the sensitivity of the stable lyapunov equation we present an analysis of the sensitivity of the solution of the lyapunov equation a x xa w where a is stable this analysis leads to a spectral norm bound on the relative perturbation of the solution which is optimal for a certain class of estimates and which is essentially equivalent to the frobenius norm bound obtained from the associated kronecker product system the latter bound can be expressed in terms of operatorname sep a a and is known to accurately reflect the sensitivity of the lyapunov problem but it is hard to interpret in terms of the original matrix a in contrast the spectral norm bound which we derive is directly related to the minimal l 2 damping of the dynamical system dot z az moreover this dynamical link with the sensitivity problem leads to a new method of systematically investigating the norm behavior of e at as well as providing a wealth of information about control theoretic aspects of dot z az when a is the closed loop state matrix

two conditions concerning common quadratic lyapunov functions for linear systems concerning a pair of linear systems some triangle conditions for the existence of a common quadratic lyapunov function are presented in this paper these conditions are derived from a criterion for judging the semipositiveness of a linear map defined on symmetric matrices

random pullback attractor of a non autonomous local modified stochastic swift hohenberg equation with multiplicative noise in this paper we study the existence of the random d pullback attractor of a non autonomous local modified stochastic swift hohenberg equation with multiplicative noise in the stratonovich sense we show that a random d pullback attractor exists in h02 d when its external force has exponential growth due to the stochastic term a priori estimates are delicate we used the ornstein uhlenbeck transformation and its properties to overcome the difficulty that the stochastic term brings to us

scaling variables and asymptotic expansions in damped wave equations abstract we study the long time behavior of small solutions to the nonlinear damped wave equationeu u a u n u u u r 0 whereeis a positive not necessarily small parameter we assume that the diffusion coefficienta converges to positive limitsa as and that the nonlinearity n u u u vanishes sufficiently fast asu 0 introducing scaling variables and using various energy estimates we compute an asymptotic expansion of the solutionu in powers of 1 2as and we show that this expansion is entirely determined up to the second order by a linear parabolic equation which depends only on the limiting valuesa in particular this implies that the small solutions of the damped wave equation behave for large like those of the parabolic equation obtained by settinge 0

long time dynamics of the swift hohenberg equations abstract we consider the initial boundary value problems for the swift hohenberg equation and its hyperbolic relaxation under the optimal conditions on the nonlinearity we prove the well posedness of the both problems and uniform with respect to the initial data global boundedness of the solutions then we show that the both problems possess the global attractors of optimal regularity

laplace transformations of hydrodynamic type systems in riemann invariants periodic sequences the conserved densities of hydrodynamic type systems in riemann invariants satisfy a system of linear second order partial differential equations for linear systems of this type darboux introduced laplace transformations generalizing the classical transformations in the scalar case it is demonstrated that laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic type systems we discuss periodic laplace sequences of hydrodynamic type systems with emphasis on the simplest nontrivial case of period 2 for systems in riemann invariants a complete discription of closed quadruples is proposed they turn out to be related to a special quadratic reduction of the 2 1 dimensional 3 wave system which can be reduced to a triple of pairwise commuting monge ampere equations in terms of the lame and rotation coefficients laplace transformations have a natural interpretation as the symmetries of the dirac operator associated with the 2 1 dimensional n wave system the 2 component laplace transformations can also be interpreted as the symmetries of the 2 1 dimensional integrable equations of davey stewartson type laplace transformations of hydrodynamic type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space

upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation abstract for a smooth bounded domain r n n 3 and e a real parameter consider the hyperbolic equation eu tt u t u u g in with dirichlet boundary conditions under certain conditions on this equation has a compact attractor a e in h 0 1 l 2 for e 0 the parabolic equation also has a compact attractor which can be naturally embedded into a compact set a 0 in h 0 1 l 2 it is shown that for any neighborhood u of a 0 the set a e u for e small

regularity of invariant sets in semilinear damped wave equations abstract under fairly general assumptions we prove that every compact invariant subset i of the semiflow generated by the semilinear damped wave equation u t t u t x u i j a i j x u x j x i f x u t x 0 u 0 t x 0 in h 0 1 l 2 is in fact bounded in d a h 0 1 here is an arbitrary possibly unbounded domain in r 3 a u x u i j a i j x u x j x i is a positive selfadjoint elliptic operator and f x u is a nonlinearity of critical growth the nonlinearity f x u needs not to satisfy any dissipativeness assumption and the invariant subset i needs not to be an attractor

the singular limit dynamics of semilinear damped wave equations on considere la dynamique qualitative d une equation d onde amortie semi lineaire a une dimension dans la limite singuliere correspondant a l annulation du coefficient de la derivee d ordre 2 en temps

upper semi continuity and regularity of random attractors on p times integrable spaces and applications abstract a sufficient condition for a family of random attractors to be upper semi continuous and regular is obtained when the state space is a p times integrable space with p 2 it is shown that both upper semi continuity and regularity under the topology of p norms do not rely on continuity convergence and absorption of the associated random dynamical systems in l p since the corresponding l 2 properties can offer all as an application of the abstract result it is shown that the family of random attractors for the stochastic reaction diffusion equations on a unbounded domain is upper semi continuous and regular at any point under the topology of p norms

pullback attractors of nonautonomous reaction diffusion equations abstract in this paper firstly we introduce the concept of norm to weak continuous cocycle in banach space and give a technical method to verify this kind of continuity then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle using the measure of noncompactness as an application we prove the existence of pullback attractors in h 0 1 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction diffusion equations the attractor pullback attracts all bounded subsets of h 0 1 in the norm of h 0 1

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

upper semicontinuity of attractors for random lattice systems perturbed by small white noises abstract in this paper we first present some conditions for the upper semicontinuity of perturbed random attractors to a limiting random attractor then we apply this result to establish the upper semicontinuity of random attractors for the first order stochastic lattice differential equations with random coupled coefficients and multiplicative additive white noises in the weighted space of infinite sequences as the coefficient of the white noise term tends to zero

upper semicontinuous attractors for a 2d mindlin timoshenko thermo viscoelastic model with memory abstract a nonlinear problem for a thermo viscoelastic mindlin timoshenko plate with hereditary heat conduction is considered here we prove the existence of a compact global attractor whose fractal dimension is finite the main aim of the work is to show the upper semicontinuity of the attractor as the relaxation kernels fade in a suitable sense

pullback attractor for a non autonomous modified swift hohenberg equation this paper is concerned with a non autonomous modified swift hohenberg equation u t d 2u 2 du au b u 2 u 3 g x t it is shown that a pullback attractor exists when its external force has exponential growth due to the nonlinear terms b u 2 and u 3 the estimates are delicate we overcome this difficulty by imposing the exponential growth condition on the external forcing term g x t

continuity of non autonomous attractors for hyperbolic perturbation of parabolic equations abstract in this work we prove the lower and upper semicontinuity of pullback uniform and cocycle attractors for the non autonomous dynamical system given by hyperbolic equation on a bounded domain r 3 u t t u t u f t u for each 0 this equation has uniform pullback and cocycle attractors in h 0 1 l 2 and for 0 the limit parabolic equation u t u f 0 u has a global attractor a 0 in h 0 1 which can be naturally embedded into a compact set a 0 in h 0 1 l 2 we prove that all three types of non autonomous attractors converge both upper and lower semicontinuously to a 0 the study of the detailed structure of the non autonomous attractors under perturbation plays the crucial role in the arguments

upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations abstract we consider the following lattice nonclassical diffusion equation with delays v i t 0 v i t 1 p p v i t e 1 p p v i t f i v i t t g i t i z where e 0 1 0 is a positive constant with 0 1 p is any positive integer and is the discrete one dimensional laplace operator under suitable conditions on f and g we prove the existence of pullback attractors for the multi valued process associated with the e small perturbed systems for which the uniqueness of solutions need not hold moreover we compare the dynamics of the original systems and the e small perturbed systems and show that their attractors are close in the sense of hausdorff semidistance

longtime dynamics of boussinesq type equations with fractional damping abstract the paper investigates the well posedness and longtime dynamics of boussinesq type equations with fractional damping u t t 2 u u t f u g x with 1 2 the main results focus on the relations among the dissipative exponent the growth exponent p of nonlinearity f u and the well posedness and the longtime dynamics of the equations we find a new critical exponent p n 2 2 1 n 2 2 1 rather than p n 2 n 2 n 3 as known before and show that when 1 p p i the equations are like parabolic that is not only the ibvp of the equations are well posedness but also their weak solutions are of higher global regularity as t 0 ii the related solution semigroup has a global attractor a in natural energy space and also has an exponential attractor a e x p in the sense of partially strong topology in particular when 1 p p n 2 n 2 p the partially strong topology becomes the strong one iii for any 0 1 2 the family of global attractors a is upper semicontinuous at the point 0

existence of secondary bifurcations or isolas for pdes abstract in this paper we introduce a method to conclude about the existence of secondary bifurcations or isolas of steady state solutions for parameter dependent nonlinear partial differential equations the technique combines the global bifurcation theorem knowledge about the non existence of nontrivial steady state solutions at the zero parameter value and explicit information about the coexistence of multiple nontrivial steady states at a positive parameter value we apply the method to the two dimensional swift hohenberg equation

exponential attractors and inertial manifolds for singular perturbations of the cahn hilliard equations abstract we consider the singular perturbations of two boundary value problems concerning respectively the viscous and the nonviscous cahn hilliard equations in one dimension of space we show that the dynamical systems generated by these two problems admit exponential attractors and inertial manifolds in the phase space h01 0 h 1 0

global attractors for singular perturbations of the cahn hilliard equations abstract we consider the singular perturbations of two boundary value problems concerning respectively the viscous and the nonviscous cahn hilliard equations in one dimension of space we show that the dynamical systems generated by these two problems admit global attractors in the phase space h 0 1 0 h 1 0 and that these global attractors are at least upper semicontinuous with respect to the vanishing of the perturbation parameter

continuous families of exponential attractors for singularly perturbed equations with memory for a family of semigroups s epsilon t h epsilon h epsilon depending on a perturbation parameter epsilon is an element of 0 1 where the perturbation is allowed to become singular at epsilon 0 we establish a general theorem on the existence of exponential attractors epsilon epsilon satisfying a suitable holder continuity property with respect to the symmetric hausdorff distance at every epsilon is an element of 0 1 the result is applied to the abstract evolution equations with memorypartial derivative t x t integral infinity 0 k epsilon s b 0 x t s ds b1 x t 0 epsilon is an element of 0 1 where k epsilon s 1 epsilon k s epsilon is the resulting of a convex summable kernel k with unit mass such a family can be viewed as a memory perturbation of the equationpartial derivative t x t b 0 x t b 1 x t 0 formally obtained in the singular limit epsilon 0

systems of conservation laws within the framework of the projective theory of congruences the levy transformations of semi hamiltonian systems we continue the investigation of the recently proposed geometric correspondence between systems of conservation laws and congruences of lines in projective space the relationship between the additional conservation laws and hypersurfaces conjugate to a congruence is established thus providing a basis for the levy transformations of semi hamiltonian systems similarly the correspondence between commuting flows and certain families of planes containing the lines of the congruence gives rise to the adjoint levy transformations

charged free fermions vertex operators and the classical theory of conjugate nets we show that the quantum field theoretical formulation of the function theory has a geometrical interpretation within the classical transformation theory of conjugate nets in particular we prove that i the partial charge transformations preserving the neutral sector are laplace transformations ii the basic vertex operators are levy and adjoint levy transformations and iii the diagonal soliton vertex operators generate fundamental transformations we also show that the bilinear identity for the multicomponent kadomtsev petviashvili hierarchy becomes through a generalized miwa map a bilinear identity for the multidimensional quadrilateral lattice equations

hyperbolic singular perturbations for integrodifferential equations we study the convergence of solutions email protected 2u t e u t e e 2a b u t e email protected 0 tk t s e 2a b u s e ds f t e t 0 u 0 e u 0 e u 0 e u 1 e to solutions ofw t bw t email protected 0 tk t s bw s ds f t t 0 w 0 w 0 when e 0 here a and b are linear unbounded operators in a banach space x k t is a linear bounded operator for each t 0 in x and f t e and f t are given x valued functions our result extends the studies in fattorini j diff eq 70 1987 1 for equations without the integral term and in liu proc am math soc 122 1994 791 for parabolic singular perturbation problems

upper semicontinuity of the global attractor for the gierer meinhardt model we are concerned with the following gierer meinhardt model on a bounded domain r n n 1 2 3 with smooth boundary which is a biological pattern formation model proposed by a gierer and h meinhardt

global attractor of a modified swift hohenberg equation in hk spaces abstract in this paper by using an iteration procedure regularity estimates for the linear semigroups and a classical existence theorem of global attractor we prove that a modified swift hohenberg equation u t 2 u g u possesses a global attractor in sobolev space h k for all k 0 which attracts any bounded subset of h k in the h k norm

upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions the motivations to study the problem considered in this paper come from the theory of composite materials where the heat diffusion properties can change from one part of the domain to another mathematically this leads to a nonlinear second order parabolic equation for which the diffusion coefficient becomes large in a subdomain 0 the equation is supplemented by a nonlinear boundary condition on and an initial condition the authors determine the form of the limit problem the so called shadow system which involves an evolution equation for the averages of the density over 0 the main results include global in time existence of solutions and upper semicontinuity of the associated global attractors when the system approaches the shadow system

on convergent rate of the attractor for a singularly perturbed wave equation abstract by applying the pullback property of attractor the paper obtains the continuity and the convergent rate p 1 2 of the attractor for a singularly perturbed wave equation without help of the hyperbolic property of stationary points

darboux transformations for schrodinger equations in two variables darboux transformations in one variable form the basis for the factorization methods and have numerous applications to geometry nonlinear equations and susy quantum mechanics in spite of this wide range of applications the theory of darboux transformations in two variables and its elegant relationship to analytic complex functions has not been recognized in the literature to close this gap we develop in this paper the theory of darboux transformation in the context of schrodinger equations in two variables this yields a constructive algorithm to determine the relationship between potential functions which are related by darboux transformations

random attractors for damped non autonomous wave equations with memory and white noise abstract we first prove the existence of random attractors for the continuous random dynamical systems generated by stochastic damped non autonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent then we study the upper semicontinuity of random attractors when the coefficient of random term approaches zero

stochastic stability properties of jump linear systems jump linear systems are defined as a family of linear systems with randomly jumping parameters usually governed by a markov jump process and are used to model systems subject to failures or changes in structure the authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties it is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability and a testable necessary and sufficient condition for second moment stability is derived the lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the lyapunov exponents of jump linear systems is presented finally for one dimensional jump linear system it is proved that the region for delta moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0 sup

pullback attractors for the norm to weak continuous process and application to the nonautonomous reaction diffusion equations abstract in this paper we introduce the concept of norm to weak continuous process in a banach space and obtain the existence of pullback attractors for this kind of process then we give a new method for proving the existence of the pullback attractors as an application we obtain the existence of pullback attractors for nonautonomous reaction diffusion equation in h 0 1 with exponential growth of the external force

limiting dynamics for stochastic wave equations abstract in this paper relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered by introducing almost surely d contracting property for random dynamical systems we obtain a global random attractor of the stochastic wave equation u t t u t u f u w endowed with dirichlet boundary condition for any 0 1 the upper semicontinuity of this global random attractor and the global attractor of the heat equation z t z f z 0 with dirichlet boundary condition as goes to zero is investigated furthermore we show the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation

optimal controls of multidimensional modified swift hohenberg equation in this paper we deal with the optimal control problem governed by multidimensional modified swift hohenberg equation after showing the relationship between the control problem and its approximation we derive the optimality conditions for an optimal control of our original problem by using one of the approximate problems

attractors for damped semilinear wave equations with a robin acoustic boundary perturbation abstract under consideration is the damped semilinear wave equation u t t u t u u f u 0 on a bounded domain in r 3 with a perturbation parameter e 0 occurring in an acoustic boundary condition limiting e 0 to a robin boundary condition with minimal assumptions on the nonlinear term f the existence and uniqueness of global weak solutions is shown for each e 0 1 also the existence of a family of global attractors is shown to exist after proving a general result concerning the upper semicontinuity of a one parameter family of sets the result is applied to the family of global attractors because of the complicated boundary conditions for the perturbed problem fractional powers of the laplacian are not well defined moreover because of the restrictive growth assumptions on f the family of global attractors is obtained from the asymptotic compactness method developed by j ball for generalized semiflows with more relaxed assumptions on the nonlinear term f we are able to show the global attractors possess optimal regularity and prove the existence of an exponential attractor for each e 0 1 this result insures that the corresponding global attractor inherits finite fractal dimension however the dimension is not necessarily uniform in e

upper semicontinuity of attractors for nonclassical diffusion equations in h1 r3 abstract in this paper we mainly investigate upper semicontinuity of global attractors in h 1 r 3 for a perturbed nonclassical diffusion equations for this purpose some necessary abstract results are established

regularity of trajectory attractor and upper semicontinuity of global attractor for a 2d non newtonian fluid abstract there are two results within this paper the one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non newtonian fluid on 2d bounded domains for which the solution to each initial value could be non unique the other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from m to r l l where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m as m that is let a and a m be the global attractors of the fluid corresponding to and m respectively we establish that for any neighborhood o a of a the global attractor a m enters o a if m is large enough

reduction restrictions of darboux and laplace transformations for the goursat equation we study darboux and laplace transformations of the solutions and potentials of the goursat equation which is equivalent to one of the lax pair equations for the 2d mkdv hierarchy the reduction restrictions for these transformations are considered the derived reduction equations are generalizations of the liouville and sinh gordon equation the integrability of these equations by the st method is proved the binary darboux transformation for the goursat equation is suggested we find exact rational nonsingular solutions of the 2d mkdv equations via the moutard transformation for the goursat equation

box counting dimensions and upper semicontinuities of bi spatial attractors for stochastic degenerate parabolic equations on an unbounded domain abstract the paper contributes box counting dimensions and upper semicontinuities of random bi spatial attractors for stochastic degenerate parabolic equations on the whole euclid space under some weak assumptions for the force and the nonlinearity we first prove the existence of a unique l 2 d 0 1 2 l q random attractor for any q 2 p 2 i 2 where p 1 is the order of the nonlinearity and i is a given integer such that the force is i 1 times integrable by using truncation and splitting techniques and also induction methods we then prove that a priori estimate is uniform with respect to the density of noise which leads to the upper semicontinuity result of the obtained attractors as the density tends to a constant including zero under the topology of the terminative space furthermore we give a new framework to discuss the bound of l q box counting dimensions of random attractors for spde on an unbounded domain

attractivity properties of α contractions abstract it is known that if t x x is completely continuous where x is a banach space then point dissipative and compact dissipative are equivalent and imply the existence of a maximal compact invariant set which is uniformly asymptotically stable and attracts bounded sets uniformly if t is an contraction it is not known whether point dissipative and compact dissipative are equivalent however it is known that if t is an contraction and compact dissipative then there exists a maximal compact invariant set which is uniformly asymptotically stable and attracts a neighborhood of any compact set uniformly in this paper we show that for most practical examples which give rise to contraction point dissipative and compact dissipative are equivalent for example we show this is true for stable neutral functional differential equations retarded functional differential equations of infinite delay and strongly damped nonlinear wave equations we conjecture that this should be true for almost any practical application which gives rise to an contraction

novel integrable reductions in nonlinear continuum mechanics via geometric constraints the nonlinear equations that describe solitonic behavior in physical systems have to date typically been derived by approximation or expansion methods here by contrast hidden integrable structure is revealed in diverse areas of nonlinear continuum mechanics through natural geometric constraints

upper semicontinuity of attractors for lattice systems under singular perturbations abstract consider the following first order lattice system u m 2 u m u m 1 u m 1 m u m f m u m g m m z which is perturbed by the small two order term u m u m 2 u m u m 1 u m 1 m u m f m u m g m m z under certain conditions on f m m and g m the original systems and the small perturbed systems have global attractors a in l 2 and a in l 2 l 2 respectively and a can be naturally embedded into a compact set a 0 in l 2 l 2 we prove the upper semicontinuity of a 0 with respect to the attractors a at zero by showing that for any neighborhood o a 0 of a 0 a enters o a 0 if is small enough

compatible poisson brackets of hydrodynamic type some general properties of compatible poisson brackets of hydrodynamic type are discussed in particular a an invariant differential geometric criterion of the compatibility based on the nijenhuis tensor which is slightly different from those existing in the literature b the lax pair with a spectral parameter governing compatible poisson brackets in the diagonalizable case c the connection of this problem with the class of surfaces in euclidean space which possess non trivial deformations preserving the weingarten operator

transformations of quadrilateral lattices motivated by the classical studies on transformations of conjugate nets we develop the general geometric theory of transformations of their discrete analogs the multidimensional quadrilateral lattices i e lattices x zn rm n m whose elementary quadrilaterals are planar our investigation is based on the discrete analog of the theory of the rectilinear congruences which we also present in detail we study in particular the discrete analogs of the laplace combescure levy radial and fundamental transformations and their interrelations the composition of these transformations and their permutability is also investigated from a geometric point of view the deep connections between transformations and discretizations is also investigated for quadrilateral lattices we finally interpret these results within the formalism

on the sensitivity of the coupled continuous time lyapunov equation the sensitivity of coupled continuous time lyapunov equations is analyzed through three types of norms of the inverse lyapunov like operator this analysis leads to bounds on the relative perturbation of the solution the results extend the bounds obtained for the standard lyapunov equation in hewcer et al 1988

pullback attractors for a non autonomous generalized 2d parabolic system in an unbounded domain abstract the existence of a pullback attractor is proven for a non autonomous generalized 2d parabolic system in an unbounded domain the asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions

higher dimensional darboux transformations abstract we give conditions for a first order partial differential operator to serve as an n dimensional darboux transformation between two second order partial differential operators we show that some of these conditions are given by first order nonlinear partial differential equations which may be solved using the method of characteristics we also discuss the well known example of the boiti leon pempinelli equation and some others

new bound on the sensitivity of the solution of the lyapunov equation abstract a new bound on the size of the variation of the solution of the normal lyapunov equation under variation of the data is derived in contrast to a first order bound for differential perturbation the new bound is exact and still holds under large data variation it is shown that the new bound is less conservative than those currently available especially in the case of a high dimensional lyapunov equation

existence and continuity of uniform exponential attractors of the singularity perturbed hodgkin huxley system abstract in order to show that the standard partially dissipative hodgkin huxley system approximates the hyperbolic hodgkin huxley system for large time we are interested in establishing the existence of exponential attractors for the hyperbolic system with estimates of the fractal dimension and the rate of attraction of trajectories which are uniform with respect to the perturbation furthermore we obtain the continuity of exponential attractors

on the sensitivity of the coupled discrete time lyapunov equation the sensitivity of coupled discrete time lyapunov equations is analyzed through three types of norms of the inverse lyapunov like operator this analysis leads to bounds on the relative perturbation of the solution these results extend the results for standard lyapunov equation presented by gahinet et al 1990

sensitivity of the stable discrete time lyapunov equation the sensitivity of the stable discrete time lyapunov equation is analyzed through the spectral norm of the inverse lyapunov operator this leads to a directly computable easy to interpret sensitivity measure which involves only the open loop state matrix and also provides insight into the connection between sensitivity stability radius and conditioning of the eigenproblem of the open loop state matrix these results are an extension to the discrete time case of analogous results for the continuous time lyapunov equation

attractors for gradient flows of nonconvex functionals and applications this paper addresses the long time behaviour of gradient flows of nonconvex functionals in hilbert spaces exploiting the notion of generalized semiflows by j m ball we provide some sufficient conditions for the existence of a global attractor the abstract results are applied to various classes of nonconvex evolution problems in particular we discuss the long time behaviour of solutions of quasistationary phase field models and prove the existence of a global attractor

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

principal component analysis of flexible systems open loop case a generic class of flexible systems characterized by finitely many lightly damped harmonic oscillators is analyzed by means of the open loop principal component analysis that is singular value analysis and gramian balancing as the main result it is shown that as the damping ratio goes to zero the balanced state coordinates are decoupled and coincide with the modal coordinates further simple formulas expressing the asymptotic singular values as functions of the modal parameters are derived

finite dimensionality and upper semicontinuity of the global attractor of singularly perturbed hodgkin huxley systems abstract in an effort to show that the standard hodgkin huxley system approximates the hyperbolic hodgkin huxley system we view the hyperbolic system as a singular perturbation of the standard system we establish the existence of global attractors for the singularly perturbed hyperbolic system and show that they have finite fractal and hausdorff dimensions we then show that the global attractors of the singularly perturbed system converge to the global attractor for the standard system as the small perturbation parameter decreases to zero

singular dynamics of strongly damped beam equation abstract the paper is devoted to the dynamics of the model for a beam with strong damping pe e 2 u t t e u t u x x x x u t x x x x g 0 l u 2 d e 0 l u t u d u x x 0 x t 0 l 0 where g r r is continuously differentiable r and l e 0 subject to boundary conditions corresponding to hinged or clamped ends we show that for e 0 the dynamics of the equation is close to the dynamics of equation p0 u t u g 0 l u 2 d a 1 2 u where a u u x x x x with the domain determined by one of the above boundary conditions specifically we show that isolated invariant sets of p0 continue to isolated invariant sets of pe e 0 small having the same conley index moreover isolated morse decompositions with respect to p0 continue to isolated morse decompositions of pe e 0 small having isomorphic homology index braids under some additional assumptions we establish existence and upper semicontinuity results for attractors of e 0 small extending previous results by sevcovic

delay partial differential equations with some large diffusion it is often the case in infinite dimensional dynamical systems that a certain unbounded operator depends upon a parameter v its eigenvalues and eigenfunctions will also depend upon v in many cases see for example l 7 there is a finite number of eigenvalues of the unbounded operator that stay bounded as the parameter varies the remaining eigenvalues blow up at a certain rate in this case one expects the dynamics to be dictated by a system of ordinary differential equations this is indeed the case in several applications and its proof strongly uses the invariant manifold theory in the cases considered in this paper we study the dynamics of infinite dimensional dynamical systems for which a countable number of eigenvalues of the unbounded operator converges to the eigenvalues of an unbounded operator in a lower dimensional domain whereas the remaining blow up this is the case for example in parabolic equations in thin domains studied by hale and raugel s we will not be able to address the general case of thin domains but only the case of a thin straight channel however the property that allows this theory to work is present in the general case that is there is a sequence of eigenvalues that stay bounded and a sequence of eigenvalues that blow up see also arrieta 9 the theory of invariant manifolds will no longer work in this case since it is centered around the existence of backward solutions which cannot be expected in this situation the main tools in this work are a converse theorem on existence of compact attractors and an eigenvalue problem a physical interpretation of a parabolic problem in a thin straight channel is that there is a very high diffusion in one of the directions from which our intuition would say that in that direction spatial changes will not be observed the same interpretation can be given to a damped wave problem in a thin straight channel the wave speed can be interpreted as being very large in one direction this would intuitively imply that there would be no spatial changes for the amplitude of the wave in that direction we also consider the problem called shadow systems where the diffusion is very large for some of the substances in the reaction and not so large for the remaining substances in this

a damped hyperbolic equation on thin domains for a damped hyperbolic equation in a thin domain in r3 over a bounded smooth domain in r2 it is proved that the global attractors are upper semicontinuous it is shown also that a global attractor exists in the case of the critical sobolev exponent

pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on two dimensional unbounded domains this paper studies the pullback asymptotic behavior of solutions for a non autonomous non newtonian fluid on r l l for some l 0 we first use the technique of truncation functions together with the decomposition of spatial domain to prove the existence of a pullback attractor in then we discuss the upper semicontinuity of the pullback attractors when the spatial domains vary from m to where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m 1 m let a a r and am am r be the pullback attractors corresponding to and m respectively we establish for each r that for any neighborhood o a of a am enters o a if m is large enough

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found

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