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

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

redundancy in logic i cnf propositional formulae a knowledge base is redundant if it contains parts that can be inferred from the rest of it we study some problems related to the redundancy of a cnf formula in particular any cnf formula can be made irredundant by deleting some of its clauses what results is an irredundant equivalent subset we study the complexity of problems related to irredundant equivalent subsets verification checking existence of an irredundant equivalent subset with a given size checking necessary and possible presence of clauses in irredundant equivalent subsets and uniqueness we also consider the problem of redundancy with different definitions of equivalence

on the local bump friedberg l function let f be a p adic field with residue field of cardinality q to each irreducible representation of gl n f we attach a local euler factor l bf q s q t pi via the rankin selberg method and show that it is equal to the expected factor l s t 1 2 phi pi l 2s lambda 2 circ phi pi of the langlands parameter phi pi of pi the corresponding local integrals were introduced in bump and friedberg the exterior square automorphic l functions on mathrm gl n 47 65 1990 and studied in matringe j reine angew math doi 10 1515 crelle 2013 0083 this work is in fact the continuation of matringe j reine angew math doi 10 1515 crelle 2013 0083 the result is a consequence of the fact that if delta is a discrete series representation of gl 2m f and chi is a character of levi subgroup l gl m f times gl m f which is trivial on gl m f embedded diagonally then delta is l chi distinguished if an only if it admits a shalika model this result was only established for chi mathbf 1 before

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

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

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

hopf bifurcation analysis for a model of genetic regulatory system with delay this paper deals with a mathematical model that describe a genetic regulatory system the model has a delay which affects the dynamics of the system we investigate the stability switches when the delay varies and show that hopf bifurcations may occur within certain range of the model parameters by combining the normal form method with the center manifold theorem we are able to determine the direction of the bifurcation and the stability of the bifurcated periodic solutions finally some numerical simulations are carried out to support the analytic results

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr

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

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 stability and hopf bifurcation analysis of a gene expression model abstract in this paper we investigate a model for gene expression unlike the models mathematically analyzed previously we have both transcriptional and translational time delays the stability and hopf bifurcation of the equilibrium point are investigated different to previous papers a multiple time scale mts technique is employed to calculate the normal form on the center manifold of system of delay differential equations which is much easier to implement in practice than the conventional method center manifold reduction our results show that when time delay is small the equilibrium is stable when it is at its critical value hopf bifurcation happens and while for very large value of time delay the oscillation sustains which has been confirmed by the published data and proved mathematically by using the global continuity of the hopf bifurcation in this paper

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

stability and bifurcations of equilibria in a multiple delayed differential equation the influence of multiple negative delayed feedback loops on the stability of a single action mechanism are considered a characteristic equation for the linearized stability of the equilibrium is completely analyzed as a function of two parameters describing a delay in one loop and a ratio of the gains in the two feedback loops the bifurcations occurring as the linear stability is lost are analyzed by the construction of a centre manifold in particular the nature of hopf and more degenerate higher codimension bifurcations are explicitly determined

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

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

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

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

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

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

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

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

dynamics of nonlinear parabolic systems with time delays abstract the dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system the same conclusions are obtained for a coupled system of parabolic ordinary equations with time delays applications are given to three model problems arising from ecology and nuclear engineering these model problems possess multiple steady state solutions and sufficient conditions are given to ensure the stability and instability of these solutions

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

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

sat based maxsat algorithms many industrial optimization problems can be translated to maxsat although the general problem is np hard like sat many practical problems may be solved using modern maxsat solvers in this paper we present several algorithms specially designed to deal with industrial or real problems all of them are based on the idea of solving maxsat through successive calls to a sat solver we show that this sat based technique is efficient in solving industrial problems in fact all state of the art maxsat solvers that perform well in industrial instances are based on this technique in particular our solvers won the 2009 partial maxsat and the 2011 weighted partial maxsat industrial categories of the maxsat evaluation we prove the correctness of all our algorithms we also present a complete experimental study comparing the performance of our algorithms with latest maxsat solvers

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

understanding the complexity of axiom pinpointing in lightweight description logics abstract lightweight description logics are knowledge representation formalisms characterised by the low complexity of their standard reasoning tasks they have been successfully employed for constructing large ontologies that model domain knowledge in several different practical applications in order to maintain these ontologies it is often necessary to detect the axioms that cause a given consequence this task is commonly known as axiom pinpointing in this paper we provide a thorough analysis of the complexity of several decision counting and enumeration problems associated to axiom pinpointing in lightweight description logics

the local exterior square l function holomorphy non vanishing and shalika functionals abstract let be a smooth irreducible square integrable representation of gl m f where f is a non archimedean local field of characteristic zero we prove that the exterior square l function l j s s 2 defined via an integral representation due to jacquet and shalika is regular and non vanishing in the region re s 0 we also investigate the behavior of the l function l j s s 2 at s 0 and show that if the function l j s s 2 has a pole at s 0 then has a non zero shalika functional

identifying the minimal transversals of a hypergraph and related problems the paper considers two decision problems on hypergraphs hypergraph saturation and recognition of the transversal hypergraph and discusses their significance for several search problems in applied computer science hypergraph saturation i e given a hypergraph cal h decide if every subset of vertices is contained in or contains some edge of cal h is shown to be co np complete a certain subproblem of hypergraph saturation the saturation of simple hypergraphs i e sperner families is shown to be under polynomial transformation equivalent to transversal hypergraph recognition i e given two hypergraphs cal h 1 cal h 2 decide if the sets in cal h 2 are all the minimal transversals of cal h 1 the complexity of the search problem related to the recognition of the transversal hypergraph the computation of the transversal hypergraph is an open problem this task needs time exponential in the input size it is unknown whether an output polynomial algorithm exists for several important subcases for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs output polynomial algorithms are presented computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice which is pointed out by identifying important applications in database theory boolean switching theory logic and artificial intelligence ai particularly in model based diagnosis

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

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

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

derivatives and exceptional poles of the local exterior square l function for gl_m let pi be an irreducible admissible representation of gl m f where f is a non archimedean local field of characteristic zero in 1990 s jacquet and shalika established an integral representation for the exterior square l function we complete following the method developed by cogdell and piatetski shapiro the computation of the local exterior square l function l s pi wedge 2 via the integral representation in terms of l functions of supercuspidal representations by a purely local argument with this result we show the equality of the local analytic l functions l s pi wedge 2 via the integral representation for the irreducible admissible representation pi for gl m f and the local arithmetic l functions l s wedge 2 phi pi of its langlands parameter phi pi through local langlands correspondence

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

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

time delay induced instabilities in reaction diffusion systems time delay in the kinetic terms of reaction diffusion systems has been investigated it has been shown that short delay beyond a critical threshold may induce spatiotemporal instabilities for unequal diffusivities and appropriate parameter space delay may induce turing instability resulting in stationary patterns and also interesting turing hopf transition with the formation of spirals the theoretical scheme has been numerically explored in two different prototypical reaction diffusion systems