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

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

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

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

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

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

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

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

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

an analysis of the linear calibration controversy from the perspective of compound estimation the competing classical and inverse point estimators in statistical calibration can be obtained by direct or inverse regression and are to some extent supported by the maximum likelihood and bayesian approaches respectively both of these approaches depend on specific distributional assumptions but these assumptions confuse the main issue because both estimators can be justified without reference to them by using a compound estimation approach it is shown that the classical estimator can be derived as a linear compound estimator satisfying the criterion of asymptotic unbiasedness while the inverse estimator is a linear compound estimator without the unbiasedness constraint this formulation requires neither specific distributional assumptions nor reference to direct or inverse regression assessments of the two estimators are made in terms of their performance in estimating the current x value it is shown that superiority of the inverse estimator can only be guaranteed if the current x value is samp

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

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

the muntz szasz theorem and the closure of translates abstract we study the closure properties in various spaces of certain sequences of the form f t c k or exp c k ti f t as an application we give another proof of the muntz szasz theorem for l 2 0 1

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