a connection between state space and doubly coprime matrix fraction descriptions of multivariable systems abstract explicit formulas for doubly coprime matrix fraction descriptions mfds of the transfer matrix of a linear time invariant state space system are given in terms of a controllable and observable state space realization of the transfer matrix these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime mfds of multivariable systems

perturbation theory for dual semigroups i the sun reflexive case soit a 0 le generateur infinitesimal d un co semigroupe d operateurs lineaires bornes to t sur un espace de banach x soit b un operateur lineaire borne sur x a 0 b engendre un co semigroupe t t sur x b agit hors de l espace x dans un espace plus grand y on donne une methode systematique pour construire un tel espace y

stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations we prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation dot y t a t y t b t y t tau f t by a periodic boundary value approach this equation arises e g in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations we obtain convergence results for the standard collocation algorithm and for two variants in particular estimates of the difference between the collocation solution and the true solution are derived for the standard collocation scheme the convergence results are unconditional that is they do not require mesh ratio restrictions numerical results that support the theoretical findings are also given

center manifolds for periodic functional differential equations of mixed type abstract we study the behaviour of solutions to nonlinear functional differential equations of mixed type mfdes that remain sufficiently close to a prescribed periodic solution under a discreteness condition on the floquet spectrum we show that all such solutions can be captured on a finite dimensional invariant center manifold that inherits the smoothness of the nonlinearity this generalizes the results that were obtained previously in h j hupkes s m verduyn lunel center manifold theory for functional differential equations of mixed type j dynam differential equations 19 2007 497 560 for bifurcations around equilibrium solutions to mfdes

translation semigroups and their linearizations on spaces of integrable functions of concern is the unbounded operator abf f with nonlinear domain d av f e w li f o 4 f which is considered on the banach space e of bochner integrable functions on an interval with values in a banach space f under the assumption that 4 is a lipschitz continuous operator from e to f it is shown that as generates a strongly continuous translation semigroup tb t 1 0 for linear operators 4 several properties such as essential compactness positivity and irreducibility of the semigroup tob t i o depending on the operator 4 are studied it is shown that if f is a banach lattice with order continuous norm then tlbl t 0 is the modulus semigroup of td t o finally spectral properties of as are studied and the spectral bound s a0p is determined this leads to a result on the global asymptotic behavior in the case where 4 is linear and to a local stability result in the case where 4 is frechet differentiable

circumventing storage limitations in variational data assimilation studies an application of pontryagin s maximum principle data assimilation is used to blend possibly incomplete or nonuniformly distributed spatio temporal observational data into geophysical models used extensively in engineering control theory applications data assimilation has been introduced relatively recently into meteorological forecasting natural resource recovery modeling and climate dynamics variational data assimilation is a promising assimilation technique in which it is assumed that the optimal state of the system is an extrema of a carefully chosen cost function provided that an adjoint model is available the required model gradient can be computed by integrating the model forward and its adjoint backward the gradient is then used to extremize the cost function with a suitable iterative or conjugate gradient solver the problem addressed in this study is the explosive growth in both on line computer memory and remote storage requirements of computing the gradient by the forward adjoint technique which characterizes large scale assimilation studies storage limitations impose severe limitation on the size of assimilation studies even on the largest computers by using a recursive strategy a schedule can be constructed that enables the forward adjoint model runs to be performed in such a way that storage requirements can be traded for longer computational times this generally applicable strategy enables data assimilation studies on significantly larger domains than would otherwise be possible given the particular hardware constraints without compromising the outcome in any way furthermore it is shown that this tradeoff is indeed viable and that when the schedule is optimized the storage and computational times grow at most logarithmically

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition the proper orthogonal decomposition pod is shown to be an efficient model reduction technique for simulating physical processes governed by partial differential equations in this paper we make an initial effort to investigate problems related to pod reduced modeling of a large scale upper ocean circulation in the tropic pacific domain we construct different pod models with different choices of snapshots and different number of pod basis functions the results from these different pod models are compared with that of the original model the main findings are 1 the large scale seasonal variability of the tropic pacific obtained by the original model is well captured by a low dimensional system of order 22 which is constructed using 20 snapshots and 7 leading pod basis functions 2 the rms errors for the upper ocean layer thickness of the pod model of order 22 are less than 1m that is less than 1 of the average thickness and the correlation between the upper ocean layer thickness with that from the pod model is around 0 99 3 retaining modes that capture 99 energy is necessary in order to construct pod models yielding a high accuracy

on duality and spectral properties of a k regularized resolvents we construct a duality theory for a k regularized resolvents extending some of the known theorems for dual semigroups we present several classes of spaces which in the semigroup case correspond to the favard class and the sun dual space by duality arguments spectral inclusions theorems for regularized resolvents are also obtained

classification and construction of quantum communication systems we consider quantum communication systems qcs s introduced by davies 10 on trace class operators on general state spaces these are different from the usual quantum information theory as they deal with input and output being continuous in time we introduce the concept of a refinement of a qcs offering the possibility of distinguishing particles for instance according to their phase or location and classify refinements of general qcs we then introduce the notions of a bounded interaction rate and a bounded modulation and give in special cases a classification of qcs s satisfying both conditions as this classification shows the existence of a class of qcs s being completely different from the examples studied by davies we are able to discuss the cases of an independent detector and a linear modulation

on the dominance of roots of characteristic equations for neutral functional differential equations we present a sufficient condition for a zero of a function that arises typically as the characteristic equation of a linear functional differential equations of neutral type to be simple and dominant this knowledge is useful in order to derive the asymptotic behaviour of solutions of such equations a simple characteristic equation arisen from the study of delay equations with small delay is analyzed in greater detail

the closure of the generalised eigenspace of a class of infinitesimal generators in this paper we study the fine geometric structure of a class of strongly continuous semigroups that satisfy the following property the resolvent of the infinitesimal generator can be represented as the quotient of entire functions of finite exponential type this class includes the solution map for functional differential equations and certain partial differential equations in particular we present necessary and sufficient conditions for one to oneness of the solution map and for completeness of the system of generalised eigenfunctions of the generator

a reduced order extrapolation central difference scheme based on pod for two dimensional fourth order hyperbolic equations the reduced order extrapolating fd schemes are just developed by luo s research team since 2013 see 47 52 and a very new technique the reduced order extrapolation central difference scheme based on pod for two dimensional fourth order hyperbolic equations is first built by us we provide theoretical analysis and examples to show the advantage of our method this paper is concerned with establishing the reduced order extrapolation central difference roecd scheme based on proper orthogonal decomposition pod for two dimensional 2d fourth order hyperbolic equations for this purpose we first develop the classical central difference cd scheme for the 2d fourth order hyperbolic equations and analyze its stability and convergence then by making use of the pod method we build the roecd scheme with fewer degrees of freedom and sufficiently high accuracy and furnish the error estimates of the roecd solutions and the algorithm procedure for solving the roecd scheme finally we employ some numerical examples to confirm the correctness of theoretical conclusions this implies that roecd scheme is feasible and efficient for seeking the numerical solutions of the 2d fourth order hyperbolic equations

a variation of constants formula for abstract functional differential equations in the phase space abstract for linear functional differential equations with infinite delay in a banach space a variation of constants formula is established in the phase space as an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line

continuation of bifurcations in periodic delay differential equations using characteristic matrices in this paper we describe a method for continuing periodic solution bifurcations in periodic delay differential equations first the notion of characteristic matrices of periodic orbits is introduced and equivalence with the monodromy operator is demonstrated an alternative formulation of the characteristic matrix is given which can be computed efficiently defining systems of bifurcations are constructed in a standard way including the characteristic matrix and its derivatives for following bifurcation curves in two parameters the pseudo arclength method is used combined with newton iteration two test examples an interrupted machining model and a traffic model with driver reaction time conclude the paper the algorithm has been implemented in the software tool pdde cont

pod deim nonlinear model order reduction of an adi implicit shallow water equations model in the present paper we consider a 2 d shallow water equations swe model on a b plane solved using an alternating direction fully implicit adi finite difference scheme on a rectangular domain the scheme was shown to be unconditionally stable for the linearized equations the discretization yields a number of nonlinear systems of algebraic equations we then use a proper orthogonal decomposition pod to reduce the dimension of the swe model due to the model nonlinearities the computational complexity of the reduced model still depends on the number of variables of the full shallow water equations model by employing the discrete empirical interpolation method deim we reduce the computational complexity of the reduced order model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the pod model to emphasize the cpu gain in performance due to use of pod deim we also propose testing an explicit euler finite difference scheme ee as an alternative to the adi implicit scheme for solving the swallow water equations model we then proceed to assess the efficiency of pod deim as a function of number of spatial discretization points time steps and pod basis functions as was expected our numerical experiments showed that the cpu time performances of pod deim schemes are proportional to the number of mesh points once the number of spatial discretization points exceeded 10000 and for 90 deim interpolation points the cpu time decreased by a factor of 10 in case of pod deim implicit swe scheme and by a factor of 15 for the pod deim explicit swe scheme in comparison with the corresponding pod swe schemes moreover our numerical tests revealed that if the number of points selected by deim algorithm reached 50 the approximation errors due to pod deim and pod reduced systems have the same orders of magnitude thus supporting the theoretical results existing in the literature

adaptive reduced order controllers for a thermal flow system using proper orthogonal decomposition an adaptive reduced order controller design is presented for flow control using proper orthogonal decomposition pod in reduced order controller design the idea is to start with an ensemble of data obtained from numerical simulation of the underlying partial differential equations pdes pod is then used to obtain a reduced set of basis functions which is then used to derive a reduced order model of the pdes via galerkin projection this reduced order model allows us to derive a reduced order controller however it is not clear a priori what is the best way to obtain an ensemble of data that would give basis functions that represent the influence of the control action on the system in this paper we explore an adaptive procedure for reduced order controller design that improves the reduced order model by successively updating the ensemble of data during the optimization iterations we illustrate this method on a control problem in thermal flow system modeled by a thermally coupled navier stokes equations numerical results are presented for a vorticity regulation problem in fluid flows using boundary temperature as control mechanism through our numerical experiments we demonstrate the feasibility and applicability of the adaptive reduced order controllers

integrated semigroups and integrated solutions to abstract cauchy problems abstract concepts which have been developed in the theory of weakly continuous semigroups on dual banach spaces are extended to integrated semigroups the relation to integrated solutions of homogeneous and inhomogeneous abstract cauchy problems is established the results are illustrated for the wave equation in l2 rm

on eigenstructure assignment in linear multivariable systems the eigenvalue assignment approach of brogan 1 2 is generalized and extended to the assignment of the entire closed loop eigenstructure of linear multivariable systems the set of assignable eigenvectors and generalized eigenvectors emerges naturally in the solution and is given moreover in an explicit parametric form two numerical examples are worked out to demonstrate the application of the procedure

a reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for burgers equation in this article a reduced optimizing finite difference scheme fds based on singular value decomposition svd and proper orthogonal decomposition pod for burgers equation is presented also the error estimates between the usual finite difference solution and the pod solution of reduced optimizing fds are analyzed it is shown by considering the results obtained for numerical simulations of cavity flows that the error between the pod solution of reduced optimizing fds and the solution of the usual fds is consistent with theoretical results moreover it is also shown that the reduced optimizing fds is feasible and efficient

a combination of proper orthogonal decomposition discrete empirical interpolation method pod deim and meshless local rbf dq approach for prevention of groundwater contamination abstract the main presented idea is to reduce the used cpu time for employing the local radial basis functions differential quadrature lrbf dq method to this end the proper orthogonal decomposition discrete empirical interpolation method pod deim has been combined with the lrbf dq technique for checking the ability of the new procedure the groundwater equation is solved this equation has been classified in category of system of advection diffusion equations the solutions of advection equations have some shock thus special numerical methods should be applied for example discontinuous galerkin and finite volume methods moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes

generalized jordan chains and two bifurcation theorems of krasnoselskii abstract given two banach spaces x and y over k r or c and a parameterized family a mu an element of l x y with mu an element of k partial and algebraic multiplicities of any value mu sub 0 an element of k such that a mu sub 0 if fredholm with index zero are defined by the means of generalized jordan chains these notions are developed in close connection with bifurcation problems and we show that partial and algebraic multiplicities are not affected by lyapunov schmidt reduction properties of invariance under equivalence are also established these general results are used to give a proof of magnus generalization of the classical bifurcation theorem by krasnoselskii through a somewhat more natural approach than his but the convincing evidence of the usefulness of the notions developed here has to be found in a new and wide extension of the boehme marino rabinowitz theorem on bifurcation for gradient operators the ancestor of which is also due to krasnoselskii

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

chaotic dynamics of coherent structures abstract a variety of chaotic flows evolving in relatively high dimensional spaces are considered it is shown through the use of an optimal choice of basis functions which are consequence of the karhunen loeve procedure that an accurate description can be given in a relatively low dimensional space particular examples of this procedure which are presented are the ginzburg landau equation turbulent convection in an unbounded domain and turbulent convection in a bounded domain

perturbation theory for dual semigroups ii time dependent perturbations in the sun reflexive case we consider time dependent perturbations of generators of strongly continuous semigroups on a banach space the perturbations map the banach space into a bigger space which is the second dual of the original space in a specific semigroup sense using the theory of dual semigroups we show that the solutions of a generalised variation of constants formuladefine an evolutionary system we investigate continuity and differentiability propertiesof this evolutionary system and its dual system and examine in what sense the perturbed generator and its adjoint generate these evolutionary systems it is shown that the results apply naturally to retarded functional differential equations and age structured population dynamics

delay and periodicity systems with time delay play an important role in modeling of many physical and biological processes in this paper we describe generic properties of systems with time delay which are related to the appearance and stability of periodic solutions in particular we show that delay systems generically have families of periodic solutions which are reappearing for infinitely many delay times as delay increases the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions we also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers we show that the spectrum of multipliers can be split into two parts pseudocontinuous and strongly unstable the pseudocontinuous part of the spectrum mediates destabilization of periodic solutions

reduced order modeling based on pod of a parabolized navier stokes equations model ii trust region pod 4d var data assimilation a reduced order model based on proper orthogonal decomposition pod 4d var four dimensional variational data assimilation for the parabolized navier stokes pns equations is derived various approaches of pod implementation of the reduced order inverse problem are studied and compared including an ad hoc pod adaptivity along with a trust region pod adaptivity the numerical results obtained show that the trust region pod 4d var provides the best results amongst all the pod adaptive methods tested in all error metrics for the reduced order inverse problem of the pns equations

linearized reduced order models for subsurface flow simulation a trajectory piecewise linearization tpwl procedure for the reduced order modeling of two phase flow in subsurface formations is developed and applied the method represents new pressure and saturation states using linear expansions around states previously simulated and saved during a series of preprocessing training runs the linearized representation is projected into a low dimensional space with the projection matrix constructed through proper orthogonal decomposition of the states determined during the training runs the tpwl model is applied to two example problems containing 24 000 and 79 200 grid blocks which are characterized by heterogeneous permeability descriptions extensive test simulations are performed for both models it is shown that the tpwl model provides accurate results when the controls bottom hole pressures of the production wells in this case applied in test simulations are within the general range of the controls applied in the training runs even though the well pressure schedules for the test runs can differ significantly from those of the training runs this indicates that the tpwl model displays a reasonable degree of robustness runtime speedups using the procedure are very significant a factor of 100 2000 depending on model size and whether or not mass balance error is computed at every time step for the cases considered the preprocessing overhead required by the tpwl procedure is the equivalent of about four high fidelity simulations finally the tpwl procedure is applied to a computationally demanding multiobjective optimization problem for which the pareto front is determined limited high fidelity simulations demonstrate the accuracy and applicability of tpwl for this optimization future work should focus on error estimation and on stabilizing the method for large models with significant density differences between phases

application of proper orthogonal decomposition to a supersonic axisymmetric jet results are presented from the application of the snapshot proper orthogonal decomposition pod method to a spatiotemporal e owe eld generated from large eddy simulations les of a mach 1 4 ideally expanded jet this is part of ongoing research in the development and use of the pod method in conjunction with advanced laser based optical measurementsin high speed e ows thepod application goal istwofold to extract dynamically signie cant information on the large scale coherent structures in a high speed jet and to facilitate low dimensional modeling of the jet it was found that the spatial eigenmodes obtained using weakly correlated snapshots but spanning tens of convective timescale and uncorrelated snapshots are similar it was also found that a shortduration temporally resolved les data simulating data obtainable from pulse burst laser based measurements could be used to calculate the time evolution coefe cients of the eigenmodes the use of a few modes namely 12 was sufe cient for a reasonable reconstruction of the spatiotemporal e owe eld the use of pod with a vector norm instead of a scalar norm did reduce the energy captured in the e rst few modes and also changed their rank order but did not substantially alter the reconstructed e ow in the early jet development region the e rst and dominant mode was found to be axisymmetric followed by either another axisymmetric or asymmetric probably helical mode whereas higher modes in this region and all ofthemodes fartherdownstream weremorecomplex and threedimensional the pod modes and their temporal coefe cients obtained at various streamwise locations suggest that the large scale jet structures undergo a process of disorganization near the end of potential core followed by reorganization farther downstream

on generalized quantum stochastic counting processes we study counting processes introduced by davies 11 on general state spaces the concept of a refinement of a counting process cp corresponding to the possibility of distinguishing particles for instance according to their energy or phase is introduced and refinements of general cp s are classified then cp s with bounded interaction rate are classified on general state spaces and sufficient conditions are given in order that the operators characterizing the interaction rate can be formulated in the schrodinger picture for cp s with unbounded interaction rates it is shown that analogous to the case of bounded interaction rates there is a family of operators characterizing the interaction rate commutation relations for such processes are derived for constructions of cp s with unbounded interaction rate it is shown that it essentially suffices to solve the semigroup perturbation problem finally refinements of these cp s are characterized by measures e j e on the set of different particles where eachj e in an unbounded operator

asynchronous exponential growth of semigroups of nonlinear operators the property of asynchronous exponential growth is analyzed for the abstract nonlinear differential equation z t az t f z t t 0 z 0 x x where a is the infinitesimal generator of a semigroup of linear operators in the banach space x and f is a nonlinear operator in x asynchronous exponential growth means that the nonlinear semigroup s t t 0 associated with this problem has the property that there exists 0 and a nonlinear operator q in x such that the range of q lies in a one dimensional subspace of x and limt e ts t x qx for all x x it is proved that if the linear semigroup generated by a has asynchronous exponential growth and f satisfies f x f x x where f r r and f r r dr then the nonlinear semigroup s t t 0 has asynchronous exponential growth the method of proof is a linearization about infinity examples from structured population dynamics are given to illustrate the results

strong ergodicity for perturbed dual semigroups and application to age dependent population dynamics abstract in this paper we first summarize the perturbation theory in a sun reflexive banach space next we shall introduce the concept of strong ergodicity for the evolutionary system and then we investigate conditions under which the evolutionary system generated by the perturbation of a c 0 semigroup in a sun reflexive banach space becomes strongly ergodic subsequently we construct the evolutionary system corresponding to lotka s renewal equation and apply it to prove strong ergodicity of the age structured population with time dependent vital rates finally we study the controllability of the age structured population controlled by changing its total fertility rate

a reduced order extrapolation algorithm based on sfve method and pod technique for non stationary stokes equations in this paper a reduced order extrapolation algorithm roea based on proper orthogonal decomposition pod technique and classical stabilized finite volume element sfve method for non stationary stokes equations is established the error estimates between the roea solutions and the classical sfve solutions and the implementation for solving roea are provided some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that roea based on sfve formulation and pod method is feasible and efficient for solving non stationary stokes equations

equations with infinite delay blending the abstract and the concrete using perturbation theory for adjoint semigroups a modification of sun star calculus we prove in the case of infinite delay the principle of linearized stability for nonlinear renewal equations delay differential equations and coupled systems of these two types of equations our results extend those of diekmann et al 1995 13 and diekmann et al 2007 14 to the case of infinite delay

bordered augmented linear systems in numerical continuation and bifurcation we present the algorithm gmbe to solve a block linear system begin gathered begin array 20 c n m 1 end array left begin array 20 c a 11 0 a 31 end array begin array 20 c a 12 a 22 a 32 end array begin array 20 c a 13 a 23 a 33 end array right left begin array 20 c x 1 x 2 x 3 end array right left begin array 20 c f 1 f 2 f 3 end array right begin array 20 c n m 1 end array hfill nm111 hfill end gathered or m z h which appears in numerical computation of folds and symmetry breaking bifurcation gmbe principally uses 1 solve witha 11 t anda 22 t each and 2 solves witha 11 anda 22 each m must be well conditioned buta 11 anda 22 may be arbitrarily ill conditioned in fact singular to machine precision the error analysis requires that the solvers fora 11 a 22 a 11 t anda 22 t are stable in the sense of backward projection of the errors and that the solvers fora 11 anda 22 are bounded in a sense to be made precise both properties are typically possessed in practice by solvers based on either direct or iterative methods gmbe is the first algorithm that solvesmz h by using the solvers fora 11 etc as black boxes and does not require prior computation of the left and right singular vectors ofa 11 anda 22

global bifurcation investigation of an optimal velocity traffic model with driver reaction time we investigate an optimal velocity model which includes the reflex time of drivers after an analytical study of the stability and local bifurcations of the steady state solution we apply numerical continuation techniques to investigate the global behavior of the system specifically we find branches of oscillating solutions connecting hopf bifurcation points which may be super or subcritical depending on parameters this analysis reveals several regions of multistability

a multiplicity theorem for hyperbolic systems abstract in this paper we show for a class of hyperbolic systems that the dimension of the range of the spectral projection corresponding to a single characteristic root 0 is equal to the multiplicity of the spectral point 0 as root of the characteristic equation

on multiplicative perturbation of integral resolvent families abstract in this paper we study multiplicative perturbations for the generator of a strongly continuous integral resolvent family of bounded linear operators defined on a banach space x assuming that a t is a creep function which satisfies a 0 0 we prove that if a a generates an integral resolvent then a i b a also generates an integral resolvent for all b b x z where z belongs to a class of admissible banach spaces in special instances of a t the space z is proved to be characterized by an extended class of favard spaces

large time behaviour for functional differential equations with dominant eigenvalues of arbitrary order the spectral theory for linear autonomous neutral functional differential equations fde yields explicit formulas for the large time behaviour of solutions our results are based on resolvent computations and dunford calculus applied to establish explicit formulas for the large time behaviour of solutions of fde we investigate in detail a class of two dimensional systems of fde

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

solution of bordered singular systems in numerical continuation and bifurcation abstract in numerical continuation and bifurcation problems linear systems with coefficient matrices in the block form m a b c t d arise naturally here a r n n b c r n m d r m m and n may be large but m is small a usually has a special structure banded block banded sparse and b c d are dense so that it is advisable to use a specialized solver for a and to solve with m by some block method unfortunately a is often also a nearly singular matrix in fact made nonsingular only by roundoff and truncation errors on the other hand m is usually nonsingular but can be ill conditioned and in certain situations will degenerate to singularity as well we describe numerical tests for this problem using the mixed block elimination method of govaerts and pryce 1993 for solving bordered linear systems with possibly nearly singular blocks a to this end we compute by newton s method a triple point bifurcation point in a parameterized reaction diffusion equation the brusselator the numerical tests show that the linear systems are solved in a stable way in spite of the use of a black box solver sgbtrs from lapack for a nearly singular matrix

a reduced finite element formulation based on pod method for two dimensional solute transport problems abstract a proper orthogonal decomposition pod method is applied to a usual finite element fe formulation for two dimensional solute transport problems with real practical applied background such that it is reduced into a reduced fe formulation with lower dimensions and high enough accuracy the error estimates between the reduced pod fe solutions and the usual fe solutions are provided it is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions moreover it is also shown that this validates the feasibility and efficiency of pod fe method

characteristic matrices and spectral properties of evolutionary systems in this paper we introduce the notion of a characteristic matrix for a large class of unbounded operators and study the precise connection between characteristic matrices and spectral properties of evolutionary systems in particular we study so called multiplicity theorems several examples will illustrate our results

stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback the bifurcation diagram of a single mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods for this we show how to cope with the special symmetry properties of the equations as the feedback strength is increased branches of modes and antimodes appear and we have found that pairs of modes and antimodes are connected by closed branches of periodic solutions bifurcation bridges such connections seem generically present as new pairs of modes and antimodes appear we subsequently investigate the behavior of the first connection as a function of the linewidth enhancement factor and the feedback phase our results extend and confirm existing results and hypotheses reported in the literature for large values of the linewidth enhancement factor alpha 5 6 bridges break through homoclinic orbits changing the feedback phase unfolds the bifurcation diagram of the modes and antimodes allowing different types of connections between modes

from sine waves to square waves in delay equations suppose f f 0 0 and the fixed point zero undergoes a generic supercritical period doubling bifurcation at 0 we characterise those small values of e 0 for which there are periodic solutions of period approximately two of the equationas e 0 these solutions approach square waves

spectral properties of finite dimensional perturbed linear semigroups abstract many interesting cauchy problems arising in applications can be treated as finite dimensional perturbations of relatively simple problems in most applications these perturbations take place via boundary conditions it is the aim of this paper to investigate such cauchy problems and especially the spectral properties of the corresponding generators some general results are obtained that unify several well known results for functional differential equations and age dependent population dynamics applications to control problems as well as some nonlinear generalizations will appear in forthcoming papers

stability of a nonlinear age dependent population model containing a control variable a very general nonlinear model of age dependent population growth which includes most of earlier presented extensions of the lotka von foerster model is considered a criterion for local asymptotic stability of equilibrium solutions of the model is given the conclusions of this paper are compared with analogous results for some important special cases considered by other authors

computation of periodic solution bifurcations in odes using bordered systems we consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous odes namely the fold the period doubling or flip bifurcation and the torus or neimark sacker bifurcation in the fold and flip cases we append one scalar equation to the standard periodic bvp that defines the periodic solution in the torus case four scalar equations are appended evaluation of these scalar equations and their derivatives requires the solution of linear bvps whose sparsity structure after discretization is identical to that of the linearization of the periodic bvp therefore the calculations can be done using existing numerical linear algebra techniques such as those implemented in the software auto and colsys

proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical pacific ocean reduced gravity model in this paper the tropical pacific ocean reduced gravity model is studied using the proper orthogonal decomposition pod technique of mixed finite element mfe method and an error estimate of pod approximate solution based on mfe method is derived pod is a model reduction technique for the simulation of physical processes governed by partial differential equations e g fluid flows or other complex flow phenomena it is shown by numerical examples that the error between pod approximate solution and reference solution is consistent with theoretical results thus validating the feasibility and efficiency of pod method

stability and instability of equilibria of an equation of size structured population dynamics in this paper we consider a quasilinear equation with a nonlinear boundary condition modelling the dynamics of a biological population structured by size we suppose vital rates depending on the total population this hypothesis introduces some nonlinearities on the equation and on the boundary condition we study the existence and uniqueness of solution of the initial value problem and the existence of stationary solutions after we calculate the spectrum of the linearization at an equilibrium and we study its local stability

on the design of a sensitivity reducing optimal dead beat controller a method is proposed for the design of constant gain feedback control laws for linear multivariable discrete time systems which reduce trajectory sensitivity to small system parameter variations and ensure the closed loop eigenvalues at the origin the given quadratic index of performance including a sensitivity term is minimized in an average sense an efficient computational procedure based on direct cost optimization using gradient type algorithm is also reported an example is worked out to illustrate the proposed technique

a reduced fve formulation based on pod method and error analysis for two dimensional viscoelastic problem abstract proper orthogonal decomposition pod method has been successfully used in the reduced order modeling of complex systems in this paper we extend the applications of pod method i e combine a classical finite volume element fve method with pod method to establish a reduced fve formulation with lower dimensions and sufficiently high accuracy for two dimensional viscoelastic problem with real practical applied background and analyze the errors between the reduced pod fve solution and the classical fve solution so as to provide scientific theoretic basis for service applications some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions moreover it is also shown that the reduced fve formulation based on pod method is feasible and efficient for solving two dimensional viscoelastic problem

equations with infinite delay numerical bifurcation analysis via pseudospectral discretization abstract we address the problem of the numerical bifurcation analysis of general nonlinear delay equations including integral and integro differential equations for which no software is currently available pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations whose properties can be numerically studied with well developed software we explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations

a pod reduced order finite difference time domain extrapolating scheme for the 2d maxwell equations in a lossy medium abstract this paper is concerned with establishing a reduced order finite difference time domain fdtd extrapolating scheme based on proper orthogonal decomposition pod method for the two dimensional 2d maxwell equations in a lossy medium for this purpose we first review the classical fdtd scheme and associated results for the 2d maxwell equations in a lossy medium and then we formulate three sets of pod bases by means of the solutions obtained from the classical fdtd scheme on the very short time interval establish the pod reduced order fdtd extrapolating scheme and furnish the error estimates among the pod reduced order fdtd solutions and the classical fdtd solutions as well as the accuracy solution for the 2d maxwell equations in a lossy medium finally we enumerate two numerical examples to confirm that the pod reduced order fdtd extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium

on the spectrum of evolution operators generated by hyperbolic systems abstract we prove that the asymptotic behavior of semigroups generated by mixed initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation the proof is based on the following reduction principle we construct a reduced system which is much simpler than the given one such that the difference of the semigroups is a compact operator the construction of the reduced system is presented in the non autonomous case because it has implifications in the study of systems which are time periodic

efficient gluing of numerical continuation and a multiple solution method for elliptic pdes numerical continuation calculations for ordinary differential equations odes are by now an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences although several excellent standard software packages are available for odes there are for good reasons no standard numerical continuation toolboxes available for partial differential equations pdes which cover a broad range of different classes of pdes automatically a natural ach to this problem is to look for efficient gluing computation approaches with independent components developed by researchers in numerical analysis dynamical systems scientific computing and mathematical modeling in this paper we shall study several elliptic pdes lane emden fowler lane emden fowler with microscopic force caginalp via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the pdes in particular we solve the initialization problem of numerical continuation for pdes via a minimax algorithm to find multiple unstable solution furthermore for the caginalp system we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the fem matlab pdetoolbox even though the approach works efficiently due to the high level programming language and without developing any new algorithms we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic pdes we study in particular with respect to symmetry breaking in particular we show for a modified lane emden fowler equation with an asymmetric microscopic force how a fully connected bifurcation diagram splits up into c shaped isolas on which localized pattern deformation appears towards two different regimes we conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient time saving gluing computation for dynamical systems analysis of pdes in the near future