upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations this paper sets out a theoretical framework for approximating the attractor a of a semigroup s t defined on a banach space x by a q step semidiscretization in time with constant steplength k using the one step theory of hale lin and raugel sufficient conditions are established for the existence of a family of attractors a k x q for the discrete semigroups s k n defined by the numerical method the convergence properties of this family are also considered full details of the theory are exemplified in the context of strictly a stable linear multistep approximations of an abstract dissipative sectorial evolution equation

a modified real ga for the sparse linear array synthesis with multiple constraints this paper describes a modified real genetic algorithm mga for the synthesis of sparse linear arrays the mga has been utilized to optimize the element positions to reduce the peak sidelobe level psll of the array and here the multiple optimization constraints include the number of elements the aperture and the minimum element spacing unlike standard ga using fixed corresponding relationship between the gene variables and their coding the mga utilized the coding resetting of gene variables to avoid infeasible solution during the optimization process also the proposed approach has reduced the size of the searching area of the ga by means of indirect description of individual the simulated results confirming the great efficiency and the robustness of this algorithm are provided in this paper

equilibria of runge kutta methods it is known that certain runge kutta methods share the property that in a constant step implementation if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system such methods are calledregular in the present paper we provide a recursive test to check whether given method is regular moreover by examining solution trajectories of linear equations we prove that the order of ans stage regular method may not exceed 2 s 2 2 and that the maximal order of regular runge kutta method with an irreducible stability function is 4

a note on multistep methods and attracting sets of dynamical systems we consider a dynamical system described by an autonomous ode with an asymptotically stable attractor a compact set of orbitrary shape for which the stability can be characterized by a lyapunov function using recent results of eirola and nevanlinna 1 we establish a uniform estimate for the change in value of this lyapunov function on discrete trajectories of a consistent strictly stable multistep method approximating the dynamical system this estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in kloeden and lorenz 3 4 for 1 step methods

generalized analytical technique for the synthesis of unequally spaced arrays with linear planar cylindrical or spherical geometry an effective method for optimizing the performance of a fixed current distribution uniformly spaced antenna array has been to adjust its element positions to provide performance improvement in comparison with the default uniform structure this approach yields performance improvements such as smaller sidelobe levels or beamwidth values additionally it provides practical advantages such as reductions in size weight and number of antenna elements the objective of this paper is to describe a unified mathematical approach to nonlinear optimization of multidimensional array geometries the approach utilizes a class of limiting properties of sinusoidal bessel or legendre functions that are dictated by the array geometry addressed the efficacy of the method is demonstrated by its generalized application to synthesis of rectangular cylindrical and spherical arrays the unified mathematical approach presented below is a synthesis technique founded on the mathematical transformation of the desired field pattern followed by the application of limiting forms of the transformation and resulting in the development of a closed form expression for the element positions the method offers the following advantages over previous techniques such as direct nonlinear optimization or genetic algorithms first it is not an iterative searching algorithm and provides element spacing values directly in a single run of the algorithm thereby saving valuable cpu time and memory storage second it permits the array designer to place practical constraints on the array geometry e g the minimum maximum spacing between adjacent elements

reducing the number of elements in a linear antenna array by the matrix pencil method the synthesis of a nonuniform antenna array with as few elements as possible has considerable practical applications this paper introduces a new non iterative method for linear array synthesis based on the matrix pencil method mpm the method can synthesize a nonuniform linear array with a reduced number of elements and can be also used to reduce the number of elements for linear arrays designed by other synthesis techniques in the proposed method the desired radiation pattern is first sampled to form a discrete pattern data set then we organize the discrete data set in a form of hankel matrix and perform the singular value decomposition svd of the matrix by discarding the non principal singular values we obtain an optimal lower rank approximation of the hankel matrix the lower rank matrix actually corresponds to fewer antenna elements the matrix pencil method is then utilized to reconstruct the excitation and location distributions from the approximated matrix numerical examples show the effectiveness and advantages of the proposed synthesis method

numerical analysis of parabolic p laplacian approximation of trajectories the long time numerical approximation of the parabolic p laplacian problem with a time independent forcing term and sufficiently smooth initial data is studied convergence and stability results which are uniform for t in 0 infty are established in the l2 w1 p norms for the backward euler and the crank nicholson schemes with the finite element method fem this result extends the existing uniform convergence results for exponentially contractive semigroups generated by some semilinear systems to nonexponentially contractive semigroups generated by some quasi linear systems

a lyapunov type theorem for dissipative numerical integrators with adaptive time stepping the asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time stepping can be fully investigated by a lyapunov type theorem on continuous and discrete dynamical systems respectively however once adaptive time stepping is involved such theories cease to work and usually the dynamics should be investigated from the past instead of the standard forward way such as in terms of pullback attractors in this paper we present a different approach we stick to a forward definition of limit sets and show that still we can establish a lyapunov type theorem which reveals the precise asymptotic behavior of adaptive time stepping integrators in the presence of a discrete lyapunov functional

on invariant closed curves for one step methods we show that a one step method as applied to a dynamical system with a hyperbolic periodic orbit exhibits an invariant closed curve for sufficiently small step size this invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit the dynamics of the one step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression our results complement those of 2 3 where one step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a hopf bifurcation

stable attracting sets in dynamical systems and in their one step discretizations we consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set of arbitrary shape under the assumption of uniform asymptotic stability of in the sense of lyapunov we show that discretized versions of the dynamical system involving one step numerical methods have nearby attracting sets h which are also uniformly asymptotically stable our proof uses the properties of a lyapunov function which characterizes the stability of

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

nonlinear elliptic systems in stochastic game theory on etudie des systemes d equations aux derivees partielles non lineaires provenant de la theorie des jeux differentiels stochastiques au sens de nash

on stability of sets for sampled data nonlinear inclusions via their approximate discrete time models and summability criteria this paper consists of two main parts in the first part we provide a framework for stabilization of arbitrary not necessarily compact closed sets for sampled data nonlinear differential inclusions via their approximate discrete time models we generalize d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 1 in several different directions we consider stabilization of arbitrary closed sets plants described as sampled data differential inclusions and arbitrary dynamic controllers in the form of difference inclusions our result does not require the knowledge of a lyapunov function for the approximate model which is a standing assumption in d nesic and a r teel ieee trans automat control 49 2004 pp 1103 1122 and d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 2 we present checkable conditions that one can use to conclude semiglobal asymptotic stability or global exponential stability ges of the sampled data system via appropriate properties of its approximate discrete time model in the second part we present sufficient conditions for stability of parameterized difference inclusions that involve various summability criteria on trajectories of the system to conclude global asymptotic stability gas or ges and they represent discrete time counterparts of results given in a r teel e panteley and a loria math control signals systems 15 2002 pp 177 201 these summability criteria are not lyapunov based and they are tailored to be used within our above mentioned framework for stabilization of sampled data differential inclusions via their approximate discrete time models we believe that these tools will be a useful addition to the toolbox for controller design for sampled data nonlinear systems via their approximate discrete time models

a global local synthesis approach for large non regular arrays a method is proposed for the synthesis of large planar non regular arrays assuming constant magnitude for the excitation of the elements the approach which combines global and local optimization is based on distinguishing aperture type and non coherent parts of the array factor the following constraints are considered minimal separation of the antennas maximum size of the array and fixed mainbeam width the level of the sidelobes is reduced via the minimization of a new type of flexible averaging cost function based on an lp norm possible applications of this model lie in the field of radio astronomy satellite communications and radar systems the proposed optimization strategy consists of three successive steps designed to be independent of each other starting with an equivalent continuous aperture the first two steps act as global transformations of the radial and azimuthal positions of the elements in the initial array while the third step performs a local optimization of individual elements of the array this last step heavily relies on computations of the gradient of the cost function which can be done quickly using an fft based procedure the method is illustrated for several large scale examples by considering as inputs four different types of non regular arrays

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

geometric properties of runge kutta discretizations for index 2 differential algebraic equations we analyze runge kutta discretizations applied to index 2 differential algebraic equations daes the asymptotic features of the numerical and the exact solutions are compared it is shown that runge kutta methods satisfying the first order constraint condition of the dae correctly reproduce the geometric properties of the continuous system the proof combines embedding techniques of index 2 daes and ordinary differential equations odes with some invariant manifolds results of nipp and stoffer attractive invariant manifolds for maps sam research report 92 11 eth zurich switzerland 1992 the results support the favorable behavior of these runge kutta methods applied to index 2 daes for t ge 0

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type part i rk methods it is shown that appropriate linear multi step methods lmms applied to singularly perturbed systems of odes preserve the geometric properties of the underlying ode if the ode admits an attractive invariant manifold so does the lmm the continuous as well as the discrete dynamical system restricted to their invariant manifolds are no longer stiff and the dynamics of the full systems is essentially described by the dynamics of the systems reduced to the manifolds these results may be used to transfer properties of the reduced system to the full system as an example global error bounds of lmm approximations to singularly perturbed odes are given

conserving first integrals under discretization with variable step size integration procedures abstract it is well known that the application of one step or linear multistep methods to an ordinary differential equation with first integrals will destroy the conserving quantities with the use of stabilization techniques similar to ascher chin reich numer math 67 1997 131 149 we derive stabilized variants of our original problem we show that variable step size one step and linear multistep methods applied to the stabilized equation will reproduce that phase portrait correctly in particular this technique will conserve first integrals over an infinite time interval within the local error of the used method

attracting sets in index 2 differential algebraic equations and in their runge kutta discretizations we analyse runge kutta discretizations applied to autonomous index 2 differential algebraic equations in the vicinity of attracting sets we compare the geometric properties of the numerical and the exact solutions and show that projected and half explicit runge kutta methods reproduce the qualitative features of the continuous system correctly the proof combines invariant manifold results of schropp siam j numer anal to appear and classical results for discretized ordinary differential equations of kloeden and lorenz siam j numer anal 23 1986 986

half explicit runge kutta methods for differential algebraic systems of index 2 half explicit runge kutta methods for differential algebraic problems of index 2 are investigated it is shown how the arising order conditions can be solved and a particular method of order 4 is constructed in addition this paper simplifies the known convergence theory for such methods and demonstrates by numerical experiments their excellent properties when applied to constrained multibody systems

statistically thinned arrays with quantized element weights the use of statistical thinning and quantized element weights to produce low sidelobe patterns using large circular array apertures is investigated the major results of the analysis show that by using either of several thinning algorithms it is possible to obtain substantial sidelobe reduction by a combination of statistical thinning and the use of discrete amplitude quantization the paper includes generalized formulas for sidelobe level eirp effective isotropic radiated power and curves to show how these parameters vary as a function of the design sidelobe level for the selected taylor pattern

continuation method for computing certain singular points for index 1 differential algebraic equations we will use the arclength continuation method to compute certain singular points for index 1 daes backward difference method will be employed to descretize the problem we will show that the resulting numerical iterative matrix is nonsingular in the presence of the singularity some numerical examples will be given to show the efficiency of the resulting augmented system

old and new results in regularity theory for diagonal elliptic systems via blowup techniques we consider quasilinear diagonal elliptic systems in bounded domains subject to dirichlet neumann or mixed boundary conditions the leading elliptic operator is assumed to have only measurable coefficients and the nonlinearities hamiltonians are allowed to be of quadratic critical growth in the gradient variable of the unknown these systems appear in many applications in particular in differential geometry and stochastic differential game theory we impose on the hamiltonians structural conditions developed between 1972 2002 and also a new condition sum coerciveness introduced in recent years in the context of the pay off functional in stochastic game theory we establish existence holder continuity liouville properties w2 q estimates etc for solutions via a unified approach through the blow up method the main novelty of the paper is the introduction of a completely new technique which in particular leads to smoothness of the solution also for dimensions d 3

general linear methods connection to one step methods and invariant curves we generalize a result of kirchgraber 1986 on multistep methods we show that every strictly stable general linear method is essentially conjugate to a one step method of the same order this result may be used to show that general properties of one step methods carry over to general linear methods as examples we treat the existence of invariant curves and the construction of attracting sets

hyperbolic trajectories of time discretizations abstract the problem of persistence of trajectories of time discretizations is addressed analogues of essential concepts in the theory of hyperbolic flows are formulated for time discretizations we show these are appropriate by proving that the underlying flow inherits a hyperbolic trajectory of the time discretization approximation properties and inherited hyperbolicity properties are analyzed here explicit estimates in terms of the key problem parameters are provided an auxiliary result on roughness of exponential dichotomies for time discretizations may be of independent interest directions of future research are indicated

spatial discretization of mappings state space discretization occurs in the discrete finite arithmetic of a computer when a dynamical system is simulated by numerical computations it consequently evolves in a discretized space of this kind where attractors are seen in the simulation what is their relation to the theoretical structures if a theoretical attractor occurs should we expect always to see a computational attractor we address these questions by giving sufficient conditions for a discretized attractor to be present and show that it converges to the true attractor in the sense of convergence of compact sets in the hausdorff metric

on structural stability of ordinary differential equations with respect to discretization methods on compact m dimensional discs morse smale differential systems having no periodic orbits are considered the main result is that they are correctly reproduced by one step discretization methods for methods of order p and stepsize h sufficiently small the time h map of the induced local flow and the h discretized system are joined by a conjugacy o h p near to the identity the paper fits well in the rapidly growing list of results stating that hyperbolic transversal structures are preserved by discretization the proof relies heavily on techniques elaborated by robbin 1971 in establishing his structural stability theorem on self diffeomorphisms of compact c infty manifolds

design of subarrayed linear and planar array antennas with sll control based on an excitation matching approach in this paper the synthesis of subarrayed antennas with subarrays having different weights and dimensions is considered the problem is formulated as the definition of the element clustering into spatially contiguous subarrays and the corresponding weights to obtain a pattern with the lowest i e minimum achievable sidelobe level sll the problem is addressed with an unconstrained approach based on an excitation matching method where the synthesis of subarray weights and sizes is simultaneously carried out successively a hybrid version of such an approach is proposed to directly enforce into the optimization procedure the sll constraints a set of representative results concerned with both sum and difference patterns are shown furthermore some comparisons with state of the art hybrid evolutionary based techniques are reported as well

on the numerical approximation of phase portraits near stationary points we show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one step methods or multi step methods satisfying a strong root condition this means that any continuous trajectory can be approximated by an appropriate discrete trajectory and vice versa to the correct order of convergence and uniformly on arbitrarily large time intervals in particular the stable and unstable manifolds of the discretization converge to their continuous counterparts

unstable manifolds for rfdes under discretization the euler method abstract a generalization of the well known unstable manifold theorem near hyperbolic equilibrium points of functional differential equations is formulated the result says that the unstable manifold of the functional differential equation is close to its discretized counterpart if the stepsize of the discretization is sufficiently small

persistence of invariant sets for dissipative evolution equations we show that results concerning the persistence of invariant sets of ordinary differential equations under perturbation may be applied directly to a certain class of partial differential equations our framework is particularly well suited to encompass numerical approximations of these partial differential equations specifically we show that for a class of pdes with ac 1 inertial form certain natural numerical approximations possess an inertial form close to that of the underlying pde in thec 1 norm

on spurious fixed points of runge kutta methods in this paper we investigate the onset of spurious fixed points when runge kutta methods are applied to study the dynamics of differential equations it is shown computationally that the spurious equilibria of griffithset al 14 are connected at infinity with fixed points inherited from the differential equation we introduce and study the concept of b regularity which is in connection to the concept of regularity introduced by iserles principium cuius hinc nobis exordia sumet nullam rem e nilo gigni diuinitus umquam nam si de nilo fierent ex omnibus rebus omne genus nasci posset nil semine egeret de rerum natura liber primus 149 lucrecio

attraction rates robustness and discretization of attractors we investigate necessary and sufficient conditions for the convergence of attractors of discrete time dynamical systems induced by numerical one step approximations of odes to an attractor for the approximated ode we show that both the existence of uniform attraction rates i e uniform speed of convergence toward the attractors and uniform robustness with respect to perturbations of the numerical attractors are necessary and sufficient for this convergence property in addition we can conclude estimates for the rate of convergence in the hausdorff metric