invariant manifolds of partial functional differential equations this paper is concerned with the existence smoothness and attractivity of invariant manifolds for evolutionary processes on general banach spaces when the nonlinear perturbation has a small global lipschitz constant and locally ck smooth near the trivial solution such a nonlinear perturbation arises in many applications through the usual cut off procedure but the requirement in the existing literature that the nonlinear perturbation is globally ck smooth and has a globally small lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut off function our general results are illustrated by and applied to partial functional differential equations for which the phase space c r 0 x where r 0 and x being a banach space has no smooth inner product structure and for which the validity of variation of constants formula is still an interesting open problem

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

bifurcations for a predator prey system with two delays in this paper a predator prey system with two delays is investigated by choosing the sum of two delays as a bifurcation parameter we show that hopf bifurcations can occur as 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 j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 we may show the global existence of periodic solutions

stability of cohen grossberg neural networks with distributed delays global asymptotic stability and global exponential stability of the cohen grossberg neural networks and hopfield neural networks with infinite and finite distributed delays are investigated in this paper for hopfield neural networks with distributed delays local stability is also discussed the existence and global stability of periodic solutions of hopfield neural networks with distributed delays and periodic inputs are achieved by using the theory of dissipative systems

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

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

restrictions and unfolding of double hopf bifurcation in functional differential equations abstract the normal form of a vector field generated by scalar delay differential equations at nonresonant double hopf bifurcation points is investigated using the methods developed by faria and magalhaes j differential equations 122 1995 181 we show that 1 there exists linearly independent unfolding parameters of classes of delay differential equations for a double hopf point which generically map to linearly independent unfolding parameters of the normal form equations ordinary differential equations 2 there are generically no restrictions on the possible flows near a double hopf point for both general and z 2 symmetric first order scalar equations with two delays in the nonlinearity and 3 there always are restrictions on the possible flows near a double hopf point for first order scalar delay differential equations with one delay in the nonlinearity and in nth order scalar delay differential equations n 2 with one delay feedback

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

bifurcation analysis of a class of first order nonlinear delay differential equations with reflectional symmetry abstract we consider a general class of first order nonlinear delay differential equations ddes with reflectional symmetry and study completely the bifurcations of the trivial equilibrium under some generic conditions on the taylor coefficients of the dde our analysis reveals a hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two takens bogdanov point in parameter space we compute the normal form coefficients of the reduced vector field on the centre manifold in terms of the taylor coefficients of the original dde and in contrast to many previous bifurcation analyses of ddes we also compute the unfolding parameters in terms of these coefficients for application purposes this is important since one can now identify the possible asymptotic dynamics of the dde near the bifurcation points by computing quantities which depend explicitly on the taylor coefficients of the original dde we illustrate these results using simple model systems relevant to the areas of neural networks and atmospheric physics and show that the results agree with numerical simulations

the asymptotic shapes of periodic solutions of a singular delay differential system let f r r be given so that f 0 0 and f x 1 x ax2 bx3 o x3 as x 0 we characterize those small values of e 0 and r for which there are periodic solutions of periods approximately 2k with k n of the following system arising from a network of neurons formula the periodic solutions are synchronized if k is even and phase locked if k is odd we show that as e 0 these periodic solutions approach square waves if a 0 and b 0 or if a 0

stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays abstract this paper is devoted to the analysis of a maturity structured system of hematopoietic stem cell hsc populations in the bone marrow the model is a system of differential equations with several time delays we discuss the stability of equilibria and perform the analysis of hopf bifurcation more precisely we first obtain a set of improved sufficient conditions ensuring the global asymptotical stability of the zero solution using the lyapunov method and the embedding technique of asymptotically autonomous semiflows then we prove that there exists at least one positive periodic solution for the n dimensional system as a time delay varies in some region this result is established by combining hopf bifurcation theory the global hopf bifurcation theorem due to wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 and a continuation theorem of coincidence degree theory some numerical simulations are also presented to illustrate the analytic results

global dynamics of nicholsonʼs blowflies equation revisited onset and termination of nonlinear oscillations abstract we revisit nicholson s blowflies model with natural death rate incorporated into the delay feedback we consider the delay as a bifurcation parameter and examine the onset and termination of hopf bifurcations of periodic solutions from a positive equilibrium we show that the model has only a finite number of hopf bifurcation values and we describe how branches of hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals the bifurcation analysis and the matlab package dde biftool developed by engelborghs et al guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions

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

attractor minimal sets for non autonomous delay functional differential equations with applications for neural networks the dynamics of a class of non autonomous convex or concave and monotone delay functional differential systems is studied in particular we provide an attractivity result when two completely st

domain decomposition method for the global dynamics of delay differential equations with unimodal feedback x t zkmx t c f x tkt with unimodal feedback is studied the existence of the global attractor is shown and bounds of the attractor are given we find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f 0 is negative with respect to a positive equilibrium so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions in particular the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established numerical examples using nicholson s blowflies equation and the mackey glass equation are provided to illustrate the main results

bifurcation and global periodic solutions in a delayed facultative mutualism system abstract a facultative mutualism system with a discrete delay is considered by analyzing the associated characteristic equation its linear stability is investigated and hopf bifurcations are demonstrated some explicit formulae are obtained by applying the normal form theory and center manifold reduction such formulae enable us to determine the stability and the direction of the bifurcating periodic solutions bifurcating from hopf bifurcations furthermore a global hopf bifurcation result due to wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 is employed to study the global existence of periodic solutions it is shown that the local hopf bifurcation implies the global hopf bifurcation after the third critical value 1 1 of delay finally numerical simulations supporting the theoretical analysis are given

barrier functions and interior point algorithms for linear programming with zero one or two sided bounds on the variables this study examines two different barrier functions and their use in both path following and potential reduction interior point algorithms for solving a linear program of the form minimize ctx subject to ax b and l x u where components of l and u can be nonfinite so the variables xj can have 0 1 or 2 sided bounds j 1 n the barrier functions that we study include an extension of the standard logarithmic barrier function and an extension of a barrier function introduced by nesterov in the case when both l and u have all of then components finite these barrier functions are psi x sum j left ln u j x j ln x j l j right and psi x sum j left ln left min u j x j x j l j right min u j x j x j l j u j l j 2 right each of these barrier functions gives rise to suitable primal and dual metrics that are used to develop both path following and potential reduction interior point algorithms for solving such linear programming problems

connecting orbits from synchronous periodic solutions to phase locked periodic solutions in a delay differential system we consider a system of delay differential equations modelling the excitatory interaction of two identical neurons assuming the delay is sufficiently large we show that the closure of the forward extension w5 of a 5 dimensional leading unstable manifold of the trivial solution contains a phase locked periodic orbit and a synchronized periodic orbit and we classify the dynamics of the semiflow restricted to w5 we also obtain the precise information about the floquet multipliers of the synchronized periodic orbit which enables us to establish the existence of heteroclinic orbits from the synchronized periodic orbit to the phase locked orbit

stability and bifurcation in a neural network model with two delays abstract a simple neural network model with two delays is considered linear stability of the model is investigated by analyzing the associated characteristic transcendental equation for the case without self connection it is found that the hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem an example is given and numerical simulations are performed to illustrate the obtained results

codimension one and two bifurcations in a symmetrical ring network with delay abstract in this paper we discussed all the possible codimension one and partial codimension two bifurcations existing in a symmetrical ring network using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of lie groups and dynamical analysis methods we have not only figured out the pattern of each bifurcation but also predicted the directions and stability analysis of the bifurcated solutions according to structure of the system numerical simulations were then given to verify our theoretical analysis and investigate the complexity of codimension two mode intersections comments were also presented to highlight some future work

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

hopf bifurcation in bidirectional associative memory neural networks with delays analysis and computation in addition to giving some delay independent conditions for global stability of the bidirectional associative memory neural networks with delays we perform analysis of local stability and hopf bifurcation we also work out an algorithm for determining the direction and stability of the bifurcated periodic solutions

attractivity multistability and bifurcation in delayed hopfieldʼs model with non monotonic feedback abstract for a system of delayed neural networks of hopfield type we deal with the study of global attractivity multistability and bifurcations in general we do not assume monotonicity conditions in the activation functions for some architectures of the network and for some families of activation functions we get optimal results on global attractivity our approach relies on a link between a system of functional differential equations and a finite dimensional discrete dynamical system for it we introduce the notion of strong attractor for a discrete dynamical system which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations our abstract setting is not limited to applications in systems of neural networks we illustrate its use in an equation with distributed delay motivated by biological models we also obtain some results for neural systems with variable coefficients

a model in a coupled system of simple neural oscillators with delays we consider a coupled system of simple neural oscillators using the symmetric functional differential equation theories of wu j wu symmetric functional differential equations and neural networks with memory transactions of the american mathematical society 350 12 1998 4799 4838 we demonstrate the multiple hopf bifurcations of the equilibrium at the origin the existence of multiple branches of bifurcating periodic solution is obtained then some numerical simulations support our analysis results

bifurcation and stability analysis of nonlinear waves in dn symmetric delay differential systems abstract in this paper a direct effective computation method is proposed on stability analysis of the bifurcating solutions in symmetric delay differential systems mechanism about multistability there coexist multiple nontrivial equilibria limit cycles is analyzed

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

affine scaling for linear programs with free variables the affine scaling modification of karmarkar s algorithm is extended to solve problems with free variables this extended primal algorithm is used to prove two important results first the geometrically elegant feasibility algorithm proposed by chandru and kochar is the same algorithm as the one obtained by appending a single column of residuals to the constraint matrix second the dual algorithm as first described by adler et al is the same as the extended primal algorithm applied to the dual

on perturbations of matrix pencils with real spectra a revisit this paper continues earlier studies by bhatia and li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra a unifying framework for creating crucial perturbation equations is developed with the help of a recent result on generalized commutators involving unitary matrices new and much sharper bounds are obtained

general criteria for asymptotic and exponential stabilities of neural network models with unbounded delays abstract for a family of differential equations with infinite delay we give sufficient conditions for the global asymptotic and global exponential stability of an equilibrium point this family includes most of the delayed models of neural networks of cohen grossberg type with both bounded and unbounded distributed delay for which general asymptotic and exponential stability criteria are derived as illustrations the results are applied to several concrete models studied in the literature and a comparison of results is given

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

formulation of the normal form of turing hopf bifurcation in partial functional differential equations abstract turing hopf bifurcation is considered as an important mechanism for generating complex spatio temporal patterns in dynamical systems in this work the normal form up to the third order for the hopf steady state bifurcation which includes the turing hopf bifurcation of a general system of partial functional differential equations pfdes in a bounded spatial region of any dimension is derived based on the center manifold and the normal form theories of pfdes the explicit formula for the coefficients in the normal form of hopf steady state bifurcation are presented concisely in a matrix form which makes it more convenient in not only symbolic derivation but also numerical implementation this provides an approach of showing the existence and stability of stationary and time periodic solutions with spatial heterogeneity the derived formulas are also applicable to partial differential equations and functional differential equations

hopf bifurcation for neutral type neural network model with two delays in this paper the dynamics of a neutral neural network model with two delays is investigated the condition to ensure the stability of the zero solution of the system is decided by choosing 1 and 2 as parameters respectively then the hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by hassard and kazarinoff global existence of periodic solution is studied by using the global hopf bifurcation theory finally some numerical simulations are carried out to illustrate the analytical results

stability and bifurcation in a two variable delay model for circadian rhythm of neurospora crassa abstract a two variable delay model for circadian rhythm of neurospora crassa is considered in this paper conditions for the global attractivity of the unique positive equilibrium are given moreover hopf bifurcation and the global continuation of the hopf bifurcation branches are addressed through a global hopf bifurcation result

global attractivity in delayed hopfield neural network models two different approaches are employed to investigate the global attractivity of delayed hopfield neural network models without assuming the monotonicity and differentiability of the activation functions liapunov functionals and functions combined with the razumikhin technique are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays in the case of monotone and smooth activation functions the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model such criteria depend on the magnitude of delays and show that self inhibitory connections can contribute to the global convergence

bifurcation analysis of a population model and the resulting sis epidemic model with delay this paper deals with the model for matured population growth proposed in cooke et al interaction of matiration delay and nonlinear birth in population and epidemic models j math biol 39 1999 332 352 and the resulting sis epidemic model the dynamics of these two models are still largely undetermined and in this paper we perform some bifurcation analysis to the models by applying the global bifurcation theory for functional differential equations we are able to show that the population model allows multiple periodic solutions for the sis model we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution

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

stable phase locked periodic solutions in a delay differential system abstract for a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function we obtain the existence stability and limiting profile of a phase locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory

complex dynamics in a delay differential equation with two delays in tick growth with diapause abstract we consider a delay differential equation for tick population with diapause derived from an age structured population model with two time lags due to normal and diapause mediated development we derive threshold conditions for the global asymptotic stability of biologically important equilibria and give a general geometric criterion for the appearance of hopf bifurcations in the delay differential system with delay dependent parameters by choosing the normal development time delay as a bifurcation parameter we analyze the stability switches of the positive equilibrium and examine the onset and termination of hopf bifurcations of periodic solutions from the positive equilibrium under some technical conditions we show that global hopf branches are bounded and connected by a pair of hopf bifurcation values this allows us to show that diapause can lead to the occurrence of multiple stability switches coexistence of two stable limit cycles among other rich dynamical behaviours

dynamics of a delayed two coupled oscillator with excitatory to excitatory connection in this paper we consider a neural network model consisting of two coupled oscillators with delayed feedback and excitatory to excitatory connection we study how the strength of the connections between the oscillators affects the dynamics of the neural network we give a full classification of all equilibria in the parameter space and obtain its linear stability by analyzing the characteristic equation of the linearized system we also investigate the spatio temporal patterns of bifurcated periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of lie groups moreover the stability and bifurcation direction of the bifurcated periodic solutions are obtained by employing center manifold reduction and normal form theory some numerical simulations are provided to illustrate the theoretical results

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

a polynomial method of approximate centers for linear programming we present a path following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour this is done both for the problem in standard form and for its dual problem we also discuss some implementation strategies

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

minimal instability and unstable set of a phase locked periodic orbit in a delayed neural network abstract the existence of a non trivial phase locked periodic orbit is established for a system of delay differential equations describing the dynamics of networks of two identical saturating neurons we discuss the instability and the unstable manifold of this phase locked orbit in particular we give detailed information about the spectrum of the related monodromy operator and establish the connection between the unstable manifold of the phase locked orbit and the boundary of the global extension of a three dimensional c 1 submanifold of the origin we obtain a smooth solid spindle contained in the global attractor separated by a disk bordered by the phase locked orbit major technical tools include a discrete lyapunov functional invariant manifolds and other recently developed geometric theory of delay differential equations

global exponential stability and periodic solutions of cohen grossberg neural networks with continuously distributed delays abstract convergence dynamics of cohen grossberg neural networks cgnns with continuously distributed delays are discussed without assuming the differentiability and monotonicity of activation functions the differentiability of amplification functions and the symmetry of synaptic interconnection weights by skilfully constructing suitable lyapunov functionals and employing inequality technique three sets of easily verifiable delay independent criteria to guarantee the global exponential stability of a unique equilibrium point are given and moreover by constructing poincare mapping other three sets of easily verifiable delay independent criteria to assure the existence and globally exponential stability of periodic solutions are obtained six examples are given to illustrate the theoretical results

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

combining phase i and phase ii in a potential reduction algorithm for linear programming this paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality the algorithm is closely related to a similar method of anstreicher the new features are that we use a two dimensional programming problem to derive better lower bounds than anstreicher that our direction finding subproblem treats phase i and phase ii more symmetrically and that we do not need an initial lower bound our method also allows for the generation of a feasible solution so that phase i is terminated during the course of the iterations and we describe two ways to encourage this behavior

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

multiple hopf bifurcations of three coupled van der pol oscillators with delay abstract a system of three coupled van der pol oscillators with delay is considered hopf bifurcations at the zero equilibrium as the delay increases are exhibited the existence and stability of multiple periodic solutions are established using a symmetric hopf bifurcation result of wu trans amer math soc 350 1998 4799 4838

stability bifurcation and global existence of a hopf bifurcating periodic solution for a class of three neuron delayed network models in this paper a system of three delay differential equations representing a hopfield type general model for three neurons with two way bidirectional time delayed connections between the neurons and time delayed self connection from each neuron to itself is studied delay independent and delay dependent sufficient conditions for linear stability instability and the occurrence of a hopf bifurcation about the trivial equilibrium are addressed the partition of the resulting parametric space into regions of stability instability and hopf bifurcation in the absence of self connection is realized to extend the local hopf branches for large delay values a particular bidirectional delayed tri neuron model without self connection is investigated sufficient conditions for global existence of multiple non constant periodic solutions are obtained for such a model using the global hopf bifurcation theorem for functional differential equations due to j wu and the bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney and following the approach developed by wei and li

solution estimates for abstract nonlinear time variant differential delay equations abstract semilinear time variant functional differential equations in a banach space are considered a solution estimate is derived by that estimate explicit conditions for boundedness of solutions and stability are established

hopf bifurcation and global periodic solutions in a delayed predator prey system this paper is concerned with a delayed predator prey system with same feedback delays of predator and prey species to their growth respectively we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases monotonously from zero by using the theory of normal norm and center manifold reduction an explicit algorithm for determining the direction of hopf bifurcations and the stability of bifurcating periodic solutions is derived in addition the global existence results of periodic solutions bifurcating from hopf bifurcations are established by using a global hopf bifurcation result due to wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 finally a numerical example supporting our theoretical prediction is also given our findings are contrasted with recent studies on a delayed predator prey system with the feedback time delay of prey species to its growth by song and wei y song j wei local hopf bifurcation and global periodic solutions in a delayed predator prey system j math anal appl 301 2005 1 21 as the feedback time delay increases monotonously from zero the positive equilibrium of the latter switches k times from stability to instability to stability in contrast the positive equilibrium of our system appears to lose the above property

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

complicated dynamics of a ring neural network with time delays this paper studies the dynamical behavior of a ring neural network with time delays on the basis of lyapunov s method the asymptotic stability of the equilibrium is first investigated and the delay dependent criteria ensuring global stability for the ring neural network are obtained moreover based on the global hopf bifurcation theorem for fde the conditions that guarantee the global existence of periodic solutions are determined it shows that periodic solutions bifurcating from the trivial equilibrium can continue when the time delay is far away from the critical value some examples are induced to illustrate our results in addition complicated dynamics of the model are investigated with the help of numerical simulation the study results show that the model exhibits period doubling bifurcations which lead eventually to chaos and the chaos can also directly occur via the bifurcations from the quasi periodic solutions

patterns in hierarchical networks of neuronal oscillators with d3 z3 symmetry abstract in this paper we consider a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators this hierarchical network is equivariant under the symmetry group d 3 z 3 using the lattice of isotropy subgroups we study the reduced equations restricted to invariant fixed point subspaces and prove that it is possible for the oscillator network to have 4 distinct equilibria or 45 distinct periodic solutions with maximal isotropy subgroups these are classified and their bifurcation directions are determined in terms of the quadratic coefficients and relevant quantities a center manifold reduction from the hierarchical network to the normal form equations is then performed in order to investigate the codimension two bifurcations using this reduction we find a great variety of equilibria periodic and quasi periodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two level pattern hierarchy

stability and hopf bifurcation analysis on a delayed leslie gower predator prey system incorporating a prey refuge in this paper a modified leslie gower predator prey system with time delays is investigated where the time delays are regarded as bifurcation parameters by analyzing the corresponding characteristic equation the local stability of a positive equilibrium is considered moreover we show that hopf bifurcations occur when time delay crosses some critical values by deriving the equation describing the flow on the center manifold we give the formula for determining the direction of the hopf bifurcation and the stability of bifurcating periodic solutions in addition we also try on the global existence of periodic solutions by using the 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 numerical simulations are carried out to illustrate the theoretical results and they show that the time delays in the system under consideration can destroy the stability of the system

polynomial time algorithms for linear programming based only on primal scaling and projected gradients of a potential function this paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function the paper contains extensions and analysis of two polynomial time algorithms for linear programming we first present an extension of gonzaga s o nl iteration algorithm that computes dual variables and does not assume a known optimal objective function value this algorithm uses only affine scaling and is based on computing the projected gradient of the potential function q ln x t s sum limits j 1 n ln x j wherex is the vector of primal variables ands is the vector of dual slack variables and q n sqrt n the algorithm takes either a primal step or recomputes dual variables at each iteration we next present an alternate form of ye s o sqrt n l iteration algorithm that is an extension of the first algorithm of the paper but uses the potential function q ln x t s sum limits j 1 n ln x j sum limits j 1 n ln s j where q n sqrt n we use this alternate form of ye s algorithm to show that ye s algorithm is optimal with respect to the choice of the parameterq in the following sense suppose thatq n n t wheret 0 then the algorithm will solve the linear program in o n r l iterations wherer max t 1 t thus the value oft that minimizes the complexity bound ist 1 2 yielding ye s o sqrt n l iteration bound

on some variational problems in the theory of unitarily invariant norms and hadamard products abstract we deal with two recent conjectures of r c li linear algebra appl 278 1998 317 326 involving unitarily invariant norms and hadamard products in the particular case of the frobenius norm the first conjecture is known to be true whereas the second is still an open problem in fact in this paper we show that the frobenius norm is essentially the only invariant norm which may comply with the two conjectures more precisely if a norm satisfies the claim of either conjecture then it can be controlled from above and from below by the frobenius norm uniformly with respect to the dimension on the other hand both conjectures remain open in the relevant case of matrices with an upper bound to the rank as a first partial result in this direction we prove the first conjecture for matrices of rank 1 and for any unitarily invariant norm

rich dynamics in a delayed htlv i infection model stability switch multiple stable cycles and torus abstract in this paper we investigate the impact of time delay in ctl immune response on a htlv i infection model by defining basic reproduction number for viral infection r 0 and basic reproduction number for ctl response r c t l we characterize the model dynamics according to whether these two threshold values are greater than one especially we obtain the global dynamics if r 0 1 or r c t l 1 r 0 as well as infection persistent result when r 0 1 however the model dynamics become much richer when r c t l 1 in this case we use the time delay as a bifurcation parameter to obtain stability switch result on the positive equilibrium and global bifurcation diagrams for the model system we also conduct higher order normal form analysis and apply center manifold theory to classify the rich model dynamics near the double hopf bifurcation points our analysis indicates that time delay in ctl immune response can induce not only hopf bifurcation and double hopf bifurcation but also quasi periodic orbits torus and coexistence of multiple stable periodic solutions

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

stability and hopf bifurcation analysis on a ring of four neurons with delays we consider a four neuron ring with self feedback and delays by analyzing the associated characteristic equation linear stability is investigated and hopf bifurcations are demonstrated as well as the stability and direction of the hopf bifurcation are determined by employing the normal form method and the center manifold reduction numerical simulations are presented to illustrate the results

neimark sacker bifurcation for periodic delay differential equations in this paper we study the delay differential equation x t a t x t f t x t 1 where is a real parameter the functions a t f t are c 4 smooth and periodic in the variable t with period 1 varying the parameter eigenvalues of the monodromy operator the derivative of the time one map at the equilibrium 0 cross the unit circle and bifurcation of an invariant curve occurs to detect the critical parameter values we use floquet theory we give an explicit formula to compute the coefficient that determines the direction of the bifurcation we extend the center manifold projection method to our infinite dimensional banach space using spectral projection represented by a riesz dunford integral the neimark sacker bifurcation theorem implies the appearance of an invariant torus in the space c s 1 we apply our results to an equation used in neural network theory

spectral variations and hadamard products some problems abstract many perturbation problems in matrix theory are related to three matrix minimization problems involving hadamard products possible solutions to the problems are conjectured the problems if solved may imply significant advances in eigenvalue generalized eigenvalue variations and perhaps in other aspects in matrix theory too

stability of nonlinear waves in a ring of neurons with delays abstract in this paper we consider a ring of identical neurons with self feedback and delays based on the normal form approach and the center manifold theory we derive some formula to determine the direction of hopf bifurcation and stability of the hopf bifurcated synchronous periodic orbits phase locked oscillatory waves standing waves mirror reflecting waves and so on in addition under general conditions such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation despite the fact that the slowly oscillatory synchronous periodic solution of the scalar equation is stable we show that the corresponding synchronized periodic solution is unstable if the number of the neurons is large or arbitrary even

polynomial affine algorithms for linear programming the method of steepest descent with scaling affine scaling applied to the potential functionq logc x i 1n logxi solves the linear programming problem in polynomial time forq n ifq n then the algorithm terminates in no more than o n2l iterations if q n sqrt n withq o n then it takes no more than o nl iterations a modified algorithm using rank 1 updates for matrix inversions achieves respectively o n4l and o n3 5l arithmetic computions

hopf bifurcating periodic orbits in a ring of neurons with delays in this paper we consider a ring of neurons with self feedback and delays the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation based on the normal form approach and the center manifold theory we derive the formula for determining the properties of hopf bifurcating slowly oscillating periodic orbits for a ring of neurons with delays including the direction of hopf bifurcation stability of the hopf bifurcating slowly oscillating periodic orbits and so on moreover by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups we not only investigate the effect of synaptic delay of signal transmission on the pattern formation but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio temporal patterns