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 stochastic kuramoto sivanshinsky equation in this article we study the solution of the kuramoto sivashinsky equation on a bounded interval subject to a random forcing term we show that a unique solution to the equation exists for all time and depends continuously on the initial data

lyapunov based boundary feedback design for parabolic pdes this paper presents a methodology for the construction of simple control lyapunov functionals clfs for boundary controlled parabolic partial differential equations pdes the proposed methodolog

the nsbeti method an extension of the feti method to non symmetrical bem fem coupled problems summary the finite element tearing and interconnecting feti method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second order elasticity problems in the boundary element tearing and interconnecting beti method a direct extension of the feti algorithm to the bem the symmetric galerkin bem formulation is used to obtain symmetric system matrices making possible to apply the same feti conjugate gradient solver in this work we propose a new beti variant labeled nsbeti that allows to couple substructures modeled with the fem and or non symmetrical bem formulations the method connects non matching bem and fem subdomains using localized lagrange multipliers and solves the associated non symmetrical flexibility equations with a bi cgstab iterative algorithm scalability issues of nsbeti in bem bem and combined bem fem coupled problems are also investigated copyright 2012 john wiley sons ltd

fredholm transform and local rapid stabilization for a kuramoto sivashinsky equation this paper is devoted to the study of the local rapid exponential stabilization problem for a controlled kuramoto sivashinsky equation on a bounded interval we build a feedback control law to force the solution of the closed loop system to decay exponentially to zero with arbitrarily prescribed decay rates provided that the initial datum is small enough our approach uses a method we introduced for the rapid stabilization of a korteweg de vries equation it relies on the construction of a suitable integral transform and can be applied to many other equations

local exact controllability to the trajectories of the 1 d kuramoto sivashinsky equation abstract we are concerned with the boundary controllability to the trajectories of the kuramoto sivashinsky equation by using a carleman estimate we obtain the null controllability of the linearized equation around a given solution from a local inversion theorem we get the local controllability to the trajectories of the nonlinear system

remarks on the kuramoto sivashinsky equation we report here a joint work in progress on the kuramoto sivashinsky equation the question we address is the analytical study of a fourth order nonlinear evolution equation this equation has been obtained by sivashinsky in the context of combustion and independently by kuramoto in the context of reaction diffusion systems both were motivated by nonlinear stability of travelling waves numerical calculations have been done on this equation all the results seem to indicate a chaotic behavior of the solution therefore the analytical study is of interest in analogy with the burger s and navier stokes equations here we give some existence and uniqueness results for the equation in space dimension one and we also study a fractional step method of numerical resolution in a forthcoming joint paper with r temam we will study the asymptotic behavior as t approaches infinity of the solution of 0 1 and give an estimate on the number of determining modes

wave structure interaction simulation driven by quantitative imaging we present a new approach to simulating wave structure interaction dynamics using proper orthogonal decomposition by directly employing eigenmodes extracted from particle image velocimetry experimental data a low dimensional galerkin model is constructed incorporating up to 12 modes and time dependent boundary conditions a penalty method is introduced to deal effectively with such boundary conditions our results suggest that this model represents accurately the flow dynamics and it is asymptotically stable without the use of any ad hoc dissipation or other stabilization schemes our findings are documented by applying this approach to studying the vorticity dynamics of wave cylinder interactions at keulegan carpenter numbers of 9 3 and 13 9

boundary control of viscosity degasperis procesi equation abstract this paper investigates the boundary control problem of the viscosity degasperis procesi equation in 0 1 results of the existence of a solution to the viscosity degasperis procesi equation in a short time interval under the boundary condition are given in the paper it proves the global exponential stability under the given boundary condition in l 2 h 1 h 2 h 3 space

stability of cellular states of the kuramoto sivashinsky equation this paper is concerned with the instability property of a particular type of solution to the kuramoto sivashinsky equation namely the symmetric time independent cellular states attention is focused on the dimension of their unstable manifolds we show that as a control parameter varies the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation further change in dimension is shown to be related to hopf bifurcations from the cellular states an asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur

linearized stability of partial differential equations with application to stabilization of the kuramoto sivashinsky equation linearization is a useful tool for analyzing the stability of nonlinear differential equations unfortunately the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations general results giving conditions for when stability or instability of the linearized equation implies the same for the nonlinear equation are given here these results are applied to stability and stabilization of the kuramoto sivashinsky equation a nonlinear partial differential equation that models reaction diffusion systems the stability of the equilibrium solutions depends on the value of a positive parameter nu it is shown that if nu 1 then the set of constant equilibrium solutions is globally asymptotically stable if nu 1 then the equilibria are unstable it is also shown that stabilizing the linearized equation implies local exponential stability of the equation stabilization of the kuramoto sivashinsky equation using a sin

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

invariant helical subspaces for the navier stokes equations three dimensional solutions with helical symmetry are shown to form an invariant subspace for the navier stokes equations uniqueness of weak helical solutions in the sense of leray is proved and these weak solutions are shown to be regular strong solutions existing for arbitrary time t the global universal attractor for the infinite dimensional dynamical system generated by the corresponding semi group of helical flows is shown to be compact and finite dimensional the hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class with the most detailed results obtained for rotating hagen poiseuille pipe flow in this case the dimension either hausdorff or fractal up to an absolute constant is bounded from above by frac operatorname re sqrt alpha 2 n 2 where is the axial wavenumber n is the azimuthal wavenumber and re is the reynolds number based on the radius of the pipe these upper bounds are independent of the rotation rate

adaptive output feedback stabilization of non local hyperbolic pdes we address the problem of adaptive output feedback stabilization of general first order hyperbolic partial integro differential equations pide such systems are also referred to as pdes with non local in space terms we apply control at one boundary take measurements on the other boundary and allow the system s functional coefficients to be unknown to deal with the absence of both full state measurement and parameter knowledge we introduce a pre transformation which happens to be based on backstepping of the system into an observer canonical form in that form the problem of adaptive observer design becomes tractable both the parameter estimator and the control law employ only the input and output signals and their histories over one unit of time prior to presenting the adaptive design we present the non adaptive baseline controller which is novel in its own right and facilitates the understanding of the more complex adaptive system the parameter estimator is of the gradient type based on a parametric model in the form of an integral equation relating delayed values of the input and output for the closed loop system we establish boundedness of all signals pointwise in space and time and convergence of the pde state to zero pointwise in space we illustrate our result with a simulation

gpbi cg generalized product type methods based on bi cg for solving nonsymmetric linear systems recently bi cgstab as a variant of bi cg has been proposed for solving nonsymmetric linear systems and its attractive convergence behavior has been confirmed in many numerical experiments bi cgstab can be characterized by its residual polynomial which consists of the product of the residual polynomial of bi cg with other polynomials generated from two term recurrence relations in this paper we propose a unified way to generalize a class of product type methods whose residual polynomials can be factored by the residual polynomial of bi cg and other polynomials with standard three term recurrence relations such product type methods which are based on bi cg can be regarded as generalizations of bi cgstab from the unified way the well known variants of the product type methods like cgs bi cgstab bi cgstab2 are reacquired again

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

convergence analysis and improvements of quantum behaved particle swarm optimization motivated by concepts in quantum mechanics and particle swarm optimization pso quantum behaved particle swarm optimization qpso was proposed as a variant of pso with better global search ability although it has been shown to perform well in finding optimal solutions for many optimization problems there has so far been little theoretical analysis on its convergence and performance this paper presents a convergence analysis and performance evaluation of the qpso algorithm and it also proposes two variants of the qpso algorithm first we investigate in detail the convergence of the qpso algorithm on a probabilistic metric space and prove that the qpso algorithm is a form of contraction mapping and can converge to the global optimum this is the first time that the theory of probabilistic metric spaces has been employed to analyze a stochastic optimization algorithm we provided a new definition for the convergence rate of a stochastic algorithm as well as definitions for three types of convergence according to the correlations between the convergence rate and the objective function values with these definitions the effectiveness of the qpso is evaluated by computing and analyzing the time complexity and the convergence rate of the algorithm then the qpso with random mean best position qpso rm and the qpso with ranking operator qpso ro are proposed as two improvements of the qpso algorithm finally some empirical studies on popular benchmark functions are performed in order to make a full performance evaluation and comparison between qpso qpso rm qpso ro and other variants of pso

global stabilization of a class of nonlinear reaction diffusion partial differential equations by boundary feedback this paper provides global exponential stabilization results by means of boundary feedback control for one dimensional nonlinear unstable reaction diffusion partial differential equations pdes wi

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

modal reduction of pde models by means of snapshot archetypes abstract a new method for constructing low dimensional reduced models of dissipative partial differential equations is proposed the original pde u t f u is projected onto a linear subspace spanned by the so called snapshot archetypes that are selected spatial profiles of u x t the selection rule of the snapshot archetypes characterizes the method two different selection methods are proposed we provide an energetic criterion for the minimum number of archetypes needed for an accurate approximation of the asymptotic dynamics this approach is tested for several pde systems such as the kuramoto sivashinsky equation the arneodo elezgaray reaction diffusion model and the self ignition dynamics of a coal stockpile the latter two systems exhibit a rich bifurcative structure and are suitable for checking the robustness of the snapshot archetype reduced models with respect to parameter variations

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

the steady states of the kuramoto sivashinsky equation abstract the kuramoto sivashinsky equation is one of the simple nonlinear partial differential equations it represents a system in which the transport of energy through nonlinear mode coupling produces a balance between long wavelength instability and short wavelength dissipation the steady states and their stabilities give a foundation for understanding the dynamics of the system they are distinguished according to their energy levels and fall into different branches that are identified by their harmonic content various types of solution include laminar states n cell states long wave distorbed n cell states the giant states and steady traveling solutions the linear stability around the steady states gives a guide to the origin of additional steady states through bifurcation and to the identification of each steady state the index rule is also useful conjectured form the numerical observation the parameter dependence of the steady traveling waves is estimated in the limit of large values of the parameter in addition a singular perturbation calculation of traveling waves near bifurcation is carried out the dynamics of a system can be guided by the systematic study of the bifurcation and stability for example if all the stationary and periodic solutions are unstable for some parameter range chaotic behavior of the system is expected since any orbit governed by the equation has to wander around forever in the phase space the orbits will then lie on strange attractors

a new global carleman estimate for the one dimensional kuramoto sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem abstract in this paper we establish a new global carleman estimate for the linearized kuramoto sivashinsky equation which is a one parameter carleman estimate based on this estimate we can obtain the local exact controllability to the trajectories and lipschitz stability in an inverse problem for the kuramoto sivashinsky equation

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

numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds we consider the numerical solution of nonlinear dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds coarse and fine grids are employed with a nonlinear mapping to relate the solutions computed on these grids the approach is illustrated by consideration of the kuramoto sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions numerical results are presented on the ability of the method to mimic the time evolution of the infinite dimensional dynamical system the results demonstrate that the coarse fine grid method is more accurate than the standard pseudospectral method on the same coarse grid however time integration studies suggest that the increased accuracy is bought at an increase in computational cost to achieve a specified accuracy the computational cost of the two grid method exceeds that of a standard pseudospectral method for this spatially periodic problem

drag reduction in transitional linearized channelow using distributed control this work focuses on feedback control of incompressible transitional newtonian channel flow described by the twodimensional linearized navier stokes equations the control objective is to use distributed feedback to achieve stabilization of the parabolic velocity profile for values of the reynolds number for which this profile is unstable and therefore to reduce the frictional drag exerted on the lower channel wall compared to the open loop values the control system uses measurements of shear stresses on the lower channel wall and the control actuation is assumed to be in the form of electromagnetic lorentz forces applied to the flow near the bottom wall galerkin s method is initially used to derive a high order discretization of the linearized flow field that captures the flow instability and accounts for the effect of control actuation on all the modes then a low order approximation of the linearized flow field is derived and used for the synthesis of a linear output feedback controller that enfor

fixed point theorems for multivalued mappings in some classes of fuzzy metic spaces abstract using the probabilistic function of noncompactness a fixed point theorem for multivalued mappings in probabilistic metric spaces is proved as an application a fixed point theorem for multivalued mappings in fuzzy metric spaces is obtained

computer assisted methods for the study of stationary solutions in dissipative systems applied to the kuramoto sivashinski equation we develop some computer assisted techniques for the analysis of stationary solutions of dissipative partial differential equations of their stability and of their bifurcation diagrams as a case study these methods are applied to the kuramoto sivashinski equation this equation has been investigated extensively and its bifurcation diagram is well known from a numerical point of view here we rigorously describe the full graph of solutions branching off the trivial branch complete with all secondary bifurcations for parameter values between 0 and 80 we also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

on traveling wave solutions of the kuramoto sivashinsky equation abstract previous numerical results indicate that the kuramoto sivashinsky equation admits three classes of non periodic traveling wave solutions namely regular shocks oscillatory shocks and solitary waves however it has been shown that regular monotonic shocks cease to exist in the weak shock limit due to the radiation of oscillatory waves of exponentially small with respect to the shock strength but growing in space amplitude here oscillatory shocks and solitary waves are constructed by asymptotic analysis it thus transpires that in the weak shock limit oscillatory shocks can only be antisymmetric otherwise oscillatory and monotonic waves of exponentially small but growing in space amplitude would inevitably be excited under certain conditions however the growing waves can link a nearly antisymmetric oscillatory shock with a radiating regular shock to form a solitary wave the predictions of the asymptotic theory are supported by numerical results

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

a class of product type krylov subspace methods for solving nonsymmetric linear systems abstract in the present paper a class of product type krylov subspace methods for solving nonsymmetric linear systems is discussed a characteristic of this class is the relationship r n h n a r n bcg where r n is the residual vector corresponding to the nth iterate x n and r n bcg is the nth residual generated in bi conjugate gradient method the polynomial hn is chosen to speed up and stabilize convergence while satisfying standard three term recurrence relations these product type methods can be regarded as unifications and generalizations of cgs and bi cgstab

a holistic finite difference approach models linear dynamics consistently i prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing for example the linear advection diffusion equation is found to be stably modelled for all advection speeds and all grid spacings the theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory when numerically solving nonlinear partial differential equations this approach can be used to systematically derive finite difference models which automatically have excellent characteristics their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems

wavelet projections of the kuramoto sivashinsky equation i heteroclinic cycles and modulated traveling waves for short systems abstract we study fairly low dimensional projections of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a short interval onto bases spanned by periodic wavelets such projections break the translation reflection symmetry o 2 replacing it by a finite dihedral group dk however we show that for the perrier basdevant wavelets used here the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved in particular we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets we use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and fourier modes which preserve full o 2 symmetry and are also optimal in that fourier truncations maximise the energy l2 norm among all finite dimensional models this study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains

local dissipativity and attractors for the kuramoto sivashinsky equation in thin 2d domains here 0 e 5 1 is assumed to be sufficiently small so that the basic periodicity cell sz 0 2n x 0 2ne is a thin domain we shall also mention other types of boundary conditions for which the same methods are applicable the dissipativity of the general two and higher dimensional problem has been open for some time the essential difficulty being the lack of a proof of the existence of an absorbing set in fact if such a set exists then there exists a global attractor and one can prove very precise regularity results excellent references for these as well as many refined results on the geometry of the attractor are the paper by nicolaenko et af l il yashenko 2 and the books by ttmam 3 and constantin et al 4 the approach we shall adopt is based on the intuitive idea that l 1 3 should be close to one dimensional and is close to the methods introduced by hale and raugel 5 61 and raugel and sell 7 there are many nontrivial results available for the periodic one dimensional problem the odd periodic case was studied by l 81 the same authors g treat a neumann type of problem without symmetry and the rigid dirichlet problem is studied by 9 again without any symmetry assumptions the general periodic one dimensional problem has only recently been proved to be dissipative 2 our main result is the following theorem

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

null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time varying feedback laws to this end on the one hand we provide a new proof of the well posedness and the optimal bound with respect to damping constants for the solutions of the kernel equations this allows one to deal with variable coefficients even with a weak regularity of these coefficients on the other hand we establish the well posedness and estimates for the heat equations with a nonlocal boundary condition at one side

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

explicit construction of an inertial manifold for a reaction diffusion equation abstract an inertial manifold is constructed for the scalar reaction diffusion equation u t vu xx u with a cubic nonlinearity uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria this uniformity leads to the global parameterization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium by the introduction of local techniques near each equilibrium we succeed in constructing an inertial manifold of lowest possible dimension

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

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

dissipation inequalities for the analysis of a class of pdes in this paper we develop dissipation inequalities for a class of well posed systems described by partial differential equations pdes we study passivity reachability induced input output norm boundedness and input to state stability iss we consider both cases of in domain and boundary inputs and outputs we study the interconnection of pde pde systems and formulate small gain conditions for stability for pdes polynomial in dependent and independent variables we demonstrate that sum of squares sos programming can be used to compute certificates for each property therefore the solution to the proposed dissipation inequalities can be obtained via semi definite programming the results are illustrated with examples

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

nonlinear galerkin method using chebyshev and legendre polynomials i the one dimensional case a new strategy stemming from the nonlinear galerkin method m marion and r temam siam j numer anal 26 1989 pp 1139 1157 for solving linear elliptic and nonlinear dissipative evolution equations by using chebyshev and legendre polynomials is presented the essential idea is to decompose the solution into a low mode part and a high mode part and to treat them separately the robustness of the method over the classical galerkin method is substantiated by rigorous error estimates and preliminary numerical experiments

the algebraic approximation of attractors the finite dimensional case abstract in this article we present a constructive method for approximating attractors the approximate sets are parts of algebraic sets i e the zeros of high degree polynomial functions and they can approximate the attractor at an arbitrary high level of accuracy

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

nonlinear stabilization of shock like unstable equilibria in the viscous burgers pde we stabilize the unstable ldquoshock likerdquo equilibrium profiles of the viscous burgers equation using control at the boundaries these equilibria are not stabilizable even locally using the standard ldquoradiation feedback boundary conditions rdquo using a nonlinear spatially scaled transformation that employs three ingredients of which one is the hopf cole nonlinear integral transformation and linear backstepping we design an explicit nonlinear full state control law that achieves exponential stability with a region of attraction for which we give an estimate the region of attraction is not the entire state space since the burgers pde is known not to be globally controllable

construction of inertial manifolds by elliptic regularization abstract in many cases an inertial manifold 2 m for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function from a finite dimensional hilbert space hp to another hilbert space hq the invariance property of m means that can be written as the solution of a first order partial differential equation d p g1 p p a p g2 p p 0 over hp where g1 and g2 are nonlinear functions which depend on the original dynamical system and a is a suitably stable linear operator in this paper we use a method introduced by sacker r j sacker a new approach to the perturbation theory of invariant surface comm pure appl math 18 1965 717 732 for the study of finite dimensional dynamical systems to find inertial manifolds in the infinite dimensional setting this method involves replacing the first order equation for by the regularized elliptic equation e d p g1 p p a p g2 p p with suitable boundary conditions it is shown that if a satisfies a spectral gap condition then the solutions e of the elliptic equation converge to a weak solution of 0 as e 0 furthermore m graph is an invariant manifold for the given dynamical system

global stabilization of the kuramoto sivashinsky equation via distributed output feedback control abstract this work addresses the problem of global exponential stabilization of the kuramoto sivashinsky equation kse subject to periodic boundary conditions via distributed static output feedback control under the assumption that the number of measurements is equal to the total number of unstable and critically stable eigenvalues of the kse and a necessary and sufficient stability condition is satisfied linear static output feedback controllers are designed that globally exponentially stabilize the zero solution of the kse the controllers are designed on the basis of finite dimensional approximations of the kse which are obtained through galerkin s method the theoretical results are confirmed by computer simulations of the closed loop system

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

a global attracting set for the kuramoto sivashinsky equation new bounds are given for the l2 norm of the solution of the kuramoto sivashinsky equation partial t u x t partial x 2 partial x 4 u x t u x t partial x u x t for initial data which are periodic with periodl there is no requirement on the antisymmetry of the initial data the result is mathop lim sup limits t to infty left u cdot t right 2 leqslant const l 8 5

steady solutions of the kuramoto sivashinsky equation abstract steady solutions of the kuramoto sivashinsky equation are studied these solutions are defined on the whole x line and propagate with a constant speed c 2 in time for large c 2 it is shown that the solution is unique and has a conical form for small c 2 there is a periodic solution and an infinite set of quasi periodic solutions as asserted by moser s twist map theorem numerical computations for intermediate values of c 2 suggest that below c 1 6 of every speed there is a continuum of odd quasi periodic solutions or a cantor set of chaotic solutions wrapped by infinite sequences of conic solutions

order and complexity in the kuramoto sivashinsky model of weakly turbulent interfaces abstract we present a large number of new geometric ergodic and statistical properties of the kuramoto sivashinsky equation modeling interfacial turbulence in various physical contexts in addition this equation has the remarkable property of inertial manifolds where some finite dimensional dynamical system is rigorously equivalent to this infinite dimensional partial differential equation in moderate size domains up to ten periods in length a low dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold the extreme numerical sensitivity of chaos in this dissipative pde requires very high precision methods despite the geometrical complexities of the bifurcation structure some statistical properties remain remarkably simple there is overwhelming evidence that for some parameter values a permanent unsteady state exists an unexpectedly simple diffusive relaxation of the large scale fluctuations is extracted from extensive numerical simulations in these calculations we observe long time tails for the correlation functions of relevant quantities we propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence

on the non homogeneous stationary kuramoto sivashinsky equation abstract in this note we give some explicit estimates for the l norm of the periodic solutions of the time independent non homogeneous kuramoto sivashinsky equation in particular we give an estimate of the michelson s upper bound of all periodic solutions of the time independent homogeneous kuramoto sivashinsky equation

approximation of attractors large eddy simulations and multiscale methods recent advances in the mathematical theory of the navier stokes equations have produced new insight in the mathematical theory of turbulence in particular the study of the attractor for the navier stokes equations produced the first connection between two approaches to turbulence that seemed far apart namely the conventional approach of kolmogorov and the dynamical systems theory approach similarly the study of the approximation of the attractor in connection with the newly introduced concept of approximate inertial manifolds has produced a new approach to large eddy simulations and the study of the interaction of small and large eddies in turbulent flows our aim in this article is to survey and describe some of the new results concerning the functional properties of the navier stokes equations and to discuss their relevance to turbulence

stability enhancement by boundary control in 2 d channel flow in this paper we stabilize the parabolic equilibrium profile in a two dimensional 2 d channel flow using actuators and sensors only at the wall the control of channel flow was previously considered by speyer and coworkers and bewley and coworkers who derived feedback laws based on linear optimal control and implemented by wall normal actuation with an objective to achieve global lyapunov stabilization we arrive at a feedback law using tangential actuation using teamed pairs of synthetic jets or rotating disks and only local measurements of wall shear stress allowing to embed the feedback in microelectromechanical systems mems hardware without need for wiring this feedback is shown to guarantee global stability in at least h sup 2 norm which by sobolev s embedding theorem implies continuity in space and time of both the flow field and the control as well as their convergence to the desired steady state the theoretical results are limited to low values of reynolds number however we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher

stability enhancement by boundary control in the kuramoto sivashinsky equation where we refer to 0 as the anti di ussion parameter note that a more general form ut 1uxxxx 2uxx 3uux 0 can always be reduced to 1 1 by appropriate rescaling of t x and u eq 1 1 was derived independently by kuramoto et al 20 22 as a model for phase turbulence in reaction di usion systems and by sivashinsky 33 as a model for plane ame propagation describing the combined in uence of di usion and thermal conduction of the gas on stability of a plane ame front so far it has been well understood that the ks equation can also serve as a mathematical model for cellular instabilities in a variety of situations the ow of thin liquid lms on inclined planes 30 in the limit of large surface tension dendritic fronts in dilute binary alloys 31 and alfven drift waves in plasmas 23 as a nonlinear saturation mechanism of the dissipative trapped ion modes

dynamics of a nonlocal kuramoto sivashinsky equation abstract in this paper we study the effects of a nonlocal term on the global dynamics of the kuramoto sivashinsky equation we show that the equation possesses a family of maximal attractors parameterized by the mean value of the initial data the dimension of the attractor is estimated as a function of the coefficient of the nonlocal term and the width of the periodic domain

the existence of steady solutions of the kuramoto sivashinsky equation abstract we investigate the existence of steady solutions of the kuramoto sivashinsky equation for wave speed c 1 we show that there are two periodic solutions and two heteroclinic orbits for the resultant third order ordinary differential equation a discussion is given concerning the possibility of using these solutions to obtain the existence of more complicated patterns

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded

chaos for a damped and forced kdv equation abstract the attractor for a damped and forced korteweg de vries equation is studied for large damping it is shown that the attractor is trivial with all solutions converging to a unique fixed point for small damping however the dynamics can be complicated it is investigated numerically how the attractor changes when the damping term decreases in particular a period doubling cascade of periodic solutions is found culminating with a chaotic regime the ratio between consecutive points in this cascade converging to feigenbaum s constant beyond this cascade the frequency spectrum of the solution is continuous and one lyapunov exponent is positive this seems to be the first work to obtain the onset of temporal chaos for this equation