analysis on the laguerre formula for approximating delay systems this paper provides a detailed analysis on the commonly employed laguerre formula for approximating delay systems convergence proofs are provided and error bounds are constructed with respect to the l spl infin and l sub 2 norms

dynamical behavior of susceptible infected recovered susceptible epidemic model on weighted networks abstract we study susceptible infected recovered susceptible epidemic model in weighted regular and random complex networks we institute a pairwise type mathematical model with a general transmission rate to evaluate the influence of the link weight distribution on the spreading process furthermore we develop a dimensionality reduction approach to derive the condition for the contagion outbreak finally we analyze the influence of the heterogeneity of weight distribution on the outbreak condition for the scenario with a linear transmission rate our theoretical analysis is in agreement with stochastic simulations showing that the heterogeneity of link weight distribution can have a significant effect on the epidemic dynamics

iterated snowdrift game among mobile agents with myopic expected reward based decision rule numerical and analytical research abstract the iterated spatial game among the mobile population is an interesting problem in the fields of biological social and economic sciences inspired by some recent works this paper concentrates on the iterated snowdrift game among movable and myopic agents two different decision making schemes namely the utility maximum rule and the fermi rule are applied and examined in the former case cooperation is found to be enhanced by moving velocity with an upper bound the analytical results of the model are deduced at two extreme cases when agents cannot move or move quickly in the latter case the influences of velocity and temptation to defect are much more complicated these results allow a deeper insight of the related model as well as the emergence of cooperation

positive correlation between strategy persistence and teaching ability promotes cooperation in evolutionary prisoner s dilemma games abstract strategy persistence and heterogeneous teaching ability are two factors that play important roles in promoting cooperation in this work we simultaneously take the strategy persistence and heterogeneous teaching ability into account in evolutionary prisoner s dilemma games we introduce the correlation between strategy persistence and teaching ability by partitioning a population into two groups in the positive negative correlation scheme individuals in one group are assigned with large and 1 reduced teaching ability 1 and those in the other with 0 and 1 1 the simulation results show that compared to negative correlation positive correlation between strategy persistence and teaching ability can greatly enhance cooperation and result in a much higher cooperation level furthermore we show that the finding is robust with respect to reduced teaching ability initial cooperator fraction f 0 amplitude of noise k and underlying network topology

effects of strategy migration direction and noise in the evolutionary spatial prisoner s dilemma spatial games are crucial for understanding patterns of cooperation in nature and to some extent society they are known to be more sensitive to local symmetries than e g spin models this paper concerns the evolution of the prisoner s dilemma game on regular lattices with three different types of neighborhoods char22 the von neumann moore and kagom e types we investigate two kinds of dynamics for the players to update their strategies that can be unconditional cooperator or defector depending on the payoff difference an individual can adopt the strategy of a random neighbor a voter model like dynamics vmld or impose its strategy on a random neighbor i e invasion process like dynamics ipld in particular we focus on the effects of noise in combination with the strategy dynamics on the evolution of cooperation we find that vmld compared to ipld better supports the spreading and sustaining of cooperation we see that noise has nontrivial effects on the evolution of cooperation maximum cooperation density can be realized either at a medium noise level in the limit of zero noise or in both these regions the temptation to defect and the local interaction structure determine the outcome especially in the low noise limit the local interaction plays a crucial role in determining the fate of cooperators we elucidate these both by numerical simulations and mean field cluster approximation methods

memory based snowdrift game on networks we present a memory based snowdrift game mbsg taking place on networks we found that when a lattice is taken to be the underlying structure the transition of spatial patterns at some critical values of the payoff parameter is observable for both four and eight neighbor lattices the transition points as well as the styles of spatial patterns can be explained by local stability analysis in sharp contrast to previously reported results cooperation is promoted by the spatial structure in the mbsg interestingly we found that the frequency of cooperation of the mbsg on a scale free network peaks at a specific value of the payoff parameter this phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation the memory effects of individuals are discussed in detail and some nonmonotonous phenomena are observed on both lattices and scale free networks our work may shed some new light on the study of evolutionary games over networks

evolutionary prisoner s dilemma game on newman watts networks maintenance of cooperation was studied for a two strategy evolutionary prisoner s dilemma game where the players are located on a one dimensional chain and their payoff comes from games with the nearest and next nearest neighbor interactions the applied host geometry makes it possible to study the impacts of two conflicting topological features the evolutionary rule involves some noise affecting the strategy adoptions between the interacting players using monte carlo simulations and the extended versions of dynamical mean field theory we determined the phase diagram as a function of noise level and a payoff parameter the peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by newman and watts

numerical oscillation and non oscillation for differential equation with piecewise continuous arguments of mixed type the paper deals with the oscillation and non oscillation of the runge kutta methods for a differential equation with piecewise continuous arguments of mixed type the conditions of the oscillation and non oscillation for the runge kutta method are obtained it is proved that oscillation of the analytic solution is not preserved by the runge kutta method under any conditions the conditions under which the non oscillation of analytic solutions is preserved by the runge kutta method are obtained some numerical experiments are given

a new approximation method for time delay systems we develop a rational approximant b sub n sup spl alpha of e sup ds and make a detailed analysis on the approximant b sub n sup spl alpha g of e sup ds g where g spl isin rl sup 2 we prove l sup p 2 spl infin error convergence and provide corresponding error bounds explicitly comparative study with other approximants is performed in order to make clear the properties of b sub n sup spl alpha making full use of its recursive structure b sub n sup spl alpha is shown to be a suitable approximant for a class of h sup spl infin problems involving time delay

global exponential stability of cohen grossberg neural networks with piecewise constant argument of generalized type and impulses in this letter we consider a model of cohen grossberg neural networks with piecewise constant argument of generalized type and impulses sufficient conditions ensuring the existence and uniqueness of solutions are obtained based on constructing a new differential inequality with piecewise constant argument and impulse and using the lyapunov function method we derive sufficient conditions ensuring the global exponential stability of equilibrium point with approximate exponential convergence rate an example is given to illustrate the validity and advantage of the theoretical results

almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence abstract we give conditions for the existence uniqueness and certain stability properties of almost periodic solutions of certain hybrid delay differential equations with almost periodic time dependence

cooperative behavior in evolutionary snowdrift games with the unconditional imitation rule on regular lattices we study an evolutionary snowdrift game with the unconditional imitation updating rule on regular lattices detailed numerical simulations establish the structure of plateaus and discontinuous jumps of the equilibrium cooperation frequency f c as a function of the cost to benefit ratio r by analyzing the stability of local configurations it is found that the transitions occur at values of r at which there are changes in the ranking of the payoffs to the different local configurations of agents using different strategies nonmonotonic behavior of f c r at the intermediate range of r is analyzed in terms of the formation of blocks of agents using the cooperative strategy that are stabilized by agents inside the block due to the updating rule for random initial condition with 50 50 agents of different strategies randomly dispersed cooperation persists in the whole range of r and the level of cooperation is higher than that in the well mixed case in a wide range of r these results are in sharp contrast to those based on the replicator updating rule the sensitivity to initial states with different fractions of cooperative agents is also discussed the results serve to illustrate that both the spatial structure and the updating rule are important in determining the level of cooperation in a competing population when extreme initial states are used where there are very few agents of a strategy in a background of the opposite strategy the result would depend on the stability of the clusters formed by the initially minority agents

effect of memory on the prisoner s dilemma game in a square lattice we have studied the effect of memory on the evolution of the prisoner s dilemma game in square lattice networks based on extensive simulations we found that the density of cooperators was enhanced by an increasing memory effect for most parameters however we also observed that the density of cooperators decreased with an increased memory effect in the case of a large memory and moderate temptation it is interesting to note that memory makes cooperators immune from temptation the strength of protection reaches its maximal value only for a moderate memory effect

phase diagrams for an evolutionary prisoner s dilemma game on two dimensional lattices the effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary prisoner s dilemma game with players located on the sites of different two dimensional lattices this system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive using monte carlo simulations and the generalized mean field approximations we have determined the phase boundaries critical points separating the two phases on the plane of the temperature noise and temptation to choose defection in the zero temperature limit the cooperation can be sustained only for those connectivity structures where three site clique percolation occurs

shift operator induced approximations of delay systems certain shift operator induced finite dimensional approximations of an important class of delay systems are studied h infty h2 and l1 error bounds are given for stable delay systems furthermore a multiple shift formula generated by a second order pade shift is analyzed further and an exact asymptotic h infty error formula is derived the studied class of shift operator induced approximations provides an appealing and most transparent approach for determining good finite dimensional models of delay systems

evolutionary prisoner s dilemma game on a square lattice a simplified prisoner s game is studied on a square lattice when the players interacting with their neighbors can follow two strategies to cooperate c or to defect d unconditionally the players updated in random sequence have a chance to adopt one of the neighboring strategies with a probability depending on the payoff difference using monte carlo simulations and dynamical cluster techniques we study the density c of cooperators in the stationary state this system exhibits a continuous transition between the two absorbing states when varying the value of temptation to defect in the limits stackrel ensuremath rightarrow c 0 and 1 we have observed critical transitions belonging to the universality class of directed percolation

linearization near the integral manifold for a system of differential equations with piecewise constant argument abstract in this paper we prove a linearization result via an integral manifold for a system of differential equations with piecewise constant argument

mean square exponential stability for stochastic control systems with discrete time state feedbacks and their numerical schemes in simulation aiming at the qualitative numerical simulation this article concerns the mean square exponential stability mses of the stochastic systems with discrete time state feedbacks and their numerical schemes a time delay dependent stability criterion is obtained for the underlying system the euler maruyama and backward euler maruyama schemes are established and their mses are proved under the same criterion which means that the numerical schemes preserve the stability of the continuous model the illustrative example at the end of this letter shows that the theoretical upper bound for the sampling period limited in the stability criterion is much greater than that obtained by the stability criteria from the literature

convergence of a class of pade approximations for delay systems this paper concerns the convergence of a class of rational approximations for delay systems of the form exp st m s where m s is a strictly proper rational transfer matrix the rationale for reducing the order of g s is to replace exp st in g s by the class of all pass low pass pade approximations the l2 and lm convergence in the frequency domain are established under mild conditions on m s and hence impulse response in the l2 case for scalar m s the convergence is achieved at an optimal rate error bounds for the approximants are obtained which provide a priori estimates for the errors

model reduction of delay systems using pade approximants this paper describes the rational approximation of a certain class of delay systems in the frequency domain using pad approximants of exp st three classes of approximants characterized by their relative degrees are considered easily computable a priori l and l l2error bounds are provided

preservation of oscillations of the runge kutta method for equation x t ax t a1x t 1 0 the paper deals with the preservation of oscillations of the runge kutta method for equation x t ax t a 1x t 1 0 it is proved that oscillations of the analytic solution are preserved by the runge kutta method special interpolation functions of the numerical solutions are given it turns out that zeros of the interpolation function of the numerical solution converge to ones of the analytic solution with the same order of accuracy as that of the corresponding runge kutta method some numerical experiments are presented

stability of differential equations with piecewise constant arguments of generalized type abstract in this paper we continue to consider differential equations with piecewise constant argument of generalized type epcag m u akhmet integral manifolds of differential equations with piecewise constant argument of generalized type nonlinear anal tma 66 2007 367 383 a deviating function of a new form is introduced the linear and quasilinear systems are under discussion the structure of the sets of solutions is specified necessary and sufficient conditions for stability of the zero solution are obtained our approach can be fruitfully applied to the investigation of stability oscillations controllability and many other problems of epcag some of the results were announced at the international conference on hybrid systems and applications university of louisiana lafayette 2006

persistence contractivity and global stability in logistic equations with piecewise constant delays abstract we establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium n 1 a i 0 m b i of the following differential equation with piecewise constant arguments dn t dt n t r t 1 an t i 0 m b i n n i n t n 0 1 2 n 0 n 0 0 and n j n j 0 j 1 2 m where r t is a nonnegative continuous function on 0 r t 0 i 0mbi 0 bi 0 i 0 1 2 m and a i 0mbi 0 these new conditions depend on a b0 and i 1mbi and hence these are other type conditions than those given by so and yu hokkaido math j 24 1995 269 286 and others in particular in the case m 0 and r t r 0 we offer necessary and sufficient conditions for the persistence and contractivity of solutions we also investigate the following differential equation with nonlinear delay terms dx t dt x t r t 1 ax t g x t x t 1 x t m t 0 x k k 0 0 k m and 0 0 where r t is a nonnegative continuous function on 0 r t 0 1 ax g x x x 0 has a unique solution x 0 and g x0 x1 xm c1 0 0 0

method of lyapunov functions for differential equations with piecewise constant delay we address differential equations with piecewise constant argument of generalized type 5 8 and investigate their stability with the second lyapunov method despite the fact that these equations include delay stability conditions are merely given in terms of lyapunov functions that is no functionals are used several examples one of which considers the logistic equation are discussed to illustrate the development of the theory some of the results were announced at the 14th international congress on computational and applied mathematics iccam2009 antalya turkey in 2009

arbitrary precision computation of the set of stabilising gains for time delay lti systems a new method is proposed to calculate the set of proportional controllers stabilising a given linear time invariant system with a fixed time delay the method is based on the utilisation of pade approximations for the time delay lam s studies on the convergence of pade approximations and a generalisation of the nyquist theorem applicable to quasi polynomials the method allows determination of the closeness of the resulting set to the exact set of stabilising gains by tracing the uncertainty introduced by the pade approximation as a result it is possible to approximate the resulting set arbitrarily close to the exact stabilising set

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

the euler maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments abstract in this paper we consider stochastic differential equations with piecewise constant arguments these equations describe hybrid dynamical systems that is combinations of continuous and discrete systems our aim is to establish the mean square convergence of the euler maruyama approximate solution under quite general conditions that is the global lipschitz condition and the linear growth condition which guarantee the existence and uniqueness of the true solution then we show that the initial equation is exponentially stable in mean square if and only if for some sufficiently small step size the euler maruyama method is exponentially stable in mean square the stability study does not involve the lyapunov functions nor functionals it should be pointed out that stochastic differential equations with piecewise constant arguments can be regarded as a class of stochastic differential equations with multiple time dependent delays which can be found in the literature however in the convergence analysis of the euler maruyama method of these equations it is required that the delay functions satisfy lipschitz continuity condition in the present paper the delay functions do not satisfy that condition so it represents an extension of results from the papers mao 2003 2007 using the technique similar to that from the first paper

integral manifolds of differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a general type of differential equations with piecewise constant argument epcag the existence of global integral manifolds of the quasilinear epcag is established when the associated linear homogeneous system has an exponential dichotomy the smoothness of the manifolds is investigated the existence of bounded and periodic solutions is considered a new technique of investigation of equations with piecewise argument based on an integral representation formula is proposed appropriate illustrating examples are given

dichotomies and asymptotic equivalence in alternately advanced and delayed differential systems abstract in this paper ordinary and exponential dichotomies are defined in differential equations with equations with piecewise constant argument of general type we prove the asymptotic equivalence between the bounded solutions of a linear system and a perturbed system with integrable and bounded perturbations

stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type this paper is concerned with the numerical properties of q methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments using two q methods namely the one leg q method and the linear q method the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained and the conditions of oscillations for the q methods are also obtained it is proved that oscillations of the analytic solution are preserved by the q methods furthermore the relationships between stability and oscillations are revealed some numerical experiments are presented to illustrate our results

differential equations with state dependent piecewise constant argument abstract a new class of differential equations with state dependent piecewise constant argument is introduced it is an extension of systems with piecewise constant argument fundamental theoretical results for the equations the existence and uniqueness of solutions the existence of periodic solutions and the stability of the zero solution are obtained appropriate examples are constructed