postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

on the large time galerkin approximation of the navier stokes equations in this article we are interested in the difficult problem of relating the large time behaviour of the navier stokes equations to the large time behaviour of their galerkin approximation we restrict ourselves to the simplest situation dealing with stationary solutions and the question is the following one if the computed approximation of the time dependent equations seems to converge to some limit as t to infty is the same true for the exact problem and are the limits related we give here some sufficient conditions reasonably easy to verify which guarantee a positive answer to this question

a connection between quantum hilbert space and classical phase space operators we develop a general classicalization procedure that links hilbert space and phase space operators using weyl s operator then we transform the time dependent schrodinger equation into a phase space picture using free parameters they include position q and momentum p we expand the phase space hamiltonian in an taylor series and fix parameters with the condition that coefficients of 0 i1 q and i1 p vanish this condition results in generalized hamilton equations and a natural link between classical and quantum dynamics while the quantum motion equation remains exact in this picture the schrodinger equation reduces in the classical limit to a generalized liouville equation for the quantum mechanical system state we modify glauber s coherent states with a suitable phase factor s q p t and use them to obtain phase space representations of quantum dynamics and quantum mechanical quantities

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

explicit dimension estimates of attractors for the mhd equations in three dimensional space abstract in this article we estimate the dimension of functional invariant sets that arise from the mhd equations in terms of some physically relevant parameters we follow the analysis of constantin foias and temam for 3d navier stokes equations and derive comparable estimates

semiclassical asymptotics of perturbed cat maps we derive an exact representation for tr u n where u is the quantum propagator associated with an anosov perturbed cat map this takes the form of a sum over the fixed points of the n th iterate of the classical transformation the contribution of each one being given by an n fold multiple integral we focus in particular on the case when n 1 an asymptotic evaluation of the integral in question then leads to a complete semiclassical series expansion the first term of which corresponds to the gutzwiller tabor trace formula it is demonstrated that this series diverges but that summing it down to its least term provides an approximation to the quantum trace that is exponentially accurate in 1 a simple universal approximation to the late terms is then derived this explains the divergence of the semiclassical expansion in terms of complex tunnelling periodic orbits and implies the existence of unusual relations between different orbit actions it also allows us to recover the semiclassical contributions from the complex orbits explicitly using borel resummation these exponentially subdominant terms are shown to exhibit the stokes phenomenon which causes them to depend sensitively on the size of the perturbation parameter finally we develop an alternative expansion based on the orbits of the unperturbed cat map rather than diverging this is shown to converge absolutely thus making possible an exact calculation of the quantum trace using only classical mechanics its properties are however distinctly anti semiclassical

semiclassical wigner functions for quantum maps on a torus a semiclassical formula is derived for the wigner representation of the quasi energy states of a quantum map on a torus it is expressed as a finite sum over the classical fixed points of the map a criterion for the appearance of scars is presented semiclassical wigner functions of the cat map are calculated and shown to reproduce the exact quantum mechanical results

on stabilization of equilibria using predictive control with and without pulses consider a chaotic difference equation x n 1 f x n we focus on the problem of control of chaos using a prediction based control pbc method if f has a unique positive equilibrium it is proved that global stabilization of this equilibrium can be achieved under mild assumptions on the map f if f has several positive equilibria we demonstrate that more than one equilibrium can be stabilized simultaneously we also show that it is still possible to stabilize an unstable equilibrium using a strategy of control with pulses that is the control is only applied after a fixed number of iterations we illustrate our main results with several examples mainly from population dynamics

epidemic outbreaks and its control using a fractional order model with seasonality and stochastic infection abstract a fractional order sir epidemic model is proposed under the influence of both parametric seasonality and the external noise the integer order sir epidemic model originally is stable by introducing seasonality and noise force to the model behaviors of the system is changed it is shown that the system has rich dynamical behaviors with different system parameters fractional derivative order and the degree of seasonality and noise complexity of the stochastic model is investigated by using multi scale fuzzy entropy finally hard limiter controlled system is designed and simulation results show the ratio of infected individuals can converge to a small enough target which means the epidemic outbreak can be under control by the implementation of some effective medical and health measures

semiclassical dynamics in the strongly chaotic regime breaking the log time barrier we investigate the behavior of the quantum baker s transformation a system whose classical analogue is completely chaotic for time scales where the classical mechanics generates phase space structures on a scale smaller than planck s constant i e past the log time t ln 1 surprisingly we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime

galerkin and subspace decomposition methods in space and time for the navier stokes equations abstract the galerkin method and the subspace decomposition method in space and time for the two dimensional incompressible navier stokes equations with the h 2 initial data are considered the subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step t and a high frequency one discretized by the large time step p t with p 1 the h 2 stability and l 2 error analysis for the subspace decomposition method are obtained finally some numerical tests to confirm the theoretical results are provided

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

noisy dynamic simulations in the presence of symmetry data alignment and model reduction abstract we process snapshots of trajectories of evolution equations with intrinsic symmetries and demonstrate the use of recently developed eigenvector based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise our illustrative examples include a one dimensional evolutionary partial differential the kuramoto sivashinsky equation with periodic boundary conditions as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear smoluchowski equation on the surface of a sphere this is a useful first step towards data mining the symmetry adjusted ensemble of snapshots in search of an accurate low dimensional parametrization and the associated reduction of the original dynamical system we also demonstrate a technique vector diffusion maps that combines in a single formulation the symmetry removal step and the dimensionality reduction step

controlling one dimensional unimodal population maps by harvesting at a constant rate controlling chaos and periodic oscillations in dynamical systems is a well known problem it has been studied by several types of algorithms some of which were only demonstrated numerically with specific examples we study here a class of one dimensional discrete maps of the form x n 1 f x n where f x is a unimodal function satisfying a few natural conditions as a model for the dynamics of a single species population for managing the population we seek to suppress any possible chaotic or periodic behavior that may emerge the paper proposes a simple rigorously proved algorithm for controlling unimodal maps implemented by harvesting the population at a constant rate this forces the orbits of the map to converge to a limit that we can compute a priori the result holds for any initial conditions within an interval that we specify our control algorithm is easy to implement requires no updated information on the population and no changes in the parameters of the system which in general are fixed by the properties of the population it can therefore be useful for exploiting a population while maintaining it at a fixed density

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

a trace formula for the semiclassical limit of some hermitian operators the aim of this paper is to provide an algorithm for calculating the leading order contributions for the semiclassical series of a functional of trace class hermitian operators we base our study on the linear space structure of the space of bounded hermitian operators on quantum hilbert space we work with a basis of operators with natural leading order weyl symbols performing stationary phase calculations with no assumption on their smoothness the stationary points are tilted orbits near the periodic ones in phase space we manage to get control of the error in the technique which scales like some power a of 1 n with 1 2 a 1 the calculations are directly applicable to quantum maps and we provide some examples in finite dimension with quantum perturbed cat maps

comparison between constant feedback and limiter controllers using symbolic dynamics of the one dimensional unimodal map the chaos stabilization mechanics of the feedback and limiter control schemes are considered for feedback control it is found that the control strength can be efficiently obtained from the superstable parameter of the embedded periodic orbits and the scaling of the control space period doubling bifurcation cascade still obeys the feigenbaum law for sarkovskii orbits the scaling is also consistent with that of the original chaotic system for limiter control a single critical point in the unimodal map is extended to a superstable periodic window and a simple approach for determining the value of the control plateau is found the scaling in the control space of the period doubling bifurcation cascade is indeed superexponential a different scaling for the fine structure of the sarkovskii sequence is also found simple one dimensional unimodal maps can also be used to generate maximum length shift register sequences

unsteady two dimensional flows in complex geometries comparative bifurcation studies with global eigenfunction expansions we present a bifurcation study of the incompressible navier stokes equations in a model complex geometry a spatially periodic array of cylinders in a channel the dynamics of the flow include a hopf bifurcation from steady to oscillatory flow at an approximate reynolds number r of 350 and the appearance of a second frequency at approximately r simeq 890 the multiple frequency dynamics include a substantial increase in spatial and temporal scales with reynolds number as compared with the simple limit cycle oscillation present close to r 350 numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions the first basis set is derived through principal factor analysis karhunen loeve expansion of snapshots from accurate direct spectral element numerical solutions of the navier stokes equations the second set is obtained from the eigenfunctions of the stokes operator for this geometry finally eigenfunctions are derived from a singular stokes operator i e the stokes operator modified to include a variable coefficient which vanishes at the domain boundaries truncated systems of sim 100 odes are obtained through projection of the navier stokes equations onto the basis sets and a comparative study of the resulting dynamical models is performed

asymmetric unimodal maps some results from q generalized bit cumulants in this study using q generalized bit cumulants q is the nonextensivity parameter of the recently introduced tsallis statistics we investigate the asymmetric unimodal maps x t 1 1 ensuremath a x t z i i 1 2 correspond to x t g0 and x t l0 respectively z i g1 0lal 2 t 0 1 2 dots the study of the q generalized second cumulant c 2 q of these maps allows us to determine the dependence of the nonextensivity parameter q on the inflection parameter pairs z 1 z 2 the slope of the c 2 q versus c 2 1 plot where c 2 1 is the standard second cumulant provides the necessary tool to accomplish this task the slope behaves exactly the same as the proper q values say q that were obtained for logisticlike maps z 1 z 2 z by costa et al phys rev e 56 245 1997 it appears that as z 2 ensuremath z 1 ensuremath rightarrow ifmmode pm else textpm fi ensuremath infty this slope approaches unity this behavior is very similar to the behavior of q as a function of the inflection parameter for some z dependent maps namely as stackrel ensuremath rightarrow z ensuremath infty q approaches 1

a semiclassical quantization of area preserving maps abstract a semiclassical path integral formalism is developed for a class of area preserving maps the quasi energy spectrum of these systems is represented as a sum over closed orbits in the manner of gutzwiller the method is illustrated with some numerical tests of the standard map

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

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

on approximate inertial manifolds for two and three dimensional turbulent flows abstract in this paper we present some results concerning finite dimensional analytic manifolds that approximate the global attractor and the structure of the small eddies for the navier stokes equations under periodic boundary conditions we show that the smoother the external force the better the approximation

the geometric phase for chaotic systems the geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two form through a surface in the system s parameter space we obtain the classical limit of this two form in a form applicable to systems whose classical dynamics is chaotic for integrable systems the expression is equivalent to the hannay two form we discuss various properties of the classical two form derive semiclassical corrections to it associated with classical periodic orbits and consider implications for the semiclassical density of degeneracies

a semiclassical sum rule for matrix elements of classically chaotic systems in the semiclassical limit the sum s e delta e sigma nm mod anm mod 2 delta e 1 2 en em delta delta e en em of matrix elements of an arbitrary operator a can be related to the classical correlation function of the weyl symbol a q p of a ca e t integral d alpha delta e h alpha a alpha a alpha t s e delta e is proportional to the fourier transform of ca e t over t plus a set of correction terms associated with periodic trajectories in phase space if the system has a chaotic classical limit the matrix elements are independently gaussian distributed with mean value zero and s e delta e gives the variance of this distribution

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

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time