization of the asymmetric exclusion process asep to particles of arbitrary length l stationary and dynamical properties of the l asep with periodic boundary conditions are derived in the hydrodynamic limit from microscopic properties of the underlying stochastic many body system in particular the hydrodynamic equation for the local density evolution and the time dependent diffusion constant of a tracer particle are calculated as a fundamental algebraic property of the symmetric exclusion process the su 2 symmetry is generalized to the case of extended particles

effect of detachments in asymmetric simple exclusion processes asymmetric simple exclusion processes are important for understanding low dimensional multi particle dynamic phenomena the effect of irreversible detachments of particles on dynamics of asymmetric simple exclusion processes is studied using analytical and computer simulation techniques in the simplest model where particles can only detach from a single site in the bulk of the system a theory is presented and used to calculate explicitly phase diagrams and particle density profiles the complexity of the phase behavior is discussed in terms of a recent domain wall theory for driven lattice systems the theoretical results qualitatively and quantitatively agree with computer monte carlo simulations

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

integrated semigroups and integrated solutions to abstract cauchy problems abstract concepts which have been developed in the theory of weakly continuous semigroups on dual banach spaces are extended to integrated semigroups the relation to integrated solutions of homogeneous and inhomogeneous abstract cauchy problems is established the results are illustrated for the wave equation in l2 rm

existence of quasiperiodic solutions of elliptic equations on rn 1 via center manifold and kam theorems abstract we consider elliptic equations on r n 1 of the form 1 x u u y y g x u 0 x y r n r where g x u is a sufficiently regular function with g 0 0 we give sufficient conditions for the existence of solutions of 1 which are quasiperiodic in y and decaying as x uniformly in y such solutions are found using a center manifold reduction and results from the kam theory we discuss several classes of nonlinearities g to which our results apply

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

existence and uniqueness of traveling waves for non monotone integral equations with applications abstract a class of integral equations without monotonicity is investigated it is shown that there is a spreading speed c 0 for such an integral equation and that its limiting integral equation admits a unique traveling wave up to translation with speed c c and no traveling wave with c c these results are also applied to some nonlocal reaction diffusion population models

a reaction diffusion model for a single species with age structure i travelling wavefronts on unbounded domains in this paper we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment we show that if the mature deat

an augmented lagrangian function with improved exactness properties in this paper we introduce a new exact augmented lagrangian function for the solution of general nonlinear programming problems for this lagrangian function a complete equivalence between its unconstrained minimization on an open set and the solution of the original constrained problem can be established under mild assumptions and without requiring the boundedness of the feasible set of the constrained problem moreover we describe an unconstrained algorithmic model which is globally convergent toward kkt pairs of the original constrained problem the algorithmic model can be endowed with a superlinear rate of convergence by a proper choice of the search direction in the unconstrained minimization without requiring strict complementarity

asymptotic behaviour of travelling waves for the delayed fisher kpp equation abstract in this work we study the behaviour of travelling wave solutions for the diffusive hutchinson equation with time delay using a phase plane analysis we prove the existence of travelling wave solution for each wave speed c 2 we show that for each given and admissible wave speed such travelling wave solutions converge to a unique maximal wavetrain as a consequence the existence of a nontrivial maximal wavetrain is equivalent to the existence of travelling wave solution non converging to the stationary state u 1

a center stable manifold theorem for differential equations in banach spaces we prove a center stable manifold theorem for a class of differential equations in infinite dimensional banach spaces

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

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

understanding totally asymmetric simple exclusion process transport on networks generic analysis via effective rates and explicit vertices in this paper we rationalize relevant features of totally asymmetric simple exclusion processes on topologies more complex than a single segment we present a mean field framework exploiting the previously introduced notion of effective rates which we express in terms of the average particle density on explicitly introduced junction sites it allows us to construct the phase behavior as well as the current density characteristic from well known results for a linear totally asymmetric simple exclusion process segment in a very systematic and generic way we validate the approach by studying a fourfold vertex in all variations in the number of entering exiting segments and compare our predictions to simulation data generalizing the notion of particle hole symmetry to take into account the topology at a junction shows that the average particle density at the junction constitutes a relevant directly observable parameter which gives detailed insight into the transport process this is illustrated by a complete study of a simple network with figure of eight topology finally we generalize the approach to handle rate bias at a junction and discuss the surprisingly rich phenomenology of a biased figure of eight structure this example highlights that the proposed framework is generic and readily extends to other topologies

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

on the equations of age dependent population dynamics the aim of this work is to give a direct and constructive proof of existence and uniqueness of a global solution to the equations of age dependent population dynamics introduced and considered by m e gurtin r c maccamy in 3 the linear theory was developed by f r sharpe a j lotka 10 and a g mckendrick 8 see also 1 9 and extended to the nonlinear case by m e gurtin r c maccamy in 3 see also 4 5 6 in 3 the key of the proof of existence and uniqueness was to reduce the problem to a pair of integral equations in fact as we shall see the problem can also be solved by a simple fixed point argument to outline more clearly the ideas of the proof we will first discuss the setting and the resolution of the linear case and then we will generalize the results of 3

normal forms for semilinear equations with non dense domain with applications to age structured models normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems in this paper using the center manifold theorem associated with the integrated semigroup theory we develop a normal form theory for semilinear cauchy problems in which the linear operator is not densely defined and is not a hille yosida operator and present procedures to compute the taylor expansion and normal form of the reduced system restricted on the center manifold we then apply the main results and computation procedures to determine the direction of the hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza a drift and an age structured population model

totally asymmetric exclusion process with extended objects a model for protein synthesis the process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles have spatial extent covering more than one lattice site we expand the well studied totally asymmetric exclusion process in which particles typically cover a single lattice site to include cases with extended objects exact solutions can be determined for a uniform closed system we analyze the uniform open system through two approaches first a continuum limit produces a modified diffusion equation for particle density profiles second an extremal principle based on domain wall theory accurately predicts the phase diagram and currents in each phase finally we briefly consider approximate approaches to a nonuniform open system with quenched disorder in the particle hopping rates and compare these approaches with monte carlo simulations

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

spontaneous symmetry breaking in totally asymmetric simple exclusion processes on two intersected lattices this paper studies totally asymmetric simple exclusion processes taseps on two intersected lattices three different models are introduced model a for molecular motor motion and models b and c for vehicle traffic extensive monte carlo simulations are carried out phase diagrams and density profiles of the three models are investigated it is shown that phase diagrams of all the three models are divided into three regions the phase boundaries are calculated by an approximate mean field approach it is found that the analytic solutions are in good agreement with the results of monte carlo simulations for models b and c spontaneous symmetry breaking is identified the particle density histograms and qualitative domain wall explanation are presented to describe the spontaneous symmetry breaking phenomenon finally in model c the effect of lane changing probability p on spontaneous symmetry breaking is investigated a threshold of p for the occurrence of symmetry breaking is obtained