long time dynamics of kirchhoff wave models with strong nonlinear damping we study well posedness and long time dynamics of a class of quasilinear wave equations with a strong damping we accept the kirchhoff hypotheses and assume that the stiffness and damping coefficients are functions of the l2 norm of the gradient of the displacement we prove the existence and uniqueness of weak solutions and study their properties for a wide class of nonlinearities which covers the case of possible degeneration or even negativity of the stiffness coefficient and the case of a supercritical source term our main results deal with global attractors for strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global finite dimensional attractor in the non supercritical case this attractor is strong in this case we also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals

lyapunov exponents from observed time series we examine the question of accurately determining lyapunov exponents for a time series we find that it is advantageous to use local mappings with higher order taylor series rather than linear maps as done earlier we demonstrate this procedure for the ikeda map and the lorenz system we present methods for identifying spurious exponents by analyzing data set singularities and by determining the lyapunov direction vectors the behavior of spurious exponents in the presence of noise is also investigated and found to be different from that of the true exponents

strongly damped wave equation with exponential nonlinearities abstract in this paper we study the initial boundary value problem for two dimensional strongly damped wave equation with exponentially growing source and damping terms we first show the well posedness of this problem and then prove the existence of the global attractor in h 0 1 l l 2

calculation of lyapunov exponents avoiding spurious elements abstract a new approach to evaluate lyapunov exponents from time series is discussed with the help of this approach the number of calculated lyapunov exponents can be reduced to the physically relevant ones calculations in high dimensional embedding spaces are performed on smaller data bases with improved speed more direct information is obtained from the dynamics of the underlying dynamical system

attractors for the degenerate kirchhoff wave model with strong damping existence and the fractal dimension abstract in this paper we study the long time behavior of the kirchhoff type wave equation in the space h 0 1 l 2 we prove the existence of the global attractor for the equation which covers the case of possible generation of the stiffness coefficient we also consider the geometrical property of the global attractor by means of the z 2 index we provide under suitable assumptions the fractal dimension of the global attractor is infinite

an accelerated exponential time integrator for semi linear stochastic strongly damped wave equation with additive noise abstract this paper is concerned with the strong approximation of a semi linear stochastic wave equation with strong damping driven by additive noise based on a spatial discretization performed by a spectral galerkin method we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise under appropriate assumptions we provide error bounds for the proposed full discrete scheme it is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem which allows for higher convergence rate than usual time stepping schemes particularly for the space time white noise case in two or three spatial dimensions the scheme still exhibits a good convergence performance another striking finding is that even for the velocity with low regularity the scheme always promises first order strong convergence in time numerical examples are finally reported to confirm our theoretical findings

on the strongly damped wave equation we prove the existence of the universal attractor for the strongly damped semilinear wave equation in the presence of a quite general nonlinearity of critical growth when the nonlinearity is subcritical we prove the existence of an exponential attractor of optimal regularity having a basin of attraction coinciding with the whole phase space as a byproduct the universal attractor is regular and of finite fractal dimension moreover we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient

infinite energy solutions for critical wave equation with fractional damping in unbounded domains abstract this work is devoted to infinite energy solutions of semi linear wave equations in unbounded smooth domains of r 3 with fractional damping of the form x 1 1 2 t u the work extends previously known results for bounded domains in finite energy case furthermore well posedness and existence of locally compact smooth attractors for the critical quintic non linearity are obtained under less restrictive assumptions on non linearity relaxing some artificial technical conditions used before this is achieved by virtue of new type lyapunov functional that allows to establish extra space time regularity of solutions of strichartz type

analysis and application of finite volume element methods to a class of partial differential equations abstract the finite volume element fve methods for a class of partial differential equations are discussed and analyzed in this paper the new initial values are introduced in the finite volume element schemes and we obtain optimal error estimates in l p and w 1 p 2 p as well as some superconvergence estimates in w 1 p 2 p the main results in this paper perfect the theory of the finite volume element methods

an existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach abstract we discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic pdes precisely we focus on the wave like equations that present also a nonzero source term and a first order in time linear term the paper carries on the research program initiated in 14 and developed in 15 21 on the de giorgi approach to hyperbolic equations

on properties of parisi measures we investigate the structure of parisi measures the functional order parameters of mixed p spin models in mean field spin glasses in the absence of external field we prove that a parisi measure satisfies the following properties first at all temperatures the support of any parisi measure contains the origin if it contains an open interval then the measure has a smooth density on this interval next we give a criterion on temperature parameters for which a parisi measure is neither replica symmetric nor one replica symmetry breaking finally we show that in the sherrington kirkpatrick model slightly above the critical temperature the largest number in the support of a parisi measure is a jump discontinuity an analogue of these results is discussed in the spherical mixed p spin models as a tool to establish these facts and of independent interest we study functionals of the associated parisi pdes and derive regularity properties of their solutions

strongly damped wave problems bootstrapping and regularity of solutions abstract the aim of the article is to present a unified approach to the existence uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in banach spaces our results are applied next to a number of examples appearing in literature which fall into the class of strongly damped semilinear wave equations the present work essentially extends the results on the existence and regularity of solutions to such problems previously these problems have been considered mostly within the hilbert space setting and with the main part operators being selfadjoint in this article we present a more general approach involving sectorial operators in reflexive banach spaces

strongly damped wave equation in uniform spaces abstract the cauchy problem for a semilinear strongly damped wave equation is considered in the whole of r n under suitable conditions imposed on nonlinear term which are much like for equations in bounded domains a dissipative semigroup t t of global solutions to this problem is constructed in a reach phase space being the product of the locally uniform spaces existence of an attractor for t t is then established

complex bi orthogonal decomposition of a chain of coupled wakes abstract this work is devoted to the experimental analysis of the flow created by a row of cylinders as each cylinder wake appears by the hopf bifurcation of a global oscillating mode this experiment is similar to a chain of coupled non linear oscillators and this dynamic system can be modelled using a ginzburg landau type equation a video analysis technique is performed on visualization pictures in order to obtain the spatio temporal evolution of the wakes this diagram is then analysed using the bi orthogonal decomposition extended to its complex form it is shown that only a few eigenmodes are excited giving a low dimensional dynamic system that enables the different parameters of the model to be estimated

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

finite dimensional attractors for the kirchhoff equation with a strong dissipation abstract the paper studies the existence of the finite dimensional global attractors and exponential attractors for the dynamical system associated with the kirchhoff type equation with a strong dissipation u t t m u 2 u u t h u t g u f x it proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor for application the fact shows that for the concerned viscoelastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity here a is the two dimensional laplacian g is a scalar function and uo u f are given functions equation 1 occurs in the description of antiplane shear motions of certain viscoelastic solids for background material the reader is referred to 20 let y t denote the position of a particle with reference position at time t and consider an infinite column 2 r consisting of a homogeneous isotropic incompressible solid with the semilinear constitutive equation

stability of exponential attractors for a family of semilinear wave equations with gentle dissipation abstract the paper investigates the stability of exponential attractors for a family of semilinear wave equations with gentle dissipation u t t u u t f u g x with 0 1 2 i we propose a new criterion on the existence and stability of a family of exponential attractors depending on the perturbation parameters ii by applying this criterion to the equations we construct a family of exponential attractors a e x p and show their stability on the dissipative exponent

a note on a strongly damped wave equation with fast growing nonlinearities a strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered the main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space these results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global lyapunov function

comparison of tests for embeddings it is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test the classical tests which depend on real numbers fractal dimensions lyapunov exponents averaged over an attractor are compared with a topological test that depends on integers the comparison can only be done for mappings into three dimensions we find that the classical tests fail to predict when a mapping is an embedding and when it is not we point out the reasons for this failure which are not restricted to three dimensions

long time behavior of strongly damped nonlinear wave equations abstract we consider the long time behavior of a strongly damped nonlinear wave equation we will show that the initial boundary value problem has a global solution and that there exists a compact global attractor with finite dimension

continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities abstract in this paper we discuss continuation properties and asymptotic behavior of regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions a necessary and sufficient condition for global in time existence of regular solutions is given we also formulate sufficient conditions to construct a piecewise regular solutions continuation beyond maximal time of existence for regular solutions applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed in particular global solvability and the existence of a global attractor for u t t d 1 2 u t d u f u in h 0 1 l 2 is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition similar result is obtained for a 2 m th order semilinear parabolic initial boundary value problem in a hilbert space h 2 b j m

topological analysis and synthesis of chaotic time series abstract we have developed a topological procedure for analyzing chaotic time series which identifies the stretching and squeezing mechanisms responsible for chaotic behavior in low dimensional dynamical systems these mechanisms quantitatively described by a template or knot holder can then be used to model the processes which generate the original chaotic data set

smooth attractors of finite dimension for von karman evolutions with nonlinear frictional damping localized in a boundary layer abstract in this paper dynamic von karman equations with localized interior damping supported in a boundary collar are considered hadamard well posedness for von karman plates with various types of nonlinear damping are well known and the long time behavior of nonlinear plates has been a topic of recent interest since the von karman plate system is of hyperbolic type with critical nonlinearity noncompact with respect to the phase space this latter topic is particularly challenging in the case of geometrically constrained nonlinear damping in this paper we first show the existence of a compact global attractor for finite energy solutions and we then prove that the attractor is both smooth and finite dimensional thus the hyperbolic like flow is stabilized asymptotically to a smooth and finite dimensional set

stability of attractors for the kirchhoff wave equation with strong damping and critical nonlinearities abstract the paper investigates longtime dynamics of the kirchhoff wave equation with strong damping and critical nonlinearities u t t 1 u 2 u u t h u t g u f x with 0 1 the well posedness and the existence of global and exponential attractors are established and the stability of the attractors on the perturbation parameter is proved for the ibvp of the equation provided that both nonlinearities h s and g s are of critical growth

large time behavior via the method of l trajectories abstract the method of l trajectories is presented in a general setting as an alternative approach to the study of the large time behavior of nonlinear evolutionary systems it can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics in both cases we prove the existence of a finite dimensional global attractor and the existence of an exponential attractor

global attractors for a third order in time nonlinear dynamics abstract long time behavior of a third order in time nonlinear pde equation is considered this type of equations arises in the context of nonlinear acoustics 12 20 22 24 where modeling accounts for a finite speed of propagation paradox the latter results in hyperbolic nature of the dynamics it will be proved that the underlying pde generates a well posed dynamical system which admits a global and finite dimensional attractor the main difficulty associated with the problem studied is the lack of lyapunov function along with the lack of compactness of trajectories which fact prevents applicability of standard tools in the area of dynamical systems

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

optimal approximation to a class of nonlinear evolution equations abstract in this paper we have developed and discussed a generalized difference method gdm for the approximation of a class of nonlinear evolution equations with integral items in one space dimension the main idea of the paper is to introduce a generalized volterra projection operator and we design a generalized difference formulation with a new approximate value of initial function which is of special importance for the error analysis then some theories of the generalized volterra projection operator are analyzed finally we deduce the optimal error estimates moreover some conclusions are obtained and our further research work is proposed

cauchy problem for quasi linear wave equations with viscous damping abstract the paper studies the global existence and asymptotic behavior of weak solutions to the cauchy problem for quasi linear wave equations with viscous damping it proves that when p max m where m 1 1 and p 1 are respectively the growth orders of the nonlinear strain terms the nonlinear damping term and the source term the cauchy problem admits a global weak solution which decays to zero according to the rate of polynomial as t as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition especially in the case of space dimension n 1 the solutions are regularized and so generalized and classical solution both prove to be unique comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms the states of the initial energy and the existence asymptotic behavior and nonexistence of global solutions of the cauchy problem

global existence and exponential stability of small solutions to nonlinear viscoelasticity the global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn dimensional euclidean space with smooth boundary dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to is described time periodic solutions are also studied as an application of our main theorem nonlinear viscoelasticity strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated

regularity of solutions on the global attractor for a semilinear damped wave equation we consider attractors a 0 1 corresponding to a singularly perturbed damped wave equation utt 2 a 1 2 ut aut au f u

longtime dynamics of the kirchhoff equations with fractional damping and supercritical nonlinearity abstract the paper investigates the global well posedness and the longtime dynamics of the kirchhoff equations with fractional damping and supercritical nonlinearity u t t m u 2 u u t f u g x with 1 2 1 the main results are focussed on the relationships among the growth exponent p of the nonlinearity f u the global well posedness and the longtime dynamics of the equations we show that i even if p is up to the supercritical range that is 1 p n 4 n 4 the well posedness and the longtime behavior of the solutions of the equation are of the characters of the parabolic equation ii when n 4 n 4 p n 4 n 4 the corresponding subclass g of the limit solutions exists and possesses a weak global attractor

finite fractal dimension of random attractor for stochastic non autonomous strongly damped wave equation abstract in this paper we first prove the existence of a random attractor for stochastic non autonomous strongly damped wave equations with additive white noise then we apply a criteria to obtain an upper bound of fractal dimension of the random attractor of considered system

asymptotic behaviour and changes of phase in one dimensional nonlinear viscoelasticity in this paper we study the asymptotic behaviour as t co of solutions u x t to the nonlinear partial differential equation ut m4 a o o 1 1 with initial conditions u 0 q x u x u x o 0 1 3a

a non autonomous strongly damped wave equation existence and continuity of the pullback attractor abstract in this paper we consider the strongly damped wave equation with time dependent terms u t t u t u t e t u t f u in a bounded domain r n under some restrictions on e t t and growth restrictions on the nonlinear term f the function e t depends on a parameter e e t e 0 0 we will prove under suitable assumptions local and global well posedness using the uniform sectorial operators theory the existence and regularity of pullback attractors a e t t r uniform bounds for these pullback attractors characterization of these pullback attractors and their upper and lower semicontinuity at 0

nonlinear second order equations with applications to partial differential equations where a and b are usually unbounded linear operators in a banach space these problems arise often in the study of partial differential equations as is usual we control a nonlinear perturbation possibly involving spatial derivatives by the linear terms which contain higher order spatial derivatives but contrary to the usual method of reducing the problem to a first order system in some energy norm space as in 9 5 we use the factoring method of lo this method allows the equation to be written as an integral equation containing a double integral involving the nonlinearity reflecting the fact that the equation is second order there are several advantages to this approach which will be illustrated fully in the examples first we show that the equation is locally well posed if loosely the nonlinearity satisfies the local lipschitz condition

finite dimensional attractors for the quasi linear strongly damped wave equation abstract we present a new method of investigating the so called quasi linear strongly damped wave equations t 2 u t x u x u f u x x u g in bounded 3d domains this method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non linearity is less than 6 and f may have arbitrary polynomial growth rate moreover the existence of a finite dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established in a particular case 0 which corresponds to the so called semi linear strongly damped wave equation our result allows to remove the long standing growth restriction f u c 1 u 5

global attractor for damped abstract nonlinear hyperbolic systems this work is concerned with the long time dynamics of a class of abstract nonlinear second order in time systems with damping this class of systems describes nonlinear dissipative elastic models with the nonlinear term produced by neo hookean type stress strain relationships in the present work we use an analysis based in part on the results of h t banks d s gilliam and v i shubov on the existence and uniqueness of the weak solutions to show that these systems generate a strong dynamical system also more importantly we are able to prove the existence of a compact strong global attractor finally we make several comments concerning the regularity of this attractor and present some examples

free energy of the spherical mean field model we compute at any temperature the free energy of the multi p spin spherical model when only terms for p even are considered

convergence of solutions to the equation of quasi static approximation of viscoelasticity with capillarity the long time behavior of solutions to the quasi steady approximation of the equation of viscoelasticity with capillarity is investigated we consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one dimensional we prove that any solution converges to an equilibrium point we admit some kind of initial data with infinite energy we use tools based on the assumption that the nonlinear term is real analytic

multiple levels of symmetry breaking in the previous paper in this volume we have studied the p spin interaction model just below the critical temperature and we have rigorously proved several aspects of the physicists prediction that this model exhibits one level of symmetry breaking in the present paper we show how to construct systems that exhibit an arbitrarily large but finite number of levels of symmetry breaking as the temperature decreases such systems exhibit many phase transitions as the structure of the overlaps gains complexity this phenomenon does not seem to have been described previously even in the physics literature

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

convergence analysis and numerical simulation for a hyperbolic system abstract in this paper we develop and analyze a finite volume element method for the approximation of the hyperbolic type equations arising in mathematical physics in one space dimension in recent years there has been increasing attention in the literatures to the development analysis and implementation of the solution of hyperbolic equations in this work we construct the discrete finite volume element schemes and give the optimal error estimates between the finite volume element solution and the exact solution moreover numerical experiments are also provided to demonstrate the behavior of the methods and confirm the efficiency and robustness of the presented approach and method finally some conclusions are obtained and our further research work is proposed

embeddings of low dimensional strange attractors topological invariants and degrees of freedom when a low dimensional chaotic attractor is embedded in a three dimensional space its topological properties are embedding dependent we show that there are just three topological properties that depend on the embedding parity global torsion and knot type we discuss how they can change with the embedding finally we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings these results apply only to chaotic attractors of genus one which covers the majority of cases in which experimental data have been subjected to topological analysis this means that the conclusions drawn from previous analyses for example that the mechanism generating chaotic behavior is a smale horseshoe mechanism a reverse horseshoe a gateau roul e an s template branched manifold etc are not artifacts of the embedding chosen for the analysis

cauchy problem for quasi linear wave equations with nonlinear damping and source terms abstract the paper studies the existence and non existence of global weak solutions to the cauchy problem for a class of quasi linear wave equations with nonlinear damping and source terms it proves that when max m p where m 1 1 and p 1 are respectively the growth orders of the nonlinear strain terms the nonlinear damping term and the source term under rather mild conditions on initial data the cauchy problem admits a global weak solution especially in the case of space dimension n 1 the weak solutions are regularized and so generalized and classical solution both prove to be unique on the other hand if 0 1 and the initial energy is negative then under certain opposite conditions any weak solution of the cauchy problem blows up in finite time and an example is shown

dynamics of strongly damped wave equations in locally uniform spaces attractors and asymptotic regularity this paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces after proving the global well posedness we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded in h 2 lu r n x h 1 lu r n subset which attracts exponentially every initial h 1 lu r n x l 2 lu r n bounded set with respect to the h 1 lu r n x l 2 lu r n norm then we show there is a h 1 lu r n x l 2 lu r n h 1 r n x h 1 r n global attractor which reflects the strongly damped property of u t to some extent

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

global topology from an embedding an embedding of chaotic data into a suitable phase space creates a diffeomorphism of the original attractor with the reconstructed attractor although diffeomorphic the original and reconstructed attractors may not be topologically equivalent in a previous work we showed how the original and reconstructed attractors can differ when the original is three dimensional and of genus one type in the present work we extend this result to three dimensional attractors of arbitrary genus this result describes symmetries exhibited by the lorenz attractor and its reconstructions

quasilinear equations for viscoelasticity of strain rate type we consider the quasilinear m m system of partial differential equations that governs the motion of a viscoelastic material of strain rate type on a bounded domain in mathbb r n the dependence of the stress on both the strain and strain rate is nonlinear and our hypotheses allow for a potential energy which is a nonconvex function of the strain the critical hypothesis is that the dependence of the stress function on the strain rate is uniformly strictly monotone in the sense of minty and browder we prove the existence and uniqueness of weak solutions to a natural initial boundary value problem for a large class of constitutive functions we then treat the question of h2 regularity of solutions and show that while regularity in the initial data is preserved solutions do not in general become more regular than their initial data this generalizes a result for the semilinear case due to rybka

limiting behavior for strongly damped nonlinear wave equations in this paper we investigate both the existence and the limiting behavior for the equation utt aut au t u ut where a is a sectorial operator is periodic in t and satisfies certain regularity and growth assumptions in most results on limiting behavior we will assume a has compact resolvent we consider the equation as an abstract ode defined on a paired space x x 0 1 with regard to the limiting behavior one of our principal results will be to show that if there is a bounded set in one of the spaces considered for which all points or trajectories enter into and remain then there is a set j consisting of very smooth functions defined on all of the spaces considered which is the maximum compact invariant set uniformly asymptotically stable connected and having very strong attractivity properties in all these spaces we will often show it attracts all points in a bounded set uniformly we will give a few sharper results for the case where a the work is motivated by recent papers of webb and fitzgibbon and applies techniques found in recent papers by the author

global existence for a quasi linear evolution equation with a non convex energy we establish the existence of global in time weak solutions to the initial boundary value problem related to the dynamics of coherent solid solid phase transitions in viscoelasticity the class of the stored energy functionals includes the double well potential and a general convolution damping term is considered

finite dimensional attractors for the kirchhoff models with critical exponents the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models utt u m 1 u ut 2u h ut g u f x it proves that for the subcritical and critical cases 1 m n 2 n 2 with n 6 the above mentioned dynamical system possesses in phase spaces x1 a finite dimensional global attractor and a weak exponential attractor respectively

existence of strong global attractors for hyperbolic equation with linear memory we prove that the existence of global attractors of strong solutions for the hyperbolic equations with linear memory using the semigroup approach

finite dimensional attractors for the kirchhoff models the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models arising in elasto plastic flow utt div u m 1 u ut 2u h ut g u f x by using the method of l trajectories and the operator technique it proves that under subcritical case 1 m n 2 n 2 the above mentioned dynamical system possesses in different phase spaces a finite dimensional weak global attractor and a weak exponential attractor respectively for application the fact shows that for the concerned elasto plastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the fractal dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

well posedness and attractor on the 2d kirchhoff boussinesq models abstract the paper studies the well posedness and the existence of attractors for a class of 2d kirchhoff boussinesq models u t t k u t 2 u d i v u 1 u 2 g u with 0 0 0 we show that i the ibvp of the equations is well posed in natural energy space x 2 and strong solution space x 4 respectively provided that g s c 1 s 2 ii the related solution semigroup has a global and an generalized exponential attractor in x 2 provided that the damping parameter k is suitably large and g s c iii in particular when 0 the corresponding boussinesq model has a subclass j of limit solutions and the subclass j has a weak global attractor in energy space x 1 without any upper bound restriction for the growth exponent of g u iv in the cases that either 0 or 0 the corresponding model has a global attractor in x 4 provided that g s c 1 s and without any restriction for the damping parameter k 0 especially when 0 the corresponding results extend those in grassell et al 2009

a reduced order energy stability preserving finite difference iterative scheme based on pod for the allen cahn equation abstract we focus on proposing and analyzing a reduced order finite difference rofd iterative scheme based on a proper orthogonal decomposition pod technique for allen cahn equations with a small perturbation parameter and strong nonlinearity we prove the discrete maximum principle dmp preserving and discrete energy stability des preserving of the rofd iterative solutions under a reasonable restriction on the time step size we also analyze the convergence of the rofd iterative solutions for allen cahn equations numerical tests are presented to verify our proposed rofd method such as error estimates convergence rates dmp preserving and des preserving moreover it is shown that the computing time of rofd iterative schemes with very few unknowns is much less than that of usual finite difference fd iterative schemes

when are projections also embeddings we study an autonomous four dimensional dynamical system used to model certain geophysical processes this system generates a chaotic attractor that is strongly contracting with four lyapunov exponents ensuremath lambda i that satisfy ensuremath lambda 1 ensuremath lambda 2 ensuremath lambda 3 l0 so the lyapunov dimension is d l 2 ensuremath mid ensuremath lambda 3 ensuremath mid ensuremath lambda 1 l3 in the range of coupling parameter values studied as a result it should be possible to find three dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior we study mappings into r 3 to determine which can be used as embeddings to reconstruct the dynamics we find dramatically different behavior in the two simplest mappings projections from r 4 to r 3 in one case the one parameter family of attractors studied remains topologically unchanged for all coupling parameter values in the other case during an intermediate range of parameter values the projection undergoes self intersections while the embedded attractors at the two ends of this range are topologically mirror images of each other

longtime dynamics of the kirchhoff equation with strong damping and critical nonlinearity on rn abstract the paper studies the longtime dynamics of the kirchhoff equation with strong damping and critical nonlinearity on r n u t t u t m u 2 u u t g x u f x we establish the well posedness the existence of the global and exponential attractors in natural energy space h h 1 r n l 2 r n in critical nonlinearity case these results improve not only the recent ones achieved by yang 2007 32 but also ones achieved by conti pata and squassina 2005 9 to some extent