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

oscillator death in coupled functional differential equations near hopf bifurcation abstract the stability of the equilibrium solution is analyzed for coupled systems of retarded functional differential equations near a supercritical hopf bifurcation necessary and sufficient conditions are derived for asymptotic stability under general coupling conditions it is shown that the largest eigenvalue of the graph laplacian completely characterizes the effect of the connection topology on the stability of diffusively and symmetrically coupled identical systems in particular all bipartite graphs have identical stability characteristics regardless of their size furthermore bipartite graphs and large complete graphs provide respectively lower and upper bounds for the parametric stability regions for arbitrary connection topologies generalizations are given for networks with asymmetric coupling the results characterize the connection topology as a mechanism for the death of coupled oscillators near hopf bifurcation

synchronization dynamics of coupled van der pol systems abstract we investigate the dynamic processes of synchronizing chaotic van der pol systems driven by periodic force for a given set of rate parameters a driven oscillator could possess two types of chaotic attractor different initial condition results in the appearance of two trajectories which have inversion symmetry with respect to each other attraction basins of these two degenerate attractors are derived numerically possibilities of synchronizing same and different chaotic trajectories are probed we investigate carefully the transient processes preceding to synchronization with appropriate criterion we define and obtain the synchronization domain in the coupling parameter space

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

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

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

dissipative non self adjoint sturm liouville operators and completeness of their eigenfunctions abstract in this paper non self adjoint sturm liouville operators in weyl s limit circle case are studied we first determine all the non self adjoint boundary conditions yielding dissipative operators for each allowed sturm liouville differential expression then using the characteristic determinant the completeness of the system of eigenfunctions and associated functions for these dissipative operators is proved

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

dissipative operators generated by the sturm liouville differential expression in the weyl limit circle case abstract in this paper using livsic s theorem we investigate the problem of completeness of the system of eigenfunctions and associated functions of dissipative operators generated by the sturm liouville differential expression on the semi axis in weyl s limit circle case

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

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

symmetry and approximability of submodular maximization problems a number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem these results hold typically in the value oracle model where the objective function is accessible via a black box returning f s for a given s we present a general approach to deriving inapproximability results in the value oracle model based on the notion of symmetry gap our main result is that for any fixed instance that exhibits a certain symmetry gap in its multilinear relaxation there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries this unifies several known hardness results for submodular maximization e g the optimality of 1 1 e approximation for monotone submodular maximization under a cardinality constraint and the impossibility of frac12 epsilon approximation for unconstrained nonmonotone submodular maximization as a new applica

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

fundamental solutions and eigenvalues of one boundary value problem with transmission conditions in this study we consider one boundary value problem type sturm liouville with eigenparameter dependent boundary conditions and with transmission conditions at one inner point of considered finite interval we give operator theoretical formulation construct fundamental solutions and investigate some properties of the eigenvalues and corresponding eigenfunctions of the considered problem

graph cuts with interacting edge weights examples approximations and algorithms we study an extension of the classical graph cut problem wherein we replace the modular sum of edge weights cost function by a submodular set function defined over graph edges special cases of this problem have appeared in different applications in signal processing machine learning and computer vision in this paper we connect these applications via the generic formulation of cooperative graph cuts for which we study complexity algorithms and connections to polymatroidal network flows finally we compare the proposed algorithms empirically

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

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

on the perturbation determinants of a singular dissipative boundary value problem with finite transmission conditions abstract in this paper a singular dissipative boundary value problem with finite transmission conditions is investigated using livsic s theorem it is proved that the system of all eigen and associated functions of this problem is complete in the hilbert space

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

maximizing stochastic monotone submodular functions we study the problem of maximizing a stochastic monotone submodular function with respect to a matroid constraint because of the presence of diminishing marginal values in real world problems our model can capture the effect of stochasticity in a wide range of applications we show that the adaptivity gap the ratio between the values of optimal adaptive and optimal nonadaptive policies is bounded and is equal to e e 1 we propose a polynomial time nonadaptive policy that achieves this bound we also present an adaptive myopic policy that obtains at least half of the optimal value furthermore when the matroid is uniform the myopic policy achieves the optimal approximation ratio of 1 1 e this paper was accepted by dimitris bertsimas and yinyu ye optimization

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

robust monotone submodular function maximization we consider a robust formulation introduced by krause et al j artif intell res 42 427 486 2011 of the classical cardinality constrained monotone submodular function maximization problem and give the first constant factor approximation results the robustness considered is w r t adversarial removal of up to tau elements from the chosen set for the fundamental case of tau 1 we give a deterministic 1 1 e 1 vartheta m approximation algorithm where m is an input parameter and number of queries scale as o n m 1 in the process we develop a deterministic 1 1 e 1 vartheta m approximate greedy algorithm for bi objective maximization of two monotone submodular functions generalizing the ideas and using a result from chekuri et al in focs 10 ieee pp 575 584 2010 we show a randomized 1 1 e epsilon approximation for constant tau and epsilon le frac 1 tilde varomega tau making o n 1 epsilon 3 queries further for tau ll sqrt k we give a fast and practical 0 387 algorithm finally we also give a black box result result for the much more general setting of robust maximization subject to an independence system

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

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

coupling leads to chaos abstract the main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two brusselators linearly coupled by diffusion we will show that such a family contains a generic unfolding of a 4 dimensional nilpotent singularity of codimension 4 on the other hand we will prove that in any generic unfolding x of an n dimensional nilpotent singularity of codimension n there are bifurcation curves of n 1 dimensional nilpotent singularities of codimension n 1 which are in turn generically unfolded by x arguments conclude recalling that any generic unfolding of the 3 dimensional nilpotent singularity of codimension 3 exhibits strange attractors

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

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

double hopf bifurcation in a container crane model with delayed position feedback in this paper we study dynamics in a container crane model with delayed position feedback with particular attention focused on non resonant double hopf bifurcation by using multiple time scales and center manifold reduction methods we obtain the equivalent normal forms near a double hopf critical point in this neutral delayed differential system moreover bifurcations are classified in a two dimensional parameter space near the critical point numerical simulations are presented to demonstrate the applicability of the theoretical results

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

local bifurcation in symmetric coupled cell networks linear theory abstract we consider a coupled cell network of differential equations with finite symmetry group where permutes cells transitively we show how the structure of the coupled cell network represented by a directed graph whose vertices represent individual cells and edges represent couplings can be taken into account in the bifurcation analysis of a fully symmetric steady state solution we focus on the analysis of the linearized vector field at a fully symmetric equilibrium and show that in the case of active cells if is abelian the network structure does not influence the types of codimension one local bifurcations we also show that beyond this context when is not abelian cells are passive or when considering local bifurcations of higher codimensions anomalies due to the network structure may arise

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

comparing apples and oranges query trade off in submodular maximization fast algorithms for submodular maximization problems have a vast potential use in applicative settings such as machine learning social networks and economics though fast algorithms were known for some special cases only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint the known fast algorithm matches the best possible approximation guarantee while trying to reduce the number of value oracle queries the algorithm performs our main result is a new algorithm for this general case that establishes a surprising trade off between two seemingly unrelated quantities the number of value oracle queries and the number of matroid independence queries performed by the algorithm specifically one can decrease the former by increasing the latter and vice versa while maintaining the best possible approximation guarantee such a trade off is very useful since various applications might incur significantly diff

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

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

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

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

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

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