triangularly implicit iteration methods for ode ivp solvers it often happens that iteration processes used for solving the implicit relations arising in ode ivp methods only start to converge rapidly after a certain number of iterations fast convergence right from the beginning is particularly important if we want to use so called step parallel iteration in which the iteration method is concurrently applied at a number of step points in this paper we construct highly parallel iteration methods that do converge fast from the first iteration on our starting point is the pdirk method parallel diagonally implicit iterated runge kutta method designed for solving implicit runge kutta equations on parallel computers the pdirk method may be considered as a newton type iteration in which the newton jacobian is simplified to block diagonal form however when applied in a step parallel mode it turns out that its relatively slow convergence or even divergent behavior reduces the effectiveness of the step parallel scheme by replacing the block diagonal newton jacobian approximation in pdirk by a block triangular approximation we do achieve convergence right from the beginning at a modest increase of the computational costs our convergence analysis of the block triangular approach will be given for the wide class of general linear methods but the derivation of iteration schemes is limited to runge kutta based methods a number of experiments show that the new parallel triangularly implicit iterated runge kutta method ptirk method is a considerable improvement over the pdirk method

a note on the efficient implementation of implicit methods for odes the use of implicit methods for odes e g implicit runge kutta schemes requires the solution of nonlinear systems of algebraic equations of dimension sm where m is the size of the continuous differential problem to be approximated usually the solution of this system represents the most time consuming section in the implementation of such methods consequently the efficient solution of this section would improve their performance in this paper we propose a new iterative procedure to solve such equations on sequential computers

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

local random quantum circuits are approximate polynomial designs we prove that local random quantum circuits acting on n qubits composed of o t10n2 many nearest neighbor two qubit gates form an approximate unitary t design previously it was unknown whether random quantum circuits were a t design for any t 3 the proof is based on an interplay of techniques from quantum many body theory representation theory and the theory of markov chains in particular we employ a result of nachtergaele for lower bounding the spectral gap of frustration free quantum local hamiltonians a quasi orthogonality property of permutation matrices a result of oliveira which extends to the unitary group the path coupling method for bounding the mixing time of random walks and a result of bourgain and gamburd showing that dense subgroups of the special unitary group composed of elements with algebraic entries are copy tensor product expanders we also consider pseudo randomness properties of local random quantum circuits of small depth and prove that circuits of depth o t10n constitute a quantum t copy tensor product expander the proof also rests on techniques from quantum many body theory in particular on the detectability lemma of aharonov arad landau and vazirani we give applications of the results to cryptography equilibration of closed quantum dynamics and the generation of topological order in particular we show the following pseudo randomness property of generic quantum circuits almost every circuit u of size o nk on n qubits cannot be distinguished from a haar uniform unitary by circuits of size o n k 9 11 that are given oracle access to u

bifurcation analysis of the generalized stretch twist fold flow based on the stretch twist fold flow a generalized stretch twist fold flow is introduced by choosing an appropriate bifurcation parameter hopf bifurcations occur in this system when the bifurcation parameter exceeds a critical value the formulae for determining the direction of the hopf bifurcations and the stability of bifurcating periodic solutions are presented in addition the paper also investigates the bifurcations of the heteroclinic orbits for this system the existence and its associated existing regions are given for two heteroclinic orbits respectively finally some numerical simulations for justifying the theoretical analysis are presented

biomathematical analysis and extension of the new class of epidemic models proposed by satsuma et al 2004 abstract the aim of this paper is to discuss the new class of epidemic models proposed by satsuma et al which are characterized by incidence rates which are nonlinearly dependent on the number of susceptibles as follows infection rate s i g s i by adding the biologically plausible constraint g s 0 we study the sir and the seir models with vital dynamics with such infection rate and results are done on the global asymptotic stability of the disease free and of the endemic equilibria similarly to the ones of the classical models also in presence of traditional and pulse vaccination strategies relaxing the constraint g s 0 we show that the epidemic system may exhibit multiple endemic equilibria

mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times in this paper we study the use of a pulse vaccination strategy to eradicate infectious diseases modelable by epidemic model having gamma distributed infectious and latent time we demonstrate the global asymptotic stability of the eradication solution for a general model in which co presence of a continuous vaccination strategy the non permanent duration of the immunization and the emerging problem of the vaccine failures are considered

locally contractive iterated function systems an iterated function system on mathscr x subset mathbb r d is defined by successively applying an i i d sequence of random lipschitz functions from to mathscr x to mathscr x this paper shows how f n f 1 circ ldots circ f n may converge even in the absence of the strong contraction conditions for instance lipschitz constant smaller than 1 on average which earlier work has required instead it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions applications to queues to self modifying random walks and to random logistic maps are given

impulsive control in a stage structure population model with birth pulses abstract the dynamical behavior of a stage structure population model with birth pulses and impulsive pest management strategy is discussed analytically and numerically it is assumed that birth pulse and impulsive pest management strategy act with the same period but not simultaneously the existence and stability of the positive 2 t period solution are investigated by using center manifold theorem and bifurcation theorem the conditions of existence for flip bifurcation are derived moreover some detailed numerical results for phase portraits periodic solutions bifurcation diagram and chaotic attractors which are illustrated with two examples are in good agreement with the theoretical analysis

bifurcation analysis in an sir epidemic model with birth pulse and pulse vaccination the dynamical behavior of an sir epidemic model with birth pulse and pulse vaccination is discussed by means of both theoretical and numerical ways this paper investigates the existence and stability of the infection free periodic solution and the epidemic periodic solution by using the impulsive effects a poincare map is obtained the poincare map center manifold theorem and bifurcation theorem are used to discuss flip bifurcation and bifurcation of the epidemic periodic solution moreover the numerical results show that the epidemic periodic solution period one bifurcates from the infection free periodic solution through a supercritical bifurcation the period two solution bifurcates from the epidemic periodic solution through flip bifurcation and the chaotic solution generated via a cascade of period doubling bifurcations which are in good agreement with the theoretical analysis

the bim code for the numerical solution of odes in this paper we present the code bim based on blended implicit methods j comput appl math 116 2000 41 appl numer math 42 2002 29 recent trends in numerical analysis nova science publ inc new york 2001 pp 81 for the numerical solution of stiff initial value problems for odes we describe in detail most of the implementation strategies used in the construction of the code and report numerical tests comparing the code bim with some of the best codes currently available the numerical tests show that the new code compares well with existing ones moreover the methods implemented in the code are characterized by a diagonal nonlinear splitting which makes its extension for parallel computers very straightforward

blended implicit methods for the numerical solution of dae problems recently a new approach for solving the discrete problems generated by the application of block implicit methods for the numerical solution of initial value problems for odes has been devised l brugnano blended block bvms b 3vms a family of economical implicit methods for odes j comput appl math 116 2000 41 62 l brugnano c magherini blended implementation of block implicit methods for odes appl numer math 42 2002 29 45 l brugnano d trigiante block implicit methods for odes in d trigiante ed recent trends in numerical analysis nova science publishers new york 2001 pp 81 105 this approach is based on the so called blended implementation of the methods giving corresponding blended implicit methods the latter have been implemented in the computational code bim l brugnano c magherini the bim code for the numerical solution of odes j comput appl math 164 165 2004 145 158 blended implicit methods are here extended to handle the numerical solution of dae problems resulting in a straightforward generalization of the basic approach

feller processes on nonlocally compact spaces we consider feller processes on a complete separable metric space x satisfying the ergodic condition of the form lim sup n 1 n n i 1 p i x o 0 for some x x where o is an arbitrary open neighborhood of some point z e x and p is a transition function it is shown that e chains which satisfy the above condition admit an invariant probability measure some results on the stability of such processes are also presented

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

blended implicit methods for solving ode and dae problems and their extension for second order problems the use of implicit numerical methods is mandatory when solving general stiff ode dae problems their use in turn requires the solution of a corresponding discrete problem which is one of the main concerns in the actual implementation of the methods in this respect blended implicit methods provide a general framework for the efficient solution of the discrete problems generated by block implicit methods in this paper we review the main facts concerning blended implicit methods for the numerical solution of ode and dae problems in addition to this we study the extension of blended implicit methods for solving second order problems which results in a straightforward generalization of the basic theory for such methods finally a few numerical tests obtained with the computational code bimd implementing a variable order variable stepsize blended implicit method are also reported in order to confirm the effectiveness of the approach

lower bound technique in the theory of a stochastic differential equation abstract we establish a criterion for the existence of an invariant measure for markov processes acting on measures defined on an arbitrary complete separable metric space this criterion is applied to time homogeneous markov processes associated with a nonlinear heat equation driven by an impulsive noise

the effects of pulse vaccination on seir model with two time delays abstract in this paper impulsive vaccination has been applied to control the spread and transmission of an infectious disease an seir epidemic model with two time delays and pulse vaccination is formulated the exact infection free periodic solution of the impulsive epidemic system is obtained moreover we show that if the vaccination rate is larger than the infectious population disappear so the disease dies out while if the vaccination rate is less than the infectious population persist our results indicate that a large vaccination rate or a short period of pulsing or a long latent period of the disease is a sufficient condition for the eradication of the disease

dynamical behavior of an epidemic model with a nonlinear incidence rate abstract in this paper we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action by carrying out global qualitative and bifurcation analyses it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region when such a region exists it is shown that the model undergoes a bogdanov takens bifurcation i e it exhibits a saddle node bifurcation hopf bifurcations and a homoclinic bifurcation existence of none one or two limit cycles is also discussed

pest regulation by means of impulsive controls in this paper we consider an integrated pest management model which is impulsively controlled by means of biological and chemical controls these controls are assumed to act in a periodic fashion a nonlinear incidence rate being used to account for the dynamics of the disease caused by the application of the biological control the floquet theory for impulsive ordinary differential equations is employed to obtain a condition in terms of an inequality involving the total action of the nonlinear force of infection in a period under which the susceptible pest eradication solution is globally asymptotically stable if the opposite inequality is satisfied then it is shown that the system under consideration becomes uniformly persistent a biological interpretation of the persistence condition is also provided

vaccination policies and nonlinear force of infection generalization of an observation by alexander and moghadas 2004 it has been recently showed in m e alexander s m moghadas periodicity in an epidemic model with a generalized non linear incidence math biosci 189 2004 75 96 that one of the possible causes of unexpected failures of some vaccination campaigns may be the nonlinearity of the force of infection the aim of this paper is to give a simple and biologically meaningful sufficient condition for the globally stable eradication of diseases whose spreading is described via a generic class of sir and seir epidemic models ruled by nonlinear force of infection we consider a scenario in which there are both traditional and pulse vaccination strategies

existence and uniqueness of fixed points for markov operators and markov processes this paper concerns a markov operator t on a space l i and a markov process p which defines a markov operator on a space m of finite signed measures for t the paper presents necessary and suf ic ient conditions for a the existence of invariant probability densities ipds b existence of strictly positive ipds and c existence and uniqueness of ipds similar results on invariant probability measures for p are presented the basic approach is to pose a fixed point problem as the problem of solving a certain linear equation in a suitable banach space and then obtain necessary and sufficient conditions for this equation to have a solution

on strong feller recurrence and weak stabilization of regime switching diffusions focusing on two component markov processes having continuous dynamics and discrete events this work develops asymptotic properties of regime switching diffusions first strong feller property is established next classifying the underlying processes as recurrence positive recurrence or null recurrence or transience sufficient conditions for such processes are obtained in addition to providing general criteria for recurrent and transient switching diffusions for processes that are linearizable with respect to the continuous component easily verifiable conditions are established also provided are conditions for null recurrent processes furthermore conditions for controlled switching diffusions in which the feedback controls ensure weak stability or ergodicity are given finally simple examples are used for demonstration

strategies for solving index one dae with non negative constraints liquid liquid extraction modeling leads to solve an index one dae system for the sake of robustness it is desirable to account for non negative constraints based on the dassl architecture a classical index one dae solver we propose and compare three different strategies to implement these bound constraints each of these strategies corresponds to a different newton modification clipping damping or interior point method the comparisons are made on two test cases the robertson ode problem and an example from liquid liquid extraction modeling

continuous random dynamical systems abstract we study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by poisson noise the main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system since the dynamical system is defined on an arbitrary banach space possibly infinite dimensional to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by lasota and yorke and extended recently to infinite dimensional spaces by szarek finally it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study hence we obtain their stability as well

continuous markov semigroups and stability of transport equations a new theorem for asymptotic stability of markov semigroups is proved this result is applied to transport equations connected with diffusion and jumping processes and randomly perturbed dynamical systems

global attractivity and permanence of a delayed sveir epidemic model with pulse vaccination and saturation incidence in this study we formulate and analyze a new sveir epidemic disease model with time delay and saturation incidence and analyze the dynamic behavior of the model under pulse vaccination using the discrete dynamical system determined by the stroboscopic map we obtain an infection free periodic solution further show that the infection free periodic solution is globally attractive for some parameters of the model under appropriate conditions the permanence of the model is investigated analytically by computer simulation it is concluded that a large vaccination rate or a short pulse of vaccination or a long latent period are each a sufficient condition for the extinction of the disease

on ergodicity of some markov processes we formulate a criterion for the existence and uniqueness of an invariant measure for a markov process taking values in a polish phase space in addition weak ergodicity that is the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution is established the principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity the latter is called the e property the general result is applied to solutions of some stochastic evolution equations in hilbert spaces as an example we consider an evolution equation whose solution describes the lagrangian observations of the velocity field in the passive tracer model the weak mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer

blended block bvms b 3 vms a family of economical implicit methods for odes we analyze some properties of block bvms for odes and introduce blended block bvms the latter are implicit methods characterized by a cheap iterative implementation which makes them computationally very appealing

dynamical behavior of computer virus on internet abstract in this paper we presented a computer virus model using an sirs model and the threshold value r 0 determining whether the disease dies out is obtained if r 0 is less than one the disease free equilibrium is globally asymptotically stable by using the time delay as a bifurcation parameter the local stability and hopf bifurcation for the endemic state is investigated numerical results demonstrate that the system has periodic solution when time delay is larger than a critical values the obtained results may provide some new insight to prevent the computer virus

on stability of some general random dynamical system we consider a new random dynamical system which generalizes markov processes corresponding to iterated function systems and poisson driven stochastic differential equations it can be used as a description of many physical and biological phenomena under the suitable assumption will be proved its stability

bifurcation of nontrivial periodic solutions for an impulsively controlled pest management model this paper investigates the onset of nontrivial periodic solutions for an integrated pest management model which is subject to pulsed biological and chemical controls the biological control consists in the periodic release of infective individuals while the chemical control consists in periodic pesticide spraying it is assumed that both controls are used with the same periodicity although not simultaneously to model the spread of the disease which is propagated through the release of infective individuals an unspecified force of infection is employed the problem of finding nontrivial periodic solutions is reduced to showing the existence of nontrivial fixed points for the associated stroboscopic mapping of time snapshot equal to the common period of controls the latter problem is in turn treated via a projection method it is then shown that once a threshold condition is reached a stable nontrivial periodic solution emerges via a supercritical bifurcation

the dynamics of a new sir epidemic model concerning pulse vaccination strategy abstract a new sir epidemic model with vertical and horizontal transmission is proposed and the dynamics of this disease model under constant and pulse vaccination are analyzed firstly global stability of the endemic equilibrium states of the model with constant vaccination is thereby established further we show that there exists a stable infection free periodic solution when the period of impulsive effect is less than some critical value the condition for the permanence of the system with pulse vaccination is also given which implies the periodic bursts of epidemic occurs numerical simulation shows system with pulse vaccination has more complex dynamic behavior for positive periodic oscillation infection free quasi periodic oscillation than system with constant vaccination finally we compare validity of the strategy of pulse vaccination with no vaccination and constant vaccination and conclude that pulse vaccination strategy is more effective than no vaccination and continuous vaccination

a study on vaccination models for a seasonal epidemic process in this paper seasonal epidemiological processes are considered and a strategy of periodic vaccination is proposed the invariant formulations associated with an n periodic system and the reproduction numbers associated with them are considered a collection of measures to study the stability of the system is introduced moreover the collection of s basic reproduction number at time j help us to establish conditions on the periodic vaccination rates in the vaccination program finally an sir model is showed and a comparison between the results obtained using constant or periodic vaccination program is analyzed

hopf bifurcation of an age structured compartmental pest pathogen model abstract this paper investigates an age structured compartmental pest pathogen model by using the theory of integrated semigroup we study the stability of the steady state of the model by analyzing the associated characteristic transcendental equation it is shown that hopf bifurcation occurs at a positive steady state as bifurcating parameter passes a sequence of critical values

a note on the efficient implementation of hamiltonian bvms we discuss the efficient implementation of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see brugnano et al 8 and references therein also sketching their blended formulation we also discuss the case of separable problems for which the structure of the problem can be exploited to gain efficiency

efficient iterations for gauss methods on second order problems we consider some important aspects about the implementation of high order implicit formulas specially the gauss methods for solving second order differential systems having high frequencies and small amplitudes superimposed the choice of an appropriate iterative scheme is discussed in detail important topics about the predictors initial guesses are analyzed and a variable order strategy to select the best predictor at each integration step is supplied a few numerical experiments on some standard test problems confirm the theory presented

species permanence and dynamical behavior analysis of an impulsively controlled ecological system with distributed time delay in this paper on the basis of the theories and methods of ecology and ordinary differential equations an ecological system with impulsive harvest and distributed time delay is established by using the theories of impulsive equations small amplitude perturbation skills and comparison techniques we get a condition which guarantees the global asymptotical stability of the prey x eradication and predator z eradication periodic solution further the influences of the impulsive perturbation on the inherent oscillation are studied numerically and shows rich dynamics such as period doubling bifurcation chaotic bands periodic windows chaotic crises etc moreover the computation of the largest lyapunov exponent shows the chaotic dynamic behavior of the model meanwhile we investigate the qualitative nature of the strange attractor by using fourier spectra all of these results may be useful in the study of the dynamic complexity of ecosystems