convergence to a steady state for asymptotically autonomous semilinear heat equations on r double struck n abstract we consider parabolic equations of the form u t u f u h x t x t r n 0 where f is a c 1 function with f 0 0 f 0 0 and h is a suitable function on r n 0 which decays to zero as t hence the equation is asymptotically autonomous we show that as t each bounded localized solution u 0 approaches a set of steady states of the limit autonomous equation u t u f u moreover if the decay of h is exponential then u converges to a single steady state we also prove a convergence result for abstract asymptotically autonomous parabolic equations

collocation software for boundary value differential algebraic equations we describe the methods and implementation of a general purpose code coldae this code can solve boundary value problems for nonlinear systems of semi explicit differential algebraic equations daes of index at most 2 fully implicit index 1 boundary value dae problems can be handled as well the code coldae is an extension of the package colnew colsys for solving boundary value odes the implemented method is piecewise polynomial collocation at gaussian points extended as needed by the projection method of ascher petzold for general semi explicit index 2 problems as well as for fully implicit index 1 problems we define a em selective projected collocation method and demonstrate its use the mesh selection procedure of colsys is modified for the case of index 2 constraints we also discuss shooting for initial guesses the power and generality of the code are demonstrated by examples

double fourier series on a sphere the solution of elliptic and vorticity equations on a sphere is studied using double fourier series as orthogonal basis functions the basis functions incorporate sine series weighted by cosine latitude as meridional basis functions for even zonal wavenumbers other than zero to meet the pole condition as to the solution of poisson s equation it is found that the new method gives improved accuracy compared to the method of yee mon weather rev 109 501 1981 due to the absence of constraints imposed on spectral coefficients with the operation number being slightly increased the new method is applied to the vorticity equation along with the use of fourier and spherical harmonics filters and its accuracy is tested for the rotated rossby haurwitz wave it is shown that the basis functions adopted here provide high accuracy for all tests used numerical integration without spherical harmonics filters indicates that they are necessary for stable and accurate time integration comparison with the spherical harmonics model reveals that the present method is more accurate by a factor of order 112 for the test case further application to the advection equation is carried out the error measure for the strong advection of the cosine bell with various rotation angles indicates that the present method is capable of producing accurate and stable calculations without the pole problem suggesting that it could be applied to the numerical weather prediction model including shallow water equations without difficulty extension to the shallow water equations with accuracy tests as in williamson et al j comput phys 102 211 1992 will be given in the future additional time could be saved by introducing the reduced grids near poles in the present method besides the advantage of applying fft to both longitudinal and latitudinal directions

on the assignment of invariant factors by time varying feedback strategies abstract in this paper the realization of arbitrary invariant factors or cyclic components of the closed loop system matrix is discussed it is shown that the use of time varying feedback strategies enables one to realize arbitrary invariant factors in contrast with time invariant feedback control

a non autonomous seirs model with general incidence rate for a non autonomous seirs model with general incidence that admits kuniya and nakata 2012 as a very particular case we obtain conditions for extinction and strong persistence of the infectives our conditions are computed for several particular settings and extend the hypothesis of several proposed non autonomous models additionally we show that our conditions are robust in the sense that they persist under small perturbations of the parameters in some suitable family we also present some simulations that illustrate our results

controllability stabilizability and matrix riccati equations for periodic systems in this note controllability stabilizahility and related concepts for periodically time varying systems are discussed especially it is proved that the definition of controllability employed in connection with the existence of periodic solutions of periodic matrix riccati equations is equivalent to kalman s original definition

skew product semiflows and morse decomposition abstract this paper is devoted to the investigation of the dynamics of non autonomous differential equations the description of the asymptotic dynamics of non autonomous equations lies on dynamical structures of some associated limiting non autonomous and autonomous differential equations one for each global solution in the attractor of the driving semigroup of the associated skew product semiflow in some cases we have infinitely many limiting problems in contrast with the autonomous or asymptotically autonomous case for which we have only one limiting problem that is the semigroup itself we concentrate our attention in the study of the morse decomposition of attractors for these non autonomous limiting problems as a mean to understand some of the asymptotics of our non autonomous differential equations in particular we derive a morse decomposition for the global attractors of skew product semiflows and thus for pullback attractors of non autonomous differential equations from a morse decomposition of the attractor for the associated driving semigroup our theory is well suited to describe the asymptotic dynamics of non autonomous differential equations defined on the whole line or just for positive times or for differential equations driven by a general semigroup

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

a harmonic framework for controllability in linear continuous time periodic systems controllability of linear continuous time periodic systems is dealt with via a harmonic analysis approach for the first time in this paper this approach reveals that controllability of continuous time periodic systems can be connected to necessary and sufficient conditions expressed explicitly with fourier coefficients of the system matrices which can be interpreted in a way similar to what we have seen in linear time invariant cases these controllability conditions shed new light upon structural characteristics of continuous time periodic systems that are hard to know by means of existing time domain controllability criteria in the literature controllability canonical decomposition of linear continuous time periodic systems is revisited through state coordinate transforms of strong analytic features the results are heuristic and significant for examining structural characteristics of continuous time periodic systems and extending controllability related techniques that are widely employed in linear time invariant systems to linear continuous time periodic systems

the control of linear time periodic systems using floquet lyapunov theory in this paper we use floquet lyapunov theory to derive the floquet factors of the state transition matrix of a given linear time periodic system we show how the periodicity of one of the factors can be determined a priori using a constant matrix which we call the yakubovich matrix based upon the signs of the eigenvalues of the monodromy matrix we then describe a method for the numerical computation of the floquet factors relying upon a boundary value problem formulation and the yakubovich matrix further we show how the invertibility of the controllability gramian and a specific form for the feedback gain matrix can be used to derive a control law for the closed loop system the controller can be full state or observer based it also allows the engineer to assign all the invariants of the system i e the full monodromy matrix deriving the feedback matrix requires solving a matrix integral equation for the periodic floquet factor of the new state transition matrix of the closed loop system this is

on the use of reachability gramians for the stabilization of linear periodic systems abstract this paper deals with the development of explicit formulas based on reachability gramians for the stabilization of linear periodic systems in contrast with the pointwise strategy already known in the literature a different intervalwise scheme is proposed involving reduced computational effort stability results are provided in both continuous and discrete time under the sole controllability assumption without requiring system reversibility

real floquet factors of linear time periodic systems abstract floquet theory plays a ubiquitous role in the analysis and control of time periodic systems its main result is that any fundamental matrix x t 0 of a linear system with t periodic coefficients will have a generally complex floquet factorization with one of the two factors being t periodic it is also well known that it is always possible to obtain a real floquet factorization for the fundamental matrix of a real t periodic system by treating the system as having 2 t periodic coefficients the important work of yakubovich in 1970 and yakubovich and starzhinskii in 1975 exhibited a class of real floquet factorizations that could be found from computations on 0 t alone here we generalize these results to obtain other such factorizations we delineate all factorizations of this form and show how they are related we give a simple extension of the lyapunov part of the floquet lyapunov theorem in order to provide one way that the full range of real factorizations may be used based on computations on 0 t only this new information can be useful in the analysis and control of linear time periodic systems

classification and characteristics of floquet factorisations in linear continuous time periodic systems this article has two purposes firstly the article is devoted to collecting basic facts about the floquet theorem and various floquet factorisation algorithms claimed for finite dimensional linear continuous time periodic fdlcp systems secondly the article presents a unified representation framework for various floquet factorisations in fdlcp systems while structural and analytic characteristics about them are examined as well more precisely the following aspects are considered i algorithms for floquet factorisations ii characteristics of floquet factors iii relationships and properties among floquet factorisations most results are reported for the first time while the others are generalised versions of existing ones

well defined steady state response does not imply cics abstract systems for which each constant input gives rise to a unique globally attracting equilibrium are considered a counterexample is provided to show that inputs which are only asymptotically constant may not result in states converging to equilibria failure of the converging input converging state or cics property

a test of a modified algorithm for computing spherical harmonic coefficients using an fft this note describes the results of testing a modified method based on an algorithm by dilts 1 for computing the spherical harmonic expansion coefficients for a function on a sphere using a two dimensional fft the new version of the dilts program eliminates problems of overflow and large storage requirements encountered when harmonic degree values of l 16 are computed thus allowing the determination of coefficients for high l values l 300 however results from timing tests indicate that the new version of the dilts program is not practical for computing spherical harmonic expansion coefficients for large l due to the very long cpu times required for execution of the program the most effective technique remains the one pioneered by brown 2

an adaptive grid scheme for the discrete hamilton jacobi bellman equation in this paper an adaptive finite difference scheme for the solution of the discrete first order hamilton jacobi bellman equation is presented local a posteriori error estimates are established and certain properties of these estimates are proved based on these estimates an adapting iteration for the discretization of the state space is developed an implementation of the scheme for two dimensional grids is given and numerical examples are discussed

pullback permanence in a non autonomous competitive lotka volterra model abstract the goal of this work is to study in some detail the asymptotic behaviour of a non autonomous lotka volterra model both in the conventional sense as t and in the pullback sense starting a fixed initial condition further and further back in time the non autonomous terms in our model are chosen such that one species will eventually die out ruling out any conventional type of permanence in contrast we introduce the notion of pullback permanence and show that this property is enjoyed by our model this is not just a mathematical artifice but rather shows that if we come across an ecology that has been evolving for a very long time we still expect that both species are represented and their numbers are bounded below even if the final fate of one of them is less happy the main tools in the paper are the theory of attractors for non autonomous differential equations the sub supersolution method and the spectral theory for linear elliptic equations

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

the linear periodic output regulation problem the problem of asymptotic output regulation for linear systems driven by time varying t periodic exosystems is considered in this paper necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential sylvester equations are derived these conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of francis a general algorithm for the synthesis of an error feedback regulator is given for the case of minimum phase systems it is shown that the regulator design can be carried out without the knowledge of the floquet decomposition of the exosystem thus extending significantly the applicability of the general result the more challenging issue of robust regulation by error feedback is also addressed and solved under a stronger observability condition

traveling fronts of the volume filling chemotaxis model with general kinetics this paper is devoted to the study of traveling fronts for the volume filling chemotaxis model with general kinetics by applying the perturbation method the proof relies on the fredholm theory and the banach contractive mapping principle the obtained results can easily detect the existence of traveling fronts for some models investigated in the known references

pitfalls of a least squares equivalent controller for linear time periodic systems we review a technique for the design of controllers for linear time periodic systems a major appeal of the technique first proposed by sinha and joseph is the use of floquet lyapunov theory to transform the periodic system to a form where classical control strategies for time invariant systems may be employed however it is normally impossible to find a completely time invariant control system that is equivalent to the original time varying system application of the floquet lyapunov transformation in fact yields a time varying control system that the technique makes equivalent to a time invariant one in the least squares sense in order to subsequently synthesize the controller via pole placement using a constant feedback matrix however classical control and floquet lyapunov theory clearly show that it is erroneous to conclude that the behaviour of the least squares equivalent time invariant system always matches the behaviour of the original timeperiodic system using an example found in the ori

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space for non autonomous systems any relevant subset of the phase space which now includes the time as one coordinate is unbounded in the t direction hence a numerical method would have to use infinitely many points to overcome this problem we introduce a transformation of the phase space restricting ourselves to exponentially asymptotically autonomous systems we can map the infinite time interval to a finite compact one the basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium for an autonomous system now we are able to generalise numerical methods from the autonomous case more precisely we characterise a lyapunov function as a solution of a suitable linear first order partial differential equation and approximate it using radial basis functions

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

a novel extended precise integration method based on fourier series expansion for the h2 norm of linear time varying periodic systems abstracta new reliable algorithm for computing the h2 norm of linear time varying periodic ltp systems via the periodic lyapunov differential equation plde is proposed by taking full advantage of the periodicity the transition matrix of the underlying ltp system associated with the plde is effectively computed by developing a novel extended precise integration method based on fourier series expansion where the time consuming work for the computation of the matrix exponential and its related integrals in every sub interval is avoided then a highly accurate and efficient algorithm for the plde is derived using the block form of the transition matrix thus the h2 norm is evaluated by solving a simple first order ordinary differential equation finally two numerical examples are presented and compared with other algorithms to verify the numerical accuracy and efficiency of the proposed algorithm

numerical determination of the basin of attraction for asymptotically autonomous dynamical systems we develop a method to numerically analyse asymptotically autonomous systems of the form dot x f t x where f t x tends to g x as t the rate of convergence is not limited to exponential but may be polynomial logarithmic or any other rate for these systems we propose a transformation of the infinite time interval to a finite compact one which reflects the rate of convergence of f to g in the transformed system the origin is an asymptotically stable equilibrium which is exponentially stable in x direction weconsider a lyapunov function in this transformed system as a solution of a suitable linear first order partial differential equation and approximate it using radial basis functions

h 2 norm computation of linear time varying periodic systems via the periodic lyapunov differential equation a new reliable algorithm for computing norm of linear time varying periodic ltp systems in continuous time domain is proposed in this article the solving of norm is firstly transformed into the solving of the periodic lyapunov differential equation plde then the key point in this article is that a new method based on the interval mixed energy matrix and the interval transition matrix computed by the structure preserving magnus series method is proposed for solving the plde with the solutions of plde the norm of ltp systems is evaluated by a simple first order ordinary differential equation finally the effectiveness and the high accuracy of the proposed algorithms are demonstrated by two numerical examples

monodromy eigenvalue assignment in linear periodic systems this note addresses the problem of assigning the eigenvalues of the monodromy matrix of a linear continuous time periodic system by periodic state feedback a controllability criterion is given which slightly strengthens known results an explicit expression for a periodic feedback gain which performs the pole assignment is given when the system is controllable over one period the whole monodromy matrix is assignable by periodic feedback an example is given

meshless collocation error estimates with application to dynamical systems in this paper we derive error estimates for generalized interpolation in particular collocation in sobolev spaces we employ our estimates in collocation problems using radial basis functions and extend and improve previously known results for elliptic problems finally we use meshless collocation to approximate lyapunov functions for dynamical systems

implementing the hamiltonian test for the h norm in linear continuous time periodic systems by the floquet similarity transformations of finite dimensional linear continuous time periodic fdlcp systems the time domain hamiltonian test for the h norm is first interpreted in terms of toeplitz operators of the system matrices based on this novel frequency domain interpretation implementing the time domain hamiltonian test can be accomplished by working on its frequency domain counterpart via truncations this gives us a bisection algorithm for evaluating the h norm of a class of fdlcp systems through finite dimensional linear time invariant continuous time models the finite dimensional hamiltonian test is necessary and sufficient in the asymptotic sense and claimed only via fourier coefficients of the system matrices without the transition matrix of the fdlcp system concerned there are numerical examples to illustrate the suggested algorithm

classification of radial solutions for semilinear elliptic systems with nonlinear gradient terms abstract we are concerned with the classification of positive radial solutions for the system u v p v f u where p 0 and f c 1 0 is a nondecreasing function such that f t 0 for all t 0 we show that in the case where the system is posed in the whole space r n such solutions exist if and only if 1 0 s f t d t p 2 p 1 d s this is the counterpart of the keller osserman condition for the case of single semilinear equation similar optimal conditions are derived in case where the system is posed in a ball of r n if f t t q q 1 using dynamical system techniques we are able to describe the behaviour of solutions at infinity in case where the system is posed in the whole r n or around the boundary in case of a ball

pullback attractors and extremal complete trajectories for non autonomous reaction diffusion problems abstract we analyse the dynamics of the non autonomous nonlinear reaction diffusion equation u t u f t x u subject to appropriate boundary conditions proving the existence of two bounding complete trajectories one maximal and one minimal our main assumption is that the nonlinear term satisfies a bound of the form f t x u u c t x u 2 d t x u where the linear evolution operator associated with c t x is exponentially stable as an important step in our argument we give a detailed analysis of the exponential stability properties of the evolution operator for the non autonomous linear problem u t u c t x u between different l p spaces

linear periodic systems eigenvalue assignment using discrete periodic feedback the eigenvalue assignment problem of a t periodic linear system using discrete periodic state feedback gains is discussed for controllable systems an explicit formula for the feedback law is given that can be used for the arbitrary assignment of the eigenvalues of phi sub c1 t 0 the closed loop state transition matrix from 0 to t for the special case of periodic systems controllable over one period this control law can be used to obtain any desired phi sub c1 t 0

h norm computation of linear continuous time periodic systems by a structure preserving algorithm a new reliable structure preserving algorithm for computing h norm of linear continuous time periodic systems is proposed in this paper in the computation of the h norm no riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic hamiltonian system will be evaluated first the monodromy matrix is expressed as the product of state transition matrices of the hamiltonian system second these state transition matrices which have been proved to be symplectic matrices are evaluated by a structure preserving magnus series method then in order to preserve the standard symplectic form of the monodromy matrix the structure preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix at last the effectiveness and the high accuracy of the proposed structure preserving algorithm are demonstrated by numerical examples

average floquet factorisations in linear continuous time periodic systems the paper examines existence and properties about average floquet factorisations determined via average integration instead of matrix logarithms for the transition matrices of finite dimensional linear continuous time periodic fdlcp systems we talk about the following aspects about average floquet factorisations existence conditions properties and relationships with the conventional floquet factorisations we reveal that average scaling and modelling may be incomplete or even incorrect for reflecting dynamics in general fdclp systems compared to the floquet factorisations defined through matrix logarithms

robustness analysis of linear periodic time varying systems subject to structured uncertainty in this paper we show how floquet theory may be combined with a technique known as lifting to cast a linear periodically time varying system subject to structured linear time invariant uncertainty in the form of a linear fractional transformation the stability and performance robustness of the resulting system may then be analysed using standard analysis methods a significant advantage of the proposed approach is that it allows the computation of a worst case destabilising uncertainty combination which may be used to estimate the conservatism of the computed robustness margin an example is given to illustrate the application of the proposed approach

computation of spherical harmonic expansion coefficients via fft s abstract a method for numerically expanding an arbitrary function on the sphere in a series of spherical harmonics which makes use of the speed of a fast fourier transform is described discussions of the operation count storage requirements accuracy and an algebraic and a numerical example are included a comparison with straightforward integration is made throughout also a new method for evaluating the spherical harmonics is discussed

h controllability and observability of linear periodic systems with reference to linear periodic systems the classical notion of controllability introduced by kalman k controllability is shown to be equivalent to the characterization of controllability h controllability first proposed by hewer as a natural extension to the periodic case of an eigenvalue eigenvector controllability condition independently introduced by several authors for time invariant systems only the proof of such an equivalence also leads to the conclusion that if a t periodic system s is k controllable and k is the degree of the minimal polynomial of the monodromy matrix associated with s then at any time t s is controllable over t t kt since k is lower than or equal to the system order a well known result due to brunovsky is slightly strengthened by duality the corresponding observability results follow

asymptotic stability of nonautonomous systems by liapunov s direct method abstract this paper discusses the asymptotic stability of nonautonomous systems by means of the direct method of liapunov we assume existence of a positive time invariant liapunov function with negative semi definite derivative we focus on the extra conditions needed in order to guarantee asymptotic stability and present a theorem which offers a new criterion for asymptotic stability the theorem is illustrated first by recovering barbashin s principle for autonomous and periodic systems and second by studying the nonlinear pendulum with time varying damping for which we derive a new asymptotic stability result

dynamics of a periodically pulsed bio reactor model abstract global attractivity and uniform persistence are established for both single species growth and two species competition in a periodically pulsed bio reactor model in terms of principal eigenvalues of the periodic parabolic eigenvalue problem by appealing to the theories of monotone discrete dynamical systems abstract persistence asymptotically periodic semiflows and perturbation of global attractors

fourier expansion based recursive algorithms for periodic riccati and lyapunov matrix differential equations combining fourier series expansion with recursive matrix formulas new reliable algorithms to compute the periodic non negative definite stabilizing solutions of the periodic riccati and lyapunov matrix differential equations are proposed in this paper first periodic coefficients are expanded in terms of fourier series to solve the time varying periodic riccati differential equation and the state transition matrix of the associated hamiltonian system is evaluated precisely with sine and cosine series by introducing the riccati transformation method recursive matrix formulas are derived to solve the periodic riccati differential equation which is composed of four blocks of the state transition matrix second two numerical sub methods for solving lyapunov differential equations with time varying periodic coefficients are proposed both based on fourier series expansion and the recursive matrix formulas the former algorithm is a dimension expanding method and the latter one uses the solutions of the homogeneous periodic riccati differential equations finally the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples

generic stabilizability for time delayed feedback control time delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment this approach feeds back only terms that depend on the difference between the current output and the output from a fixed time t ago thus any periodic orbit of period t in the feedback controlled system is also a periodic orbit of the uncontrolled system independent of any modelling assumptions it has been an open problem whether this approach can be successful in general that is under genericity conditions similar to those in linear control theory controllability or if there are fundamental restrictions to time delayed feedback control we show that in principle there are no restrictions this paper proves the following for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability one can find control gains that stabilize this orbit with extended time delayed feedback control while the paper s techniques are based on linear stability analysis they exploit the specific properties of linearizations near autonomous periodic orbits in nonlinear systems and are thus mostly relevant for the analysis of nonlinear experiments