a sharp lower bound for the hausdorff dimension of the global attractors of the 2d navier stokes equations for a special class of the navier stokes equations on the two dimensional torus we give a lower bound in the formg2 3 whereg is the grashof number for the hausdorff dimension of its global attractor which is optimal up to a logarithmic term

stability criteria for certain high even order delay differential equations abstract in this paper we study the asymptotic stability of the zero solution of even order linear delay differential equations of the form y 2 m t j 0 2 m 1 a j y j t j 0 2 m 1 b j y j t where a j and b j are certain constants and m 1 here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

variational bounds on energy dissipation in incompressible flows ii channel flow a variational principle for lower bounds on the time averaged mass flux for newtonian fluids driven by a pressure gradient in a channel is derived from the incompressible navier stokes equations when supplied with appropriate test background flow fields the variational formulation produces explicit estimates for the friction coefficient these rigorous bounds are compared with the predictions of conventional turbulence theory

on stability of systems of delay differential equations abstract in this paper we give necessary and sufficient conditions for the asymptotic stability of the zero solution of the system of linear delay differential equations of the form x t ax t 1 ax t where a is an n n matrix 0 is constant and 0 1 we reduce this to systems of first and second order problems our stability results are given in terms of the eigenvalues of a the proof of our results are carried out by an application of pontryagin s criterion for quasi polynomials to the characteristic functions of subsystems of the delay differential equations we also provide four algorithmic stability tests and include several examples

stability criteria for certain third order delay differential equations in this paper we study the asymptotic stability of the zero solution of third order linear delay differential equations of the form y t p1y t p2y t q1y t q2y t v1y t v2y t where p1 p1 q1 q2 v1 and v2 are certain constants here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

a determining form for the two dimensional navier stokes equations the fourier modes case the determining modes for the two dimensional incompressible navier stokes equations nse are shown to satisfy an ordinary differential equation ode of the form dv dt f v in the banach space x of all bounded continuous functions of the variable s r with values in certain finite dimensional linear space this new evolution ode named determining form induces an infinite dimensional dynamical system in the space x which is noteworthy for two reasons one is that f is globally lipschitz from x into itself the other is that the long term dynamics of the determining form contains that of the nse the traveling wave solutions of the determining form i e those of the form v t s v0 t s correspond exactly to initial data v0 that are projections of solutions of the global attractor of the nse onto the determining modes the determining form is also shown to be dissipative an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the grashof num

stability criteria for certain second order delay differential equations with mixed coefficients in this paper we study asymptotic stability of the zero solution of the second order linear delay differential equation y t p1y t p2y t q1y t q2y t where p1p2 q1 and q2 are constants with p1p2 0 and q1q2 0 is a constant in proving our results we make use of pontryagin s theory of quasi polynomials

on the construction of families of approximate inertial manifolds abstract the concept of approximate inertial manifold is related to the study of the longtime behaviour of partial differential equations in this paper we present a general method in order to construct sequences of such manifolds providing better and better approximations to the attractor le concept de variete inertielle approximative intervient dans l etude du comportement pour les grands temps des equations aux derivees partielles nous presentons dans cet article une methode generale pour construire des suites de telles varietes qui approximent de mieux en mieux l attracteur

rigorous estimates of small scales in turbulent flows we derive rigorous bounds on the length scale of determining local averages volume elements for the 3 d navier stokes equations these length scale estimates are related to kolmogorov s notion of a dissipation length scale in turbulent flows

bifurcation computations on an approximate inertial manifold for the 2d navier stokes equations abstract we compare the minimum number of modes needed to capture certain bifurcations of the 2d navier stokes equations by a certain nonlinear galerkin method and by the traditional galerkin method the boundary conditions are periodic and the forcing is in a single mode two different choices of the forcing mode are tested we introduce and implement a homotopy which allows continuation between approximations of different size

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

dynamical transition for a particle in a squared gaussian potential we study the problem of a brownian particle diffusing in finite dimensions in a potential given by 2 2 where is gaussian random field exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power law fashion at the dynamical transition temperature our results are confronted with numerical simulations where the gaussian field is constructed in a standard way as a sum over random fourier modes we show that when the number of fourier modes is finite the low temperature diffusion constant becomes non zero and has an arrhenius form thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition in addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model

an exponential stability criterion for nonlinear second order functional differential equations with time variable delays abstract this paper is concerned with the exponential stability for nonlinear second order functional differential equations fdes with time variable delays an exponential stability relationship between the fdes and the corresponding ordinary differential equations odes is derived it is proved under some appropriate conditions that the second order fdes can preserve the exponential stability of the corresponding odes this stability result is also illustrated with a numerical approach

stability criteria for certain high odd order delay differential equations in this paper we study the asymptotic stability of the zero solution of odd order linear delay differential equations of the formy 2 m 1 t j 02ma jy j t j 02mb jy j t t where a j and b j are certain constants and m 1 here t 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

random walk on percolation under an external field random walk on percolation under an external field has been investigated by using statistical analysis and monte carlo simulation the mean square displacement r2 as a function of time t was obtained there exist some steps in the log r2 logt plot the defect in the percolation plays the part of the trap for the walker each step corresponds to a type of trap the height of the steps indicates the mean square displacement before the walker enters the trap and the width of the step expresses the trapping time spent in the trap the calculated height and width by the probability theory are in agreement with simulation results quite well

determining modes for the surface quasi geostrophic equation abstract we introduce a determining wavenumber for the surface quasi geostrophic sqg equation defined for each individual trajectory and then study its dependence on the force while in the subcritical and critical cases this wavenumber has a uniform upper bound it may blow up when the equation is supercritical a bound on the determining wavenumber provides determining modes and measures the number of degrees of freedom of the flow or resolution needed to describe a solution to the sqg equation

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

low dimensional attractors arise from forcing at small scales abstract standard estimates of the dimension of the attractor of the 2d navier stokes equations are given in terms of a dimensionless grashof number that measures the total amount of energy injected into the flow however this result takes no account of whether the forcing is concentrated at large or small scales by a simple modification of the usual argument this paper provides a bound that is linear in a modified grashof number that reflects the spatial structure of the forcing in particular as a fixed amount of energy is injected at progressively smaller scales then the dimension of the attractor decreases

stability analysis of the zero solution for two class evolution equations abstract by constructing some suitable lyapunov type functionals and applying the theory of the definite quadratic form we obtain the stability of the zero solution of two classes of evolution equations with delays including reaction diffusion equations and damped wave equations our criteria depend on the derivatives of delays consequently when the delays are constants these criteria are independent of the magnitudes of the delays so the delays are harmless for the stability of the zero solution

stabilisation of second order nonlinear equations with variable delay for a wide class of second order nonlinear non autonomous models we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system the delays are variable which leads to more flexible controls permitting delay perturbations only delay bounds are significant for stabilisation by a delayed control the results are applied to the sunflower equation which has an infinite number of equilibrium points

linear and nonlinear response in the aging regime of the one dimensional trap model we investigate the behavior of the response function in the one dimensional trap model using scaling arguments that we confirm by numerical simulations we study the average position of the random walk at time t w t given that a small bias h is applied at time t w several scaling regimes are found depending on the relative values of t t w and h comparison with the diffusive motion in the absence of bias allows us to show that the fluctuation dissipation relation is valid even in the aging regime at least for times such that linear response is obeyed however for sufficiently long times the response always becomes nonlinear in h

discrete data assimilation in the lorenz and 2d navier stokes equations consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing in this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time this paper studies discrete data assimilation for the lorenz equations and the incompressible two dimensional navier stokes equations in both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution

stability criteria for linear time invariant systems with multiple delays two new asymptotic stability criteria for linear time invariant systems with multiple delays are established by using a quadratic inequality lemma together with frequency domain techniques one criterion is for delay dependent stability and another is for delay independent asymptotic stability each of them has only one tuning parameter matrix p 0 as the lyapunov type stability criteria for the corresponding linear time invariant systems without delays do both of the established stability criteria are given in the form of lmi problems and hence are very easy to use in applications in addition it is remarked that the established delay independent asymptotic stability criterion with one tuning parameter matrix p 0 improves and simplifies the existing criterion with more than one tuning parameter matrix in the literature

convergence of time averages of weak solutions of the three dimensional navier stokes equations using the concept of stationary statistical solution which generalizes the notion of invariant measure it is proved that in a suitable sense time averages of almost every leray hopf weak solution of the three dimensional incompressible navier stokes equations converge as the averaging time goes to infinity this system of equations is not known to be globally well posed and the above result answers a long standing problem extending to this system a classical result from ergodic theory it is also shown that from a measure theoretic point of view the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit finally any borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that borel set and for at least one leray hopf weak solution starting with that initial condition the corresponding orbit is recurrent to that borel subset and its mean sojourn time within that borel subset is strictly positive

observers for compressible navier stokes equation we consider a multidimensional model of a compressible fluid in a bounded domain we want to estimate the density and velocity of the fluid based on the observations for only velocity we build an

trap models with slowly decorrelating observables we study the correlation and response dynamics of trap models of glassy dynamics considering observables that only partially decorrelate with every jump this is inspired by recent work on a microscopic realization of such models which found strikingly simple linear out of equilibrium fluctuation dissipation relations in the limit of slow decorrelation for the barrat and mezard model with its entropic barriers we obtain exact results at zero temperature t for arbitrary decorrelation factor these are then extended to nonzero t where the qualitative scaling behaviour and all scaling exponents can still be found analytically unexpectedly the choice of transition rates glauber versus metropolis affects not just prefactors but also some exponents in the limit of slow decorrelation even complete scaling functions are accessible in closed form the results show that slowly decorrelating observables detect persistently slow out of equilibrium dynamics as opposed to intermittent behaviour punctuated by excursions into fast effectively equilibrated states

data assimilation algorithm for 3d benard convection in porous media employing only temperature measurements abstract in this paper we propose a continuous data assimilation downscaling algorithm for the benard convection in porous media using only discrete spatial mesh measurements of the temperature in this algorithm we incorporate the observables as a feedback nudging term in the evolution equation of the temperature we show that under an appropriate choice of the nudging parameter and the size of the mesh and under the assumption that the observed data is error free the solution of the proposed algorithm converges at an exponential rate asymptotically in time to the unique exact unknown reference solution of the original system associated with the observed finite dimensional projection of temperature data moreover we note that in the case where the observational measurements are not error free one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements

exponential decay of the power spectrum of turbulence the analyticity on a strip of the solutions of navier stokes equations in 2d is shown to explain the observed fast decay of the frequency power spectrum of the turbulent velocity field some subtleties in the application of the wiener khinchine method to turbulence are resolved by showing that the frequency power spectrum of turbulent velocities is in fact a measure exponentially decaying for frequency our approach also shows that the conventional procedures used in analyzing data in turbulence experiments are valid even in the absence of the ergodic property in the flow

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

estimates on enstrophy palinstrophy and invariant measures for 2 d turbulence we construct semi integral curves which bound the projection of the global attractor of the 2 d navier stokes equations in the plane spanned by enstrophy and palinstrophy of particular interest are certain regions of the plane where palinstrophy dominates enstrophy previous work shows that if solutions on the global attractor spend a significant amount of time in such a region then there is a cascade of enstrophy to smaller length scales one of the main features of 2 d turbulence theory the semi integral curves divide the plane into regions having limited ranges for the direction of the flow this allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy we also derive a sharp universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero

the behavior of solutions of second order delay differential equations in this paper we study the behavior of solutions of second order delay differential equation y t p1y t p2y t q1y t q2y t where p1 p2 q1 q2 are real numbers is positive real number a basic theorem on the behavior of solutions is established as a consequence of this theorem a stability criterion is obtained

kolmogorov theory via finite time averages abstract several relations from the kolmogorov theory of fully developed three dimensional turbulence are rigorously established for finite time averages over leray hopf weak solutions of the navier stokes equations the navier stokes equations are considered with periodic boundary conditions and an external forcing term the main parameter is the grashof number associated with the forcing term the relations rigorously proved in this article include estimates for the energy dissipation rate the kolmogorov wavenumber the taylor wavenumber the reynolds number and the energy cascade process for some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone while for others such as the kolmogorov energy dissipation law and the energy cascade the estimates depend also on the grashof number as compared with earlier works by some of the authors the more physical concept of finite time average is replacing the concept of infinite time average used before

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

microscopic realizations of the trap model monte carlo optimizations of number partitioning and of diophantine approximations are microscopic realizations of trap model dynamics this offers a fresh look at the physics behind this model and points at other situations in which it may apply our results strongly suggest that in any such realization of the trap model the response and correlation functions of smooth observables obey the fluctuation dissipation theorem even in the aging regime our discussion of the number partitioning problem may be relevant for the class of optimization problems whose cost function does not scale linearly with the size and are thus awkward from the statistical mechanic point of view

a class of numerical algorithms for large time integration the nonlinear galerkin methods the nonlinear galerkin methods are numerical schemes well adapted to the long term integration of evolution partial differential equations they consist of looking for approximate solutions lying in nonlinear manifolds that are close in some sense to the attractor the aim of this paper is to extend these schemes to a large class of manifolds convergence results are derived for the schemes introduced

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

on the number of determining nodes for the 2d navier stokes equations abstract it is known that the solutions of the 2d navier stokes equations in bounded domains are determined by a finite discrete set of nodal values that is if the large time behavior of the solutions to the navier stokes equations is known on an appropriate finite discrete set then the large time behavior of the solution itself is totally determined here an upper bound is rigorously established for the number of nodes needed to determine the solutions of the navier stokes equations in two dimensions with periodic boundary conditions

data assimilation for the navier stokes α equations abstract the well posedness of the data assimilation problem for the navier stokes equations on a bounded three dimensional domain is investigated the data assimilation procedures under consideration are the adjoint method of variational data assimilation 4d var and the method of continuous data assimilation concerning the adjoint method the existence of optimal initial conditions with respect to an observation dependent cost functional is proven the optimizers are characterized by a first order necessary condition involving the adjoint linearized navier stokes equations and conditions for the uniqueness of the initial conditions are given well posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided

accuracy and stability of filters for dissipative pdes data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system filters refer to a class of data assimilation algorithms designed to update the estimation of the state in an on line fashion as data is acquired sequentially for linear problems subject to gaussian noise filtering can be performed exactly using the kalman filter for nonlinear systems filtering can be approximated in a systematic way by particle filters however in high dimensions these particle filtering methods can break down hence for the large nonlinear systems arising in applications such as oceanography and weather forecasting various ad hoc filters are used mostly based on making gaussian approximations the purpose of this work is to study the accuracy and stability properties of these ad hoc filters we work in the context of the 2d incompressible navier stokes equation although the ideas readily generalize to a range of dissipative partial differential equations pdes by working in this infinite dimensional setting we provide an analysis which is useful for the understanding of high dimensional filtering and is robust to mesh refinement we describe theoretical results showing that in the small observational noise limit the filters can be tuned to perform accurately in tracking the signal itself filter accuracy provided the system is observed in a sufficiently large low dimensional space roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics the tuning corresponds to what is known as variance inflation in the applied literature numerical results are given which illustrate the theory the positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters can perform poorly in reproducing statistical variation about the true signal

on a criterion for locating stable stationary solutions to the navier stokes equations on donne une estimation superieure uniforme explicite a la borne de constantin foias ternam et on donne une forme applicable explicite aux conditions suffisantes on considere seulement le cas a 2 dimensions

a determining form for the damped driven nonlinear schrodinger equation fourier modes case in this paper we show that the global attractor of the 1d damped driven nonlinear schrodinger equation nls is embedded in the long time dynamics of a determining form the determining form is an ordinary differential equation in a space of trajectories x cb1 r pmh2 where pm is the l2 projector onto the span of the first m fourier modes there is a one to one identification with the trajectories in the global attractor of the nls and the steady states of the determining form we also give an improved estimate for the number of the determining modes

global in time stability and accuracy of imex fem data assimilation schemes for navier stokes equations abstract we study numerical schemes for incompressible navier stokes equations using imex temporal discretizations finite element spatial discretizations and equipped with continuous data assimilation a technique recently developed by azouani et al 2014 we analyze stability and accuracy of the proposed methods and are able to prove well posedness long time stability and long time accuracy estimates under restrictions of the time step size and data assimilation parameter we give results for several numerical tests that illustrate the theory and show that for good results the choice of discretization parameter and element choices can be critical

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

positive solutions of second order delay differential equations with a damping term in the paper the existence of positive solutions is studied for the second order delay differential equation with a damping term email protected t a t email protected t b t x h t 0 using a comparison with the integro differential equation email protected t email protected t 0 te s t a x d xb s y h s ds 0 explicit non oscillation criteria and comparison type results are derived

on the landau lifschitz degrees of freedom in 2 d turbulence we show that if the kraichnan theory of fully developed turbulence holds then the landau lifschitz degrees of freedom is bounded up to a logarithmic term by g1 2 where g is the grashof number

kraichnan turbulence via finite time averages relations central to kraichnan s theory of fully developed two dimensional turbulence are rigorously established for finite time averages in particular we prove that if the ratio of the averages of palinstrophy to enstrophy is large then a large inertial range displaying an enstrophy cascade exists moreover if this ratio is comparable up to a logarithm to the dissipation wave number a necessary condition for turbulence then the power law for the energy spectrum until now derived only heuristically is rigorously shown to provide up to a logarithm an upper bound for the energy spectrum finally we show that deep in the dissipation range the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number the averaging times needed for these relations are bounded in terms of the generalized grashof number independent of the solution for which the time averages are taken solutions are not assumed to be on the global attractor merely within the absorbing ball an easily verified condition

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

continuous data assimilation for the 2d benard convection through velocity measurements alone 2015 elsevier b v an algorithm for continuous data assimilation for the two dimensional benard convection problem is introduced and analyzed it is inspired by the data assimilation algorithm developed for the navier stokes equations which allows for the implementation of variety of observables low fourier modes nodal values finite volume averages and finite elements the novelty here is that the observed data is obtained for the velocity field alone i e no temperature measurements are needed for this algorithm we provide conditions on the spatial resolution of the observed data under the assumption that the observed data is free of noise which are sufficient to show that the solution of the algorithm approaches at an exponential rate the unique exact unknown solution of the benard convection problem associated with the observed finite dimensional projection of velocity

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

filter accuracy for the lorenz 96 model fixed versus adaptive observation operators in the context of filtering chaotic dynamical systems it is well known that partial observations if sufficiently informative can be used to control the inherent uncertainty due to chaos the purpose of this paper is to investigate both theoretically and numerically conditions on the observations of chaotic systems under which they can be accurately filtered in particular we highlight the advantage of adaptive observation operators over fixed ones the lorenz 96 model is used to exemplify our findings we consider discrete time and continuous time observations in our theoretical developments we prove that for fixed observation operator the 3dvar filter can recover the system state within a neighbourhood determined by the size of the observational noise it is required that a sufficiently large proportion of the state vector is observed and an explicit form for such sufficient fixed observation operator is given numerical experiments where the data is incorporated by use of the 3dvar and extended kalman filters suggest that less informative fixed operators than given by our theory can still lead to accurate signal reconstruction adaptive observation operators are then studied numerically we show that for carefully chosen adaptive observation operators the proportion of the state vector that needs to be observed is drastically smaller than with a fixed observation operator indeed we show that the number of state coordinates that need to be observed may even be significantly smaller than the total number of positive lyapunov exponents of the underlying system

stability for differential difference equations abstract in this paper we consider linear autonomous differential difference equations of retarded or neutral type some algebraic sufficient conditions for the zero solution to be stable hyperbolic for all the values of the delays are given and some simple criteria for the asymptotically stable cone to be convex are obtained