two parameter bifurcations in a network of two neurons with multiple delays abstract we consider a network of two coupled neurons with delayed feedback we show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena regarding eigenvalues of the connection matrix as bifurcation parameters we obtain codimension 1 bifurcations including a fold bifurcation and a hopf bifurcation and codimension 2 bifurcations including fold hopf bifurcations and hopf hopf bifurcations we also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions in particular we obtain stable or unstable equilibria periodic solutions quasi periodic solutions and sphere like surfaces of solutions we also show how to evaluate critical normal form coefficients from the original system of delay differential equations without computing the corresponding center manifolds

stability of a general hiv 1 reaction diffusion model with multiple delays and immune response abstract the aim of this paper is to study a general hiv 1 reaction diffusion model with space dependent diffusion coefficients and antibody immune response the model contains four discrete time delays and three categories of infected cells latently infected cells short lived productively infected cells and long lived productively infected cells the effect of highly active antiretroviral therapies haart used to treat hiv 1 is incorporated into the model the existence non negativity and boundedness of solutions are studied in order to verify the well posedness of the model all possible equilibria of the model are evaluated and the threshold parameters needed for their existence and stability are investigated the global asymptotic stability of the corresponding equilibria are proved by building suitable lyapunov functionals the numerical simulations are conducted in order to confirm the theoretical results and show the behavior of solutions in space and time

multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks we show that an n neuron cellular neural network with time varying delay can have 2n periodic orbits located in saturation regions and these periodic orbits are locally exponentially attractive in addition we give some conditions for ascertaining periodic orbits to be locally or globally exponentially attractive and allow them to locate in any designated region as a special case of exponential periodicity exponential stability of delayed cellular neural networks is also characterized these conditions improve and extend the existing results in the literature to illustrate and compare the results simulation results are discussed in three numerical examples

on the global convergence of a class of functional differential equations with applications in neural network theory abstract we study a system of retarded functional differential equations which generalise both the hopfield neural network model as well as hybrid network models of the cellular neural network type our main results give sufficient conditions for the global asymptotic stability of such systems and are milder than previously known conditions for the hybrid models when specialised to neural networks our models allow us to consider several different types of activation functions including piecewise linear sigmoids and unbounded activations as well as the usualc smooth sigmoids these issues are vital in the applications we also study neural network models with nonconstant delaysr t

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

a note on convergence under dynamical thresholds with delays we complement the study of the asymptotic behaviour of the dynamical threshold neuron model with delay introduced by gopalsamy and leung 1997 by providing a description of the dynamics of the system in the remaining parameters range we characterize the regions of harmless delays and those in which delay induced oscillations appear

a criterion for stability of matrices a necessary and sufficient condition for the stability ofn nmatrices with real entries is proved applications to asymptotic stability of equilibria for vector fields are considered the results offer an alternative to the well known routh hurwitz conditions

erratum to existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients by combining the theory of the exponential dichotomy and liapunov function we study the existence and attractivity of almost periodic solutions for cellular neural networks cnns with distributed delays and variable coefficientsdx idt b i t x i t j 1 na i j t f j x j t j 1 nb i j t f j m j 0 k i j u x j t u du i i t we obtain some sufficient conditions to ensure that for the networks there exists a unique almost periodic solution and all its solutions converge to such an almost periodic solution an example is given to illustrate that the conditions of our results are feasible

global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time varying delays in this paper we present the analytical results on the global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time varying delays sufficient conditions are derived for ascertaining the existence uniqueness and global exponential periodicity of the oscillatory solution of such recurrent neural networks by using the comparison principle and mixed monotone operator method the periodicity results extend or improve existing stability results for the class of recurrent neural networks with and without time delays

global continuation of nonlinear waves in a ring of neurons ms received 14 may 2003 accepted 23 march 2005 in this paper we consider a ring of neurons with self feedback and delays as a resultof our approach based on global bifurcation theorems of delay dierential equationscoupled with representation theory of lie groups the coexistence of its asynchronousperiodic solutions i e mirror re ecting waves standing waves and discrete waves bifurcated simultaneously from the trivial solution at some critical values of thedelay will be established for delay not only near to but also far away from the criticalvalues therefore we can obtain wave solutions of large amplitudes in addition weconsider the coincidence of these periodic solutions

linear stability of delayed reactiondiffusion systems a common feature of pattern formation in both space and time is the destabilization of a stable equilibrium solution of an ordinary differential equation by adding diffusion or delay or both here we study linear stability of general reactiondiffusion systems with off diagonal time delays we show that a delay stable system cannot be destabilized by diffusion and that a diffusion stable system is also stable with respect to delay if the diffusion is sufficiently fast a system with direct negative feedback which is strongly stable with respect to diffusion can be destabilized by off diagonal delay

multistability and convergence in delayed neural networks abstract we present the existence of 2 n stable stationary solutions for a general n dimensional delayed neural networks with several classes of activation functions the theory is obtained through formulating parameter conditions motivated by a geometrical observation positively invariant regions for the flows generated by the system and basins of attraction for these stationary solutions are established the theory is also extended to the existence of 2 n limit cycles for the n dimensional delayed neural networks with time periodic inputs it is further confirmed that quasiconvergence is generic for the networks through justifying the strongly order preserving property as the self feedback time lags are small for the neurons with negative self connection weights our theory on existence of multiple equilibria is then incorporated into this quasiconvergence for the network four numerical simulations are presented to illustrate our theory

bifurcations stability and monotonicity properties of a delayed neural network model abstract a delay differential equation modelling an artificial neural network with two neurons is investigated a linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions these can lose stability through either a pitchfork or a hopf bifurcation which is shown to be supercritical at appropriate parameter values an interaction takes place between the pitchfork and hopf bifurcations conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space analytic results are illustrated with numerical simulations

a numerical method for delayed partial differential equations describing infectious diseases abstract in this paper we propose a numerical method for delayed partial differential equations that describe the dynamics of viral infections such as the human immunodeficiency virus hiv and the hepatitis b virus hbv we first prove that the proposed numerical method preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem by constructing appropriate discrete lyapunov functionals we show that the proposed method also preserves the global stability of equilibria of the corresponding continuous system with no restriction on the space and time step sizes moreover the discrete model and main results presented in qin et al 2014 are extended and generalized

a generalized hbv model with diffusion and two delays an hepatitis b virus hbv model with spatial diffusion general incidence rate and time delays subject to homogeneous neumann boundary conditions is formulated the stability of the disease free equilibrium and the chronic infection equilibrium is analyzed by using the linearization method and by constructing appropriate lyapunov functionals furthermore several diffusive hbv models existing in the literature are generalized and extended

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

slowly oscillating periodic solutions for a delayed frustrated network of two neurons in this article we study a delayed frustrated network of two neurons we obtain a two dimensional closed disk bordered by a slowly oscillating periodic orbit and we give a complete description about the dynamics of the flow restricted to this closed disk

nonlinear measures a new approach to exponential stability analysis for hopfield type neural networks in this paper a new concept called nonlinear measure is introduced to quantify stability of nonlinear systems in the way similar to the matrix measure for stability of linear systems based on the new concept a novel approach for stability analysis of neural networks is developed with this approach a series of new sufficient conditions for global and local exponential stability of hopfield type neural networks is presented which generalizes those existing results by means of the introduced nonlinear measure the exponential convergence rate of the neural networks to stable equilibrium point is estimated and for local stability the attraction region of the stable equilibrium point is characterized the developed approach can be generalized to stability analysis of other general nonlinear systems

improved stability criteria for a class of neural networks with variable delays and impulsive perturbations this paper further investigates the problem of stability analysis for a kind of neural networks with variable delays and nonlinear impulses with the help of m matrix the homeomorphism theory some effective inequalities and analysis techniques certain novel criteria on the existence uniqueness and exponential stability of the equilibrium point have been established moreover these criteria possess adjustable real parameters which extend and improve many existing results in the literature in the end two numerical examples are provided to illustrate the validation of the theoretical results

on some necessary and sufficient conditions for a recurrent neural network model with time delays to generate oscillations in this paper the existence of oscillations for a class of recurrent neural networks with time delays between neural interconnections is investigated by using the fixed point theory and liapunov functional we prove that a recurrent neural network might have a unique equilibrium point which is unstable this particular type of instability combined with the boundedness of the solutions of the system will force the network to generate a permanent oscillation some necessary and sufficient conditions for these oscillations are obtained simple and practical criteria for fixing the range of parameters in this network are also derived typical simulation examples are presented

a combinatorial converse to the perron frobenius theorem abstract a sign pattern requires allows the perron property if every some matrix with that sign pattern has its special radius among its eigenvalues we characterize those sign patterns that require the perron property which in a sense provides a converse to the classical perron frobenius theorem also a large class of patterns that allow the perron property is identified but a complete characterization remains an open problem

almost periodic dynamics of a class of delayed neural networks with discontinuous activations we use the concept of the filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right hand side which contains the widely studied delayed neural network models with almost periodic self inhibitions interconnection weights and external inputs we prove that diagonal dominant conditions can guarantee the existence and uniqueness of an almost periodic solution as well as its global exponential stability as special cases we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients respectively from the proof of the existence and uniqueness of the solution we prove that the solution of a delayed dynamical system with high slope activations approximates to the filippov solution of the dynamical system with discontinuous activations

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

synchronization of diffusively coupled limit cycle oscillators we develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems we examine two classes of systems reaction diffusion pdes with neumann boundary conditions and compartmental odes where compartments are interconnected through diffusion terms with adjacent compartments in both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle we provide two time scale averaging methods for certifying stability of spatially homogeneous time periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion we highlight cases where diffusion stabilizes or destabilizes such trajectories

dynamics of deterministic and stochastic paired excitatory inhibitory delayed feedback we examine the effects of paired delayed excitatory and inhibitory feedback on a single integrate and fire neuron with reversal potentials embedded within a feedback network these effects are studied using bifurcation theory and numerical analysis the feedback occurs through modulation of the excitatory and inhibitory conductances by the previous firing history of the neuron as a consequence the feedback also modifies the membrane time constant such paired feedback is ubiquitous in the nervous system we assume that the feedback dynamics are slower than the membrane time constant which leads to a rate model formulation our article provides an extensive analysis of the possible dynamical behaviors of such simple yet realistic neural loops as a function of the balance between positive and negative feedback with and without noise and offers insight into the potential behaviors such loops can exhibit in response to time varying external inputs with excitatory feedback the system can be quiescent can be periodically firing or can exhibit bistability between these two states with inhibitory feedback quiescence oscillatory firing rates and bistability between constant and oscillatory firing rate solutions are possible the general case of paired feedback exhibits a blend of the behaviors seen in the extreme cases and can produce chaotic firing we further derive a condition for a dynamically balanced paired feedback in which there is neither bistability nor oscillations we also show how a biophysically plausible smoothing of the firing function by noise can modify the existence and stability of fixed points and oscillations of the system we take advantage in our mathematical analysis of the existence of an invariant manifold which reduces the dimensionality of the dynamics and prove the stability of this manifold the novel computational challenges involved in analyzing such dynamics with and without noise are also described our results demonstrate that a paired delayed feedback loop can act as a sophisticated computational unit capable of switching between a variety of behaviors depending on the input current the relative strengths and asymmetry of the two parallel feedback pathways and the delay distributions and noise level

a sufficient condition for additive d stability and application to reaction diffusion models abstract the matrix a is said to be additively d stable if a d remains hurwitz for all non negative diagonal matrices d in reaction diffusion models additive d stability of the matrix describing the reaction dynamics guarantees the stability of the homogeneous steady state thus ruling out the possibility of diffusion driven instabilities we present a new criterion for additive d stability using the concept of compound matrices we first give conditions under which the second additive compound matrix has non negative off diagonal entries we then use this metzler property of the compound matrix to prove additive d stability with the help of an additional determinant condition this result is then applied to investigate the stability of cyclic reaction networks in the presence of diffusion finally a reaction network structure that fails to achieve additive d stability is exhibited

fixed b subsampling and the block bootstrap improved confidence sets based on p value calibration summary subsampling and block based bootstrap methods have been used in a wide range of inference problems for time series to accommodate the dependence these resampling methods involve a bandwidth parameter such as the subsampling window width and block size in the block based bootstrap in empirical work using different bandwidth parameters could lead to different inference results but traditional first order asymptotic theory does not capture the choice of the bandwidth we propose to adopt the fixed b approach as advocated by kiefer and vogelsang in the heteroscedasticity auto correlation robust testing context to account for the influence of the bandwidth on inference under the fixed b asymptotic framework we derive the asymptotic null distribution of the p values for subsampling and the moving block bootstrap and further propose a calibration of the traditional small b based confidence intervals regions or bands and tests our treatment is fairly general as it includes both finite dimensional parameters and infinite dimensional parameters such as the marginal distribution function simulation results show that the fixed b approach is more accurate than the traditional small b approach in terms of approximating the finite sample distribution and that the calibrated confidence sets tend to have smaller coverage errors than the uncalibrated counterparts

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

activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks this technical note proposes to study the activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks the concept of activity invariant sets deeply describes the property of an invariant set by that the activity of some neurons keeps invariant all the time conditions are obtained for locating activity invariant sets under some conditions it shows that an activity invariant set can have one equilibrium point which attracts exponentially all trajectories starting in the set since the attractors are located in activity invariant sets each attractor has binary pattern and also carries analog information such results can provide new perspective to apply attractor networks for applications such as group winner take all associative memory etc

global stability for reaction diffusion equations in biology the aim of this work is to study the global stability for some diffusion equations in biology by constructing lyapunov functionals these lyapunov functionals are obtained from those for ordinary differential equations several examples from virology and epidemiology are given to illustrate our method

classes of general h matrices let m a denote the comparison matrix of a square h matrix a that is m a is an m matrix h matrices such that their comparison matrices are nonsingular are well studied in the literature in this paper we study characterizations of h matrices with either singular or nonsingular comparison matrices the spectral radius of the jacobi matrix of m a and the generalized diagonal dominance property are used in the characterizations finally a classification of the set of general h matrices is obtained

dynamics of a diffusive virus model with general incidence function cell to cell transmission and time delay abstract in this paper we revisit a diffusive virus dynamics model with general incidence function and time delay one novelty of our model is that we introduce cell to cell transmission via formation of virological synapses to reflect the fact that it may play a more important role in virus spreading in addition to virus to cell infection we justify the well posedness of the model and identify the basic reproduction number 0 for the model to be a sharp threshold value the global stability of equilibria is determined by constructing suitable lyapunov functionals in the sense that the infection free equilibrium is globally asymptotically stable if 0 1 and when 0 1 the global asymptotic stability of infection equilibrium implies that the infection will persist a significant impact of the cell to cell transmission is that they increase the basic reproduction number if one neglects either the cell to cell transmission or virus to cell infection the basic reproduction number of the model that is under evaluated last we perform numerical simulation to support our theoretic results we set the domain of the viruses to be a two dimensional square domain with the homogeneous neumann boundary conditions to reflect the spatial spreading

reaction diffusion systems with time delays monotonicity invariance comparison and convergence in a recent paper we have obtained differential inequality results developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a banach space the primary application for these results is to reaction diffusion systems in which the reaction nonlinearity contains time delays in the present paper we apply these results to reaction diffusion systems with time delays of the form u t d u f x u t x x u u n 0 x where is a bounded open set in r n with smooth boundary u r n d diag d 1 d n with d i 0

on stability of two degenerate reaction diffusion systems abstract in this paper we construct a partially degenerate reaction diffusion equation subject to the neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable in this way we solve an open problem proposed by casten and holland 1977 4 moreover we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross diffusion effects which cannot be determined by a simple spectral analysis based on the well developed semigroup theory

diagnostic checking in multivariate arma models with dependent errors using normalized residual autocorrelations in this paper we derive the asymptotic distribution of normalized residual empirical autocovariances and autocorrelations under weak assumptions on the noise we propose new portmanteau statistics

global stability of a diffusive and delayed virus dynamics model with beddington deangelis incidence function and ctl immune response in this paper we study a diffusive and delayed virus dynamics model with beddington deangelis incidence and ctl immune response by constructing lyapunov functionals we show that if the basic reproductive number is less than or equal to one then the infection free equilibrium is globally asymptotically stable if the immune reproductive number is less than or equal to one and the basic reproductive number is greater than one then the immune free equilibrium is globally asymptotically stable if the immune reproductive number is greater than one then the interior equilibrium is globally asymptotically stable

diffusion driven instability in reaction diffusion systems abstract for a stable matrix a with real entries sufficient and necessary conditions for a d to be stable for all non negative diagonal matrices d are obtained implications of these conditions for the stability and instability of constant steady state solutions to reaction diffusion systems are discussed and an example is given to show applications

a stochastic hepatitis b epidemic model driven by levy noise abstract in this paper we investigate a stochastic hepatitis b epidemic model driven by levy noise we prove the existence of a unique global positive solution and we present sufficient conditions to derive extinction and persistence in mean of the disease

stability and hopf bifurcation of a magnetic bearing system with time delays the effect of time delays occurring in the feedback control loop on the linear stability of a simple magnetic bearing system is investigated by analyzing the associated characteristic transcendental equation it is found that a hopf bifurcation can take place when time delays pass certain values the direction and stability of the hopf bifurcation are determined by constructing a center manifold and by applying the normal form method it is also found that a codimension two bifurcation can occur through a hopf and a steady state bifurcation interaction finally numerical simulations are performed to verify the analytical predictions

nonlinear stability in reaction diffusion systems via optimal lyapunov functions we define optimal lyapunov functions to study nonlinear stability of constant solutions to reaction diffusion systems a computable and finite radius of attraction for the initial data is obtained applications are given to the well known brusselator model and a three species model for the spatial spread of rabies among foxes

stabilization role of inhibitory self connections in a delayed neural network abstract in a delayed hopfield neural network that is strongly connected with non inhibitory interconnections fast and inhibitory self connections lead to global convergence to a unique equilibrium of the network by applying monotone dynamical systems theory and an embedding technique we prove that this conclusion remains true without the requirement of strong connectivity or non inhibitory interconnections

global dynamics of a diffusive and delayed viral infection model with cellular infection and nonlinear infection rate abstract a diffusive and delayed viral dynamics model which incorporates cell to cell transmission cell mediated immune responses and general nonlinear incidence is investigated by constructing lyapunov functionals it is shown that the global threshold dynamics of the model is fully determined by the basic reproduction numbers of the virus and the immune response r 0 and r 1 the result implies that both the basic reproduction number of the virus and the immune response might be underevaluated without considering either the virus to cell transmission or cell to cell transmission the main results obtained in this paper extend some known results

linear threshold discrete time recurrent neural networks stability and globally attractive sets the stability of linear threshold dynamic neural networks is studied and a series of methods to obtain globally attractive sets is proposed a sufficient condition to judge whether an invariant set is a globally attractive set is also proposed this method requires only the solution to a class of linear matrix inequalities also two direct methods to obtain globally attractive sets are given the stability criteria presented are based on the proposed globally attractive sets some numerical examples are given to illustrate the effectiveness of the obtained results

okubo s quartic trace formula for exceptional lie algebras soit l une algebre de lie simple a forme de killing k et soit 7b l endv une representation irreductible de 7b l on donne une preuve des 2 formules suivantes trace x 4 c k x x 2 x 7b l trace x 3 0 x 7b l si 7b l n est pas de type a l

global convergence of delayed dynamical systems we discuss some delayed dynamical systems investigating their stability and convergence in a critical case to ensure the stability the coefficients of the dynamical system must satisfy some inequalities in most existing literatures the restrictions on the coefficients are strict inequalities the tough question is what will happen in the case critical case the strict inequalities are replaced by nonstrict inequalities i e is replaced by spl les the purpose of the paper is to discuss this critical case and give an affirmative answer in the case that the activation functions are hyperbolic tangent

flexible lie admissible algebras abstract this paper investigates finite dimensional flexible lie admissible algebras a over fields of characteristic 0 under these hypotheses the vector space a with the lie product x y xy yx is a lie algebra denoted by a the main result of this work gives a characterization of those flexible lie admissible algebras for which the solvable radical of a is a direct summand of a included in this class of algebras are all flexible lie admissible a for which a is a reductive lie algebra our technique is to view a as a module for a certain semisimple lie algebra of derivations of a and to see what restrictions the module structure imposes on the multiplication of a a subsequent investigation will show that this module approach can also be used to determine the flexible lie admissible algebras a for which the radical of a is abelian

stability and hopf bifurcation analysis on a simplified bam neural network with delays a delay differential equation modelling a bidirectional associative memory bam neural network with three neurons is investigated its dynamics are studied in terms of local analysis and hopf bifurcation analysis by analyzing the associated characteristic equation its linear stability is investigated and hopf bifurcations are demonstrated the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold reduction numerical simulation results are given to support the theoretical predictions

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

global asymptotic stability of a class of nonautonomous integro differential systems and applications abstract in this paper we study the global asymptotic stability of a class of nonautonomous integro differential systems by constructing suitable lyapunov functionals we establish new and explicit criteria for the global asymptotic stability in the sense of definition 2 1 in the autonomous case we discuss the global asymptotic stability of a unique equilibrium of the system and in the case of periodic system we establish sufficient criteria for existence uniqueness and global asymptotic stability of a periodic solution also explored are applications of our main results to some biological and neural network models the examples show that our criteria are more general and easily applicable and improve and generalize some existing results

applications of the exponential ordering in the study of almost periodic delayed hopfield neural networks abstract this paper studies almost periodic neural networks of hopfield type described by delayed differential equations the authors introduce an exponential ordering to analyze the long term behavior of the solutions they prove some theorems of global convergence and deduce the stabilization role of the fast inhibitory self connections the proof which in the case of the neural networks considered in this paper requires uniform stability uses arguments of comparison of differential equations and methods of the non autonomous monotone theory of dynamical systems when the connections between different neurons are excitatory improved conditions of convergence are obtained and the stabilization role of strongly positive inputs is also shown the applicability of the results is illustrated with several numerical experiments based on two different families of neural networks

stability bifurcation and multistability in a system of two coupled neurons with multiple time delays a system of delay differential equations representing a model for a pair of neurons with time delayed connections between the neurons and time delayed feedback from each neuron to itself is studied conditions for the linear stability of the trivial solution of this system are represented in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements it is shown that the trivial fixed point may lose stability via a pitchfork bifurcation a hopf bifurcation or one of three types of codimension two bifurcations multistability near these latter bifurcations is predicted using center manifold analysis and confirmed using numerical simulations

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

some remarks on matrix stability with application to turing instability abstract an example of a 4 4 matrix is given that provides a counterexample to a result on turing diffusion driven instability and also answers negatively a conjecture on strong stability such instability is shown to arise from nonreal eigenvalues the relevance and the connection of our example to classes of matrix stability known in the literature are discussed

existence of periodic solutions for a system of delay differential equations abstract in this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two neuron network model of hopfield type with time delayed connections between the neurons we first examine the local stability of the trivial solution propose some sufficient conditions for the uniqueness of equilibria and then apply the poincare bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions in addition a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given numerical examples are provided to support the theoretical analysis

nonstandard finite difference scheme for a diffusive within host virus dynamics model with both virus to cell and cell to cell transmissions abstract in this paper we first propose a diffusive within host virus dynamics model with both virus to cell and cell to cell transmissions then we consider the discretization of the model by using nonstandard finite difference scheme it is then followed by the investigation of the global stability of the equilibria of the discrete model our study shows that if the basic reproduction number r 0 1 then the infection free equilibrium is globally asymptotically stable however if r 0 1 then the infection equilibrium is globally asymptotically stable numerical simulations are presented to illustrate our theoretical results

an lmi approach to asymptotical stability of multi delayed neural networks abstract in this paper some criteria are derived for global asymptotic stability of a class of neural networks with multiple constant or time varying delays based on the lyapunov krasovskii stability theory for functional differential equations and the linear matrix inequality lmi approach some delay independent criteria for neural networks with multiple constant delays and delay dependent criteria for neural networks with multiple time varying delays are provided to guarantee global asymptotic stability of these networks the main results are generalizations of some recent results reported in the literature

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

stability analysis of a virus dynamics model with general incidence rate and two delays the aim of this work is to study the dynamical behavior of a virus dynamics model with general incidence rate and two delays the first delay represents the time from the virus entry to the production of new viruses and the second delay corresponds to the time necessary for a newly produced virus to become infectious lyapunov functionals are constructed and lasalle invariance principle for delay differential equations is used to establish the global asymptotic stability of the disease free and the chronic infection equilibria the results obtained show that the global dynamics are completely determined by the value of a certain threshold parameter called the basic reproduction number r 0 and under some assumptions on the general incidence function our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection these results can be applied to a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models

existence and global exponential stability of periodic solution of cellular neural networks with time varying delays by using the continuation theorem of mawhin s coincidence degree theory and gronwall s inequality some new sufficient conditions are obtained ensuring existence and global exponential stability of periodic solution of cellular neural networks with periodic coefficients and delays these results are helpful to design globally exponentially stable and oscillatory cellular neural networks

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

travelling waves of a diffusive kermack mckendrick epidemic model with non local delayed transmission we obtain full information about the existence and non existence of travelling wave solutions for a general class of diffusive kermack mckendrick sir models with non local and delayed disease transmission we show that this information is determined by the basic reproduction number of the corresponding ordinary differential model and the minimal wave speed is explicitly determined by the delay such as the latent period and non locality in disease transmission and the spatial movement pattern of the infected individuals the difficulty is the lack of order preserving property of the general system and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying schauder s fixed point theorem

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

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

on characteristic equations trace identities and casimir operators of simple lie algebras two approaches are developed to exploit for simple complex or compact real lie algebras g the information that stems from the characteristic equations of representation matrices and casimir operators these approaches are selected so as to be viable not only for small lie algebras but also to be suitable for treatment by computer algebra a very large body of new results emerges in the forms of a identities of a tensorial nature involving structure constants etc of g b trace identities for powers of matrices of the adjoint and defining representations of g c expressions of nonprimitive casimir operators of g in terms of primitive ones the methods are sufficiently tractable to allow not only explicit proof by hand of the nonprimitive nature of the quartic casimir of g2 f4 e6 but also e g of that of the tenth order casimir of f4

dynamical behaviors of a large class of general delayed neural networks research of delayed neural networks with varying self inhibitions interconnection weights and inputs is an important issue in the real world self inhibitions interconnection weights and inputs should vary as time varies in this letter we discuss a large class of delayed neural networks with periodic inhibitions interconnection weights and inputs we prove that if the activation functions are of lipschitz type and some set of inequalities for example the set of inequalities 3 1 in theorem 1 is satisfied the delayed system has a unique periodic solution and any solution will converge to this periodic solution we also prove that if either set of inequalities 3 20 in theorem 2 or 3 23 in theorem 3 is satisfied then the system is exponentially stable globally this class of delayed dynamical systems provides a general framework for many delayed dynamical systems as special cases it includes delayed hopfield neural networks and cellular neural networks as well as distributed delayed neural networks with periodic self inhibitions interconnection weights and inputs moreover the entire discussion applies to delayed systems with constant self inhibitions interconnection weights and inputs

lmi based stability criteria for neural networks with multiple time varying delays abstract in this paper the stability of neural networks with multiple time varying delays is studied a new class of lyapunov krasovskii functionals is constructed and the s procedure and free weighting matrix method are employed to derive a delay dependent stability criterion from which a delay independent criterion is obtained as a special case moreover the result is also extended to delay dependent and rate independent stability criteria for multiple unknown time varying delays finally numerical examples are given to illustrate the effectiveness of our methods and improvement over the existing ones

frustration stability and delay induced oscillations in a neural network model the effect of time delays on the linear stability of equilibria in an artificial neural network of hopfield type is analyzed the possibility of delay induced oscillations occurring is characterized in terms of properties of the not necessarily symmetric connection matrix of the network such oscillations are possible exactly when the network is frustrated equivalently when the signed digraph of the matrix does not require the perron property nonlinear analysis centre manifold computation of a three unit frustrated network is presented giving the nature of the bifurcations taking place a supercritical hopf bifurcation is shown to occur and a codimension two bifurcation is unfolded

bifurcation of a three unit neural network abstract considered is a system of delay differential equations modeling a time delayed connecting network of three neurons without self feedback discussing the change of the number of eigenvalues with zero real part we locate the boundary of the stability region and finally determine the largest stability region of trivial solution we investigate the existence of bifurcation phenomena of codimension one two of the trivial equilibrium by considering the intersections of some parameter curves which in the a half parameter plane correspond to zero root or pure imaginary roots in particular the equivariant bifurcation is studied because of the equivariance of the system we also present numerical simulations to demonstrate the rich dynamical behavior near the equivariant pitchfork hopf bifurcation points hopf hopf bifurcation points and some higher codimension bifurcation points

convergence dynamics and pseudo almost periodicity of a class of nonautonomous rfdes with applications this paper studies the dynamics of a system of retarded functional differential equations i e rfdes which generalize the hopfield neural network models the bidirectional associative memory neural networks the hybrid network models of the cellular neural network type and some population growth model sufficient criteria are established for the globally exponential stability and the existence and uniqueness of pseudo almost periodic solution the approaches are based on constructing suitable lyapunov functionals and the well known banach contraction mapping principle the paper ends with some applications of the main results to some neural network models and population growth models and numerical simulations

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

a non standard finite difference scheme for a delayed and diffusive viral infection model with general nonlinear incidence rate abstract a non standard finite difference scheme is proposed to solve a delayed and diffusive viral infection model with general nonlinear incidence rate the results show that the discrete model preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem moreover this method preserves all equilibria of the original continuous model by constructing lyapunov functionals we show that the global stability of equilibria is completely determined by the basic reproduction number 0 which implies that the proposed discrete model can efficiently blue preserve the global stability of equilibria of the corresponding continuous model numerical experiments are carried out to support the theoretical results

combinatorial matrix analysis an overview abstract we survey recent work in some components of combinatorial matrix analysis including qualitative matrix theory matrix completion problems bounding of matrix parameters the role of the longest simple circuit and attenuation of matrix products some definitional thoughts historical perspectives and unification challenges etc are offered the style is informal following the talk on which this writeup is based

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

exponential convergence of delayed dynamical systems we discuss some delayed dynamical systems investigating their stability and convergence we prove that under mild conditions these delayed systems are global exponential convergent

stability in asymmetric hopfield nets with transmission delays sufficient conditions are derived for the delay independent stability of the equilibria in hopfield s graded response networks of the type dxi t dt bixi t nj 1aijfj jxj t ij fi t i 1 2 n when the external inputs fi are held temporally uniform a generalization to continuously distributed delays is briefly indicated several illustrative examples are numerically simulated and the results of simulations are graphically displayed