bifurcation analysis on a survival red blood cells model in this paper we consider a model described the survival of red blood cells in animal its dynamics are studied in terms of local and global hopf bifurcations we show that a sequence of hopf bifurcations occur at the positive equilibrium as the delay crosses some critical values using the reduced system on the center manifold we also obtain that the periodic orbits bifurcating from the positive equilibrium are stable in the center manifold and all hopf bifurcations are supercritical further particular attention is focused on the continuation of local hopf bifurcation we show that global hopf bifurcations exist after the second critical value of time delay

synchronized hopf bifurcation analysis in a neural network model with delays we consider the synchronized periodic oscillation in a ring neural network model with two different delays in self connection and nearest neighbor coupling employing the center manifold theorem and normal form method introduced by hassard et al we give an algorithm for determining the hopf bifurcation properties using the global hopf bifurcation theorem for fde due to wu and bendixson s criterion for high dimensional ode due to li and muldowney we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large

stability and hopf bifurcation in a delayed competition system abstract in this paper hopf bifurcation for two species lotka volterra competition systems with delay dependence is investigated by choosing the delay as a bifurcation parameter we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form i e there are periodic solutions bifurcating out from the positive equilibrium our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter further an explicit algorithm for determining the direction of the hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds and numerical simulations supporting the theoretical analysis are also given

s1 degree and global hopf bifurcation theory of functional differential equations abstract the recently developed s 1 degree and bifrucation theory are applied to provide a purely topological argument of a global hopf bifurcation theory for functional differential equations of mixed type in the special case where the equation is of retarded type the established result represents an analog of alexander and yorke s global hopf bifurcation theorem which has been obtained by chow fiedler mallet paret and nussbaum using different approaches

multistability and stable asynchronous periodic oscillations in a multiple delayed neural system abstract we consider a network of three identical neurons with multiple discrete signal transmission delays the model for such a network is a system of nonlinear delay differential equations after some consideration of the absolute synchronization of the system and the global attractivity of the zero solution we present a detailed discussion about the boundaries of the stability region of the trivial solution this allows us to determine the possible codimension one bifurcations which occur in the system in particular we show the existence of standard hopf bifurcations giving rise to synchronized periodic solutions and of d 3 equivariant hopf bifurcations giving rise to three types of periodic solutions phase locked mirror reflecting and standing waves hopf hopf and hopf steady state bifurcations interactions are shown to exist and give rise to coexistence of stable synchronized and desynchronized solutions perturbation techniques coupled with the floquet theory are used to determine the stability of the phase locked oscillations

global existence of periodic solutions in a tri neuron network model with delays we consider a delayed differential system that models a network of three neurons with memory using a global hopf bifurcation theorem for fde due to j wu and a bendixson s criterion for high dimensional ode due to li and muldowney we obtain a group of sufficient conditions for the system to have multiple periodic solutions when the sum of delays is sufficiently large

analytic integrability of cubic linear planar polynomial differential systems for the cubic linear polynomial planar differential systems with a finite singular point we classify the ones which have a local analytic first integral around the origin and the ones that have a global analytic first integral

almost periodic solutions of lasota wazewska type delay differential equation sufficient conditions are derived for the existence of a globally attractive almost periodic solution of a single species model given by the nonautonomous lasota wazewska type delay differential equations x t txt ptfxt in which 0 p t 0 t are continuous almost periodic functions and f is a decreasing positive c1 function conditions of the uniform asymptotical stability for some associated nonautonomous delay differential equations x t txt ptxt are also discussed

directional agglomeration multigrid techniques for high reynolds number viscous flows a preconditioned directional implicit agglomeration algorithm is developed for solving two and three dimensional viscous flows on highly anisotropic unstructured meshes of mixed element types the multigrid smoother consists of a pre conditioned point or line implicit solver which operates on lines constructed in the unstructured mesh using a weighted graph algorithm directional coarsening or agglomeration is achieved using a similar weighted graph algorithm a tight coupling of the line construction and directional agglomeration algorithms enables the use of aggressive coarsening ratios in the multigrid algorithm which in turn reduces the cost of a multigrid cycle convergence rates which are independent of the degree of grid stretching are demonstrated in both two and three dimensions further improvement of the three dimensional convergence rates through a gmres technique is also demonstrated

global attractivity in population dynamics we established sufficient conditions for the global attractivity of the positive equilibrium of the delay differential equation n t mu n t i 1m pi exp n t i t 0 m 1 for m 1 equation 1 was used by wazewska czyzewska and lasota as a model for the survival of red blood cells in an animal

harmless delays in cohen grossberg neural networks without assuming monotonicity and differentiability of the activation functions and any symmetry of interconnections we establish some sufficient conditions for the globally asymptotic stability of a unique equilibrium for the cohen grossberg neural network with multiple delays lyapunov functionals and functions combined with the razumikhin technique are employed the criteria are all independent of the magnitudes of the delays and thus the delays under these conditions are harmless 2002 elsevier science b v all rights reserved msc 34d23 34k20 92b20

stabilisation of discrete steady adjoint solvers a new implicit time stepping scheme which uses runge kutta time stepping and krylov methods as a smoother inside fas cycle multigrid acceleration is proposed to stabilise the flow solver and its discrete adjoint counterpart the algorithm can fully converge the discrete adjoint solver in a wide range of cases where conventional point implicit methods fail due to either physical or numerical instability this enables the discrete adjoint to be applied to a much wider range of flow regimes in addition the new algorithm offers improved efficiency when applied to stable cases for which the conventional block jacobi solver can fully converge both stable and unstable cases are presented to demonstrate the improved robustness and performance of the new scheme eigen analysis is presented to outline the mechanism of the adjoint stabilisation effect

homotopy linearization and bifurcation on presente une prise en compte essentiellement auto contenue de la relation entre bifurcation locale et la determination a l aide de la linearisation de la non trivialite de certaines classes d homotopie d operateurs non lineaires

agglomeration multigrid methods with implicit runge kutta smoothers applied to aerodynamic simulations on unstructured grids for unstructured finite volume methods an agglomeration multigrid with an implicit multistage runge kutta method as a smoother is developed for solving the compressible reynolds averaged navier stokes rans equations the implicit runge kutta method is interpreted as a preconditioned explicit runge kutta method the construction of the preconditioner is based on an approximate derivative the linear systems are solved approximately with a symmetric gauss seidel method to significantly improve this solution method grid anisotropy is treated within the gauss seidel iteration in such a way that the strong couplings in the linear system are resolved by tridiagonal systems constructed along these directions of strong coupling the agglomeration strategy is adapted to this procedure by taking into account exactly these anisotropies in such a way that a directional coarsening is applied along these directions of strong coupling turbulence effects are included by a spalart allmaras model and the additional transport type equation is approximately solved in a loosely coupled manner with the same method for two dimensional and three dimensional numerical examples and a variety of differently generated meshes we show the wide range of applicability of the solution method finally we exploit the gmres method to determine approximate spectral information of the linearized rans equations this approximate spectral information is used to discuss and compare characteristics of multistage runge kutta methods

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

global analytic first integrals for the real planar lotka volterra system we provide the complete classification of all lotka volterra systems of the form x x ax by c and y y ax by c in r2 having a global analytic first integral

normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation abstract the paper addresses for retarded functional differential equations fdes with parameters the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity the analysis is based on the theory previously developed for fdes without parameters by considering an extension of a phase space appropriate to the computation of these normal forms as an application the generic hopf bifurcation for scalar retarded fdes is considered

an index for global hopf bifurcation in parabolic systems on considere le systeme non lineaire d t u a u f u 0 et on cherche des solutions periodiques en temps u t qui bifurquent a partir d une solution stationnaire

normal forms for retarded functional differential equations and applications to bogdanov takens singularity abstract the paper addresses for retarded functional differential equations fdes the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to invariant spaces for the infinitesimal generator of the linearized equation at a singularity a phase space appropriate to the computation of these normal forms is introduced and adequate nonresonance conditions for the computation of the normal forms are derived as an application the general situation of bogdanov takens singularity and its versal unfolding for scalar retarded fdes with nondegeneracy at second order is considered both in the general case and in the case of differential delay equations of the form t x t x t 1

a critical study of agglomerated multigrid methods for diffusion agglomerated multigrid techniques used in unstructured grid methods are studied critically for a model problem representative of laminar diffusion in the incompressible limit the studied target grid discretizations and discretizations used on agglomerated grids are typical of current node centered formulations agglomerated multigrid convergence rates are presented using a range of two and three dimensional randomly perturbed unstructured grids for simple geometries with isotropic and stretched grids two agglomeration techniques are used within an overall topology preserving agglomeration framework the results show that multigrid with an inconsistent coarse grid scheme using only the edge terms also referred to in the literature as a thin layer formulation provides considerable speedup over single grid methods but its convergence deteriorates on finer grids multigrid with a galerkin coarse grid discretization using piecewise constant prolongation and a heuristic correction factor is slower and also grid dependent in contrast grid independent convergence rates are demonstrated for multigrid with consistent coarse grid discretizations convergence rates of multigrid cycles are verified with quantitative analysis methods in which parts of the two grid cycle are replaced by their idealized counterparts

snakes oriented families of periodic orbits their sources sinks and continuation abstract poincare observed that for a differential equation x x depending on a parameter each periodic orbit generally lies in a connected family of orbits in x space in order to investigate certain large connected sets denoted q of orbits containing a given orbit we introduce two indices an orbit index and a center index defined at certain stationary points we show that genetically there are two types of hopf bifurcation those we call sources 1 and sinks 1 generically if the set q is bounded in x space and if there is an upper bound for periods of the orbits in q then q must have as many source hopf bifurcations as sink hopf bifurcations and each source is connected to a sink by an oriented one parameter snake of orbits a snake is a maximal path of orbits that contains no orbits whose orbit index is 0 see fig 1 1

retracted stability and hopf bifurcations in a delayed leslie gower predator prey system in this paper we consider the following delayed leslie gower predator prey system x t r 1 x t 1 x t k mx t y t y t r 2 y t 1 y t x t by choosing the delay as a bifurcation parameter we show that hopf bifurcations can occur as the delay crosses some critical values by deriving the equation describing the flow on the center manifold we can determine the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions in addition special attention is paid to the global continuation of local hopf bifurcations using a global hopf bifurcation result of wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 for functional differential equations we may show the global existence of periodic solutions computer simulations illustrate the results

existence of inverse integrating factors and lie symmetries for degenerate planar centers abstract we prove that any possibly degenerate center of an analytic planar system admits a c inverse integrating factor and a c lie symmetry in a neighborhood of the singularity we also show that there exists an analytic degenerate center for which any c inverse integrating factor is flat at the singular point and that a sufficient condition for the existence of an analytic inverse integrating factor is the existence of an analytic first integral

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

numerical dynamics of a nonstandard finite difference method for a class of delay differential equations abstract this paper proposes a nonstandard finite difference nsfd method for a class of delay differential equations we prove that for any step size h 1 m m z this numerical method could intrinsically preserve the qualitative behavior of the dynamical system including the local stability of equilibrium the existence and the direction of hopf bifurcation and the stability of bifurcating periodic solution finally to illustrate the analytic results the nsfd method is used for a model of the survival of red blood cells in animals

circulant matrices and differential delay equations on etudie les solutions periodiques non triviales de familles parametrisees d equations differentielles a retard

kukles revisited advances in computing techniques we revisit the kukles system to show how advances in computer hardware and software have improved symbolic calculations we confirm directly that there are no non persistent centres for the kukles system we also prove an earlier conjecture regarding the isochronous centres for the extended kukles system we introduce a technique whereby modular resultants can be used to investigate the properties of resultants that cannot be readily calculated using the existing software

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

computer aided analysis of the convergence to steady state of discrete approximations to the euler equations abstract the behaviour of a centered finite volume scheme for the isoenergetic euler equations in two space dimensions is studied by numerical differentiation and approximate eigensystem analysis the entire semidiscrete approximation including boundary conditions is formulated as a large system of odes which are linearized by numerically approximating the frechet derivative an approximate eigensystem procedure that only needs the frechet derivative is used to extract the least damped eigenmodes the overall method has been applied to the case of transonic flow past an airfoil and has revealed that the most persistent transient modes are highly structured and are associated with eigenvalues of small modulus furthermore they appear to be centered around the shock region the stagnation region and the trailing edge wake region of the airfoil the beneficial effect of local time step scaling and artificial dissipation is also demonstrated by the method

hopf bifurcation analysis in a delayed nicholson blowflies equation abstract the dynamics of a nicholson s blowflies equation with a finite delay are investigated we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases explicit algorithm for determining the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions are derived using the theory of normal form and center manifold global existence of periodic solutions are established using a global hopf bifurcation result of wu trans amer math soc 350 1998 4799 and a bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney j differential equations 106 1994 27