center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with fold hopf singularity in this paper we present explicit formulas for computing the coefficients of a center manifolds for the fold hopf singularity in autonomous retarded functional differential equations as consequence normal forms associated with the flow on a center manifold up to an arbitrary order are derived the explicit formulas have been implemented using the computer algebra system maple we apply our results to a delayed system in order to show the applicability of the methodology

bifurcation analysis on a two neuron system with distributed delays abstract a general two neuron model with distributed delays is studied in this paper its local linear stability is analyzed by using the routh hurwitz criterion if the mean delay is used as a bifurcation parameter we prove that hopf bifurcation occurs for a weak kernel this means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value the direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem some numerical simulations for justifying the theoretical analysis are also given

superspaces and supersymmetries a theory of graded banach modules over a banach grassmann algebra is developed and applied to differential geometry of super manifolds the explicit structure of superspaces carrying poincare supersymmetry and extended supersymmetry including central charges is described

global hopf bifurcation in a neural netlet sufficient conditions are obtained for the global hopf bifurcation of periodic solutions in the dynamics of a neural netlet of activation and inhibition with continuously distributed delays the orbital asymptotic stability of the bifurcating periodic solutions is investigated the results are illustrated with numerical simulations a few problems are formulated in the form of conjectures

convergence and periodicity in a discrete time network of two neurons with self connections abstract in this paper we study a class of difference systems modelling a two neuron network with feedback and mcculloch pitts type activation which includes the discrete version of the two neuron model with piecewise constant argument as its special case we obtain some interesting sufficient conditions insuring the convergence of solutions as well as the existence of periodic solutions

discrete time analogues of integrodifferential equations modelling bidirectional neural networks we formulate discrete time analogues of integrodifferential equations modelling bidirectional neural networks studied by gopalsamy and he the discrete time analogues are considered to be numerical discretizations of the continuous time networks and we study their dynamical characteristics it is shown that the discrete time analogues preserve the equilibria of the continuous time networks by constructing a lyapunov type sequence we obtain easily verifiable sufficient conditions under which every solution of the discrete time analogue converges exponentially to the unique equilibrium the sufficient conditions are identical to those obtained by gopalsamy and he for the uniqueness and global asymptotic stability of the equilibrium of the continuous time network by constructing discrete time versions of halanay type inequalities we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete time analogue the latter sufficient conditions have not been obtained in the literature of continuous time bidirectional neural networks several computer simulations are provided to illustrate the advantages of our discrete time analogue in numerically simulating the continuous time network with distributed delays over finite intervals

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

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

supersymmetric distributions hilbert spaces of supersymmetric functions and quantum fields the recently investigated hilbert krein and other positivity structures of the superspace are considered in the framework of superdistributions these tools are applied to problems raised by rigorous supersymmetric quantum field theory

temporal similarity by measuring possibilistic uncertainty in cbr similarity is an essential concept in case based reasoning cbr in domains in which time plays a relevant role cbr systems require good temporal similarity measures to compare cases temporal cases are traditionally represented by a set of temporal features defining time series and temporal event sequences in the particular situation where these features are not homogeneous i e combination of qualitative and quantitative information systems find difficulties in performing the cbr cycle furthermore temporal similarity measures cannot directly apply the efficient time series techniques requiring new approaches to deal with these heterogeneous sequences to this end recent proposals are focused on direct matching between pairs of features within sequences mainly based on classical distances however three limitations to the traditional approaches have been identified 1 they do not consider the implicit temporal relations amongst all features of the sequence ignoring a large amount of temporal information 2 they ignore the uncertainty produced in any process of analogy 3 they are designed to compare pairs of sequences limiting their use to basic aspects of the retrieval step of cbr no benefits on other cbr steps temporal constraint networks have proved to be useful tools for temporal representation and reasoning and can be easily extended to manage imprecision and uncertainty an approach to solve similarity problems could be the transformation of these heterogeneous sequences into uncertain temporal relations obtaining a temporal constraint network the overall uncertainty of this network can be considered as an effective indicator of the sequences similarity therefore this paper proposes a non classical approach to measure temporal similarity of cases which are heterogeneous temporal event sequences given two or more sequences the temporal similarity is measured by describing a unique temporal scenario of possibilistic temporal relations and calculating the uncertainty produced

on a planar system modelling a neuron network with memory abstract we consider a delay differential equation modelling a network of two neurons with memory the situations without self connections and two delays and with self connections and two equal delays are addressed in terms of local stability and bifurcation analysis in the first case the dynamical behavior is studied by taking one of the delays as the bifurcating parameter namely the hopf bifurcation whose existence was proven in former works occurs as the delay crosses some critical values and is completely described conditions ensuring the stability of the periodic cycles are given when self connections are present hopf and pitchfork bifurcations co exist and the equation describing the flow on the center manifold is derived

stability and bifurcation in a tri neuron network model with discrete and distributed delays in this paper we consider a tri neuron network model with discrete and distributed delays after discussing the distributions of the roots of the characteristic equation local stability analysis of the steady state solution leads to we analyze the stability of network models with instantaneous feedback and neural interaction history delayed neural feedback and no neural interaction history and delayed neural feedback and neural interaction history we give sufficient conditions for the linear stability of the equilibrium solution we also show that a hopf bifurcation can occur when the delays take certain critical values and we further prove that there is a positive integer k such that there are k switches from stability to instability and back to stability numerical simulations are finally performed to illustrate the obtained results

superspace first order partial differential equations through the cartan kahler integration theorem in the paper first order partial differential equations are studied on superspace the key point is the use of a suitable generalization of the cartan kahler integration theorem on superspace by means of this result it is possible to investigate the characteristic fields and the structure of the solutions

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

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

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

lattices supersymmetry and kahler fermions it is shown that a graded extension of the space group of a generalised simple cubic lattice exists in any space dimension d the fermionic variables which arise admit a kahlerian interpretation each graded space group is a subgroup of a graded extension of the appropriate euclidean group e d the relevance of this to the construction of lattice theories is discussed

dynamics of a class of nonlinear discrete time neural networks in this paper we consider a class of delay difference systems with piecewise constant nonlinearity which includes the discrete version of an artificial neural network model of two neurons described by differential equations with piecewise constant arguments some interesting results are obtained for the periodicity and convergence of solutions of the systems

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

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

stability and bifurcation for discrete time cohen grossberg neural network abstract this paper investigates a discrete time cohen grossberg neural network model some sufficient criteria ensuring the asymptotic stability of the equilibrium point for this model are derived moreover by choosing the appropriate bifurcation parameter we prove that neimark sacker bifurcation occurs when the bifurcation parameter exceeds a critical value we determine the direction and stability of bifurcation by applying the normal form theory and the center manifold theorem some numerical simulations for justifying the theoretical analysis are also given

hopf bifurcation and stability for a delayed tri neuron network model a neural network model with three neurons and a single time delay is considered its linear stability is investigated and hopf bifurcations are demonstrated by analyzing the corresponding characteristic equation in particular the explicit formulae determining the stability and the direction of periodic solutions bifurcating from hopf bifurcations are obtained by applying the normal form theory and the center manifold theorem in order to illustrate our theoretical analysis some numerical simulations are also included in the end

convergence under dynamical thresholds with delays necessary and sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium of a class of delay differential equations modeling the action of a neuron with dynamical threshold effects

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

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

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

delay induced periodicity in a neural netlet of excitation and inhibition abstract the dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated it is shown that in the absence of delay the netlet relaxes to the trivial resting state if the delay is of sufficient magnitude the network is excited to a temporally periodic cyclic behaviour the analytical mechanism for the onset of cyclic behaviour is through a hopf type bifurcation approximate solusions to the periodic output of the netlet is calculated stability of the temporally periodic cycle is investigated it is shown that the bifurcation is supercritical a related discrete version of the continuous time system is formulated it is found that the discrete system also displays a cyclic behaviour results of a number of computer simulations are displayed graphically the article concludes with a brief neurobiological discussion

bifurcation analysis on a discrete time tabu learning model in this paper we consider a discrete time tabu learning single neuron model after investigating the stability of the given system we demonstrate that pichfork bifurcation flip bifurcation and neimark sacker bifurcation will occur when the bifurcation parameter exceed a critical value respectively a formula is given for determining the direction and stability of neimark sacker bifurcation by applying the normal form theory and the center manifold theorem some numerical simulations for justifying the theoretical results are also provided

convergence and periodicity of solutions for a neural network of two neurons this paper is concerned with a delay differential system with piecewise constant nonlinearity such a differential system can be regarded as a model of two neuron artificial neural network with delayed feedback some interesting results are obtained for the convergence and periodicity of solutions of the system

graded gauge theory the mathematical background for a graded extension of gauge theories is investigated after discussing the general properties of graded lie algebras and what may serve as a model for a graded lie group the graded fiber bundle is constructed its basis manifold is supposed to be the so called superspace i e the product of the minkowskian space time with the grassmann algebra spanned by the anticommuting lorentz spinors the vertical subspaces tangent to the fibers are isomorphic with the graded extension of the su n lie algebra the connection and curvature are defined then on this bundle the two different gradings are either independent of each other or may be unified in one common grading which is equivalent to the choice of the spin statistics dependence the yang mills lagrangian is investigated in the simplified case the conformal symmetry breaking is discussed as well as some other physical consequences of the model

differential equations frobenius theorem and local flows on supermanifolds the classical frobenius theorem both in its local and global formulations is generalised to superanalytic supermanifolds as an application it is proved that a coset space g h where both g and h are super lie groups is a supermanifold existence and uniqueness of local flows of tangent vector fields is proved

dynamical approach study of spurious steady state numerical solutions of nonlinear differential equations i the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics abstract the goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady state numerical solutions when using the time dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms this interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as the dynamics of numerics and the numerics of dynamics at the present time this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications this is the first of a series of research papers under the same topic our hope is to reach researchers in the fields of computational fluid dynamics cfd and in particular hypersonic and combustion related cfd by simple examples in which the exact solutions of the governing equations are known the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results one striking phenomenon is that with the same initial data the continuum and its discretized counterpart can asymptotically approach different stable solutions this behavior is especially important for employing a time dependent approach to the steady state since the initial data are usually not known and a freestream condition or an intelligent guess for the initial conditions is often used with the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart it is not easy to delineate the true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum part i concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in cfd and to serve as an introduction for parts 11 and iii of the same series of research papers

chaos in numerical analysis of ordinary differential equations abstract the discretisation of the ordinary nonlinear differential equation d y d t y 1 y by the entral difference scheme is studied for fixed mesh size in the usual numerical computation this method produces some ghost solution for the long range calculation regarding this discretisation as a dynamical system in r2 these pathological behaviors are shown to be a kind of chaos in the dynamical system for any mesh size moreover some combination of the central difference scheme and the euler s scheme is studied for the above equation it gives some motivation for henon s model the usual discretisation of a second order differential equation are studied also it gives some chaotic behaviors numerically which is similar to the behavior of the orbits of the system of differential equations proposed by henon heiles

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

turbulence in multistep methods for initial value problems solving a scalar initial value problem with a numerical k step scheme amounts to iterating a k dimensional discrete dynamical system in this paper we investigate regular and chaotic behaviour of some dynamical systems associated with adams bashforth formulas applied to a simple nonlinear test equation

bifurcation and category learning in network models of oscilating cortex abstract a genetic model of oscillating cortex which assumes minimal coupling justified by known anatomy is shown to function as an associative memory using previously developed theory the network has explicit excitatory neurons with local inhibitory interneuron feedback that forms a set of nonlinear oscillators coupled only by long range excitatory connections using a local hebb like learning rule for primary and higher order synapses at the ends of the long range connections the system learns to store the kinds of oscillation amplitude patterns observed in olfactory and visual cortex in olfaction these patterns emerge during respiration by a pattern forming phase transition which we characterize in the model as a multiple hopf bifurcation we argue that these bifurcations play an important role in the operation of real digital computers and neural networks and we use bifurcation theory to derive learning rules which analytically guarantee cam storage of continuous periodic sequences capacity n 2 fourier components for an n node network no spurious attractors

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

variational calculus and poincare cartan formalism on supermanifolds in this work we show how the poincare cartan form can be used to describe the symmetries of lagrangian supersymmetric possibly p adic field theories we show that the existence of the poincare cartan form in supermanifold theory is ensured only in a relevant class of lagrangian densities moreover we give an abstract characterization of supersymmetric invariance based on the poincare cartan form

asymptotic stability criteria for a two neuron network with different time delays in this paper the asymptotic stability of a two neuron system with different time delays has been investigated some criteria for determining the global asymptotically stability of equilibrium are derived from the theory of monotonic dynamical system and the approach of lyapunov functional for local asymptotic stability some elegant criteria are also obtained by the nyquist criteria we find that one of them depends on the length of delays while the other ones do not in the latter case the delays are sometimes called harmless delays the results obtained have leading significance in the study of neural networks composed of a large number of neurons with different time delays

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

bogdanov takens singularity in tri neuron network with time delay this brief reports a retarded functional differential equation modeling tri neuron network with time delay the bogdanov takens b t bifurcation is investigated by using the center manifold reduction and the normal form method we get the versal unfolding of the norm forms at the b t singularity and show that the model can exhibit pitchfork hopf homoclinic and double limit cycles bifurcations some numerical simulations are given to support the analytic results and explore chaotic dynamics finally an algorithm is given to show that chaotic tri neuron networks can be used for encrypting a color image

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

even sectors of lie superalgebras as locally convex lie algebras let g be a finite dimensional lie superalgebra over a graded commutative local augmented arens michael ground algebra a lie group is associated with a number of pleasant properties to the even sector g0 viewed as an ordinary complete locally convex lie algebra which is in general infinite dimensional

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

on the applicability of nonmonotonic logic to formal reasoning in continuous time the paper criticizes arguments recently advanced by shoham mcdermott and sandewall which purport to demonstrate the relevance of non monotonic logic to the formalization of reasoning about the evolution of mechanical systems in continuous time the first half of the paper examines the extended prediction problem of shoham and mcdermott reasons are given to support the claim that the problem is the product of a mistaken understanding of the the formal basis of newtonian mechanics and has no real existence an example is given showing how contrary to shoham and mcdermott s arguments it is possible to formalise reasoning about the evolution of physical systems in continuous time using only classical logic and differential calculus the second half then reviews sandewall s non monotonic logic for almost continuous systems here it is argued that the proposed framework offers only very marginal advantages in compactness of notation and generally tends to collapse back into classical logic in summary i conclude that there is as yet no good reason to believe that non monotonic logic will be a useful tool in this area

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

problems in formal temporal reasoning abstract ever since its introduction by mccarthy and hayes in 1969 the so called frame problem has been the object of much fascination and debate although it was defined in the narrow context of the situation calculus a specific temporal formalism it was clear from the start that it is in fact a manifestation of some fundamental problem in temporal reasoning our aim in this informal paper is to identify the general form of certain classes of problems that arise in formal temporal reasoning we argue that problems such as the frame problem arise from the conflicting desires to reason both rigorously and efficiently about the future this conflict does not depend on the particular underlying temporal formalism in particular we identify two formalism independent problems called the qualification problem and the extended prediction problem which subsume the frame problem to illustrate the fact that these problems are indeed inherent to the prediction task and not to a particular formalism we show that they arise in two distinct frameworks classical mechanics and hayes histories notation

chronological ignorance experiments in nonmonotonic temporal reasoning abstract we offer solutions to two problems in formal temporal reasoning the qualification problem and the extended prediction problem which as argued in 22 subsume the infamous frame problem the solutions make use of nonmonotonic logics in particular we advocate the use of a new nonmonotonic logic the logic of chronological ignorance on the practical level we demonstrate the utility of the logic in specific cases on the more theoretical level we identify classes of theories in the logic that have nice properties they each have a model that is in a certain sense unique and those models can be computed very easily the first class causal theories is sufficiently expressive to solve the qualification problem it also suggests a new account of the intuitive notion of causation although we do not discuss that in depth this class is then generalized to the class of inertial theories which are sufficiently expressive to solve also the extended prediction problem inertial theories embody the concept of potential histories which are the way the world tends to behave naturally

orthogonality relations and supercharacter formulas of u m n representations in this paper we obtain the orthogonality relations for the supergroup u m n which are remarkably different from the ones for the u n case we extend our results for ordinary representations obtained some time ago to the case of complex conjugated and mixed representations our results are expressed in terms of the young tableaux notation for irreducible representations we use the supersymmetric harish chandra itzykson zuber integral and the character expansion technique as mathematical tools for deriving these relations as a byproduct we also obtain closed expressions for the supercharacters and dimensions of some particular irreducible u m n representations a new way of labeling the u m n irreducible representations in terms of m n numbers is proposed finally as a corollary of our results new identities among the dimensions of the irreducible representations of the unitary group u n are presented

effect of delay on the boundary of the basin of attraction in a self excited single graded response neuron little attention has been paid in the past to the effects of interunit transmission delays representing a xonal and synaptic delays ontheboundary of the basin of attraction of stable equilibrium points in neural networks as a first step toward a better understanding of the influence of delay we study the dynamics of a single graded response neuron with a delayed excitatory self connection the behavior of this system is representative of that of a family of networks composed of graded response neurons in which most trajectories converge to stable equilibrium points for any delay value it is shown that changing the delay modifies the location of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria the dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins our results suggest that when dealing with networks with delay it is important to stud

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

variational calculus on supermanifolds and invariance properties of superspace field theories the foundations of a variational calculus on fibered supermanifolds are outlined giving applications to the formulation of superspace field theories utiyama theorems and conservation laws related to local gauge and general invariance of the theory are proved

all solutions of a class of discrete time systems are eventually periodic this paper is concerned with a class of discrete time systems as a model for two neurons with no internal decay such a discrete time system can be regarded as the discrete analog of a differential equation with piecewise constant nonlinearity of mcculloch pitts type base on some simple geometric facts we show that all solutions of this discrete time system are eventually periodic

analysis on the p adic superspace i generalized functions gaussian distribution the theory of generalized functions on the p adic superspace qn mp over the super commutative banach superalgebra 0 1 with the trivial 1 annihilator is proposed the p adic gaussian supermeasure is defined as a generalized function on the p adic superspace the connection between the p adic gaussian integration and the p adic gamma function of morita is considered

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

dynamics of simple electronic neural networks abstract we examine systems of one and two nonlinear threshold switching elements neurons of the kind used in electronic neural networks characteristics of these systems which deviate from standard ideal models are found to induce complex dynamics when the neurons possess a finite frequency response or a transfer characteristic with a time delay underdamped transients and instability leading to oscillation can occur inertia in the neuron connections is found to cause ringing about fixed points convoluted basin boundaries instability and spontaneous oscillation and chaotic behavior when driven furthermore the collective behavior of a network of multiple neurons can be underdamped even when the individual connections are overdamped these results imply that care should be exercised in implementing networks with electronic devices or when adding inertia to enhance the performance of optimizing networks

variational calculation of the limit cycle and its frequency in a two neuron model with delay we consider a model system of two coupled hopfield neurons which is described by delay differential equations taking into account the finite signal propagation and processing times when the delay exceeds a critical value a limit cycle emerges via a supercritical hopf bifurcation first we calculate its frequency and trajectory perturbatively by applying the poincar e lindstedt method then the perturbation series are resummed by means of the shohat expansion in good agreement with numerical values however with increasing delay the accuracy of the results from the shohat expansion worsens we thus apply variational perturbation theory vpt to the perturbation expansions to obtain more accurate results which moreover hold even in the limit of large delays

introduction to supersymmetric spin networks in this article we study the construction of supersymmetric spin networks which has a direct interpretation in context of the representation theory of the superalgebra in particular we analyze a special kind of spin network associated with superalgebra osp 1 2n it turns out that the set of corresponding spin network states forms an orthogonal basis of the hilbert space l2 a g and this argument holds even in the q deformed case the osp n 2 spin networks are also discussed briefly we expect they could provide useful techniques to quantum supergravity and gauge field theories from the point of nonperturbative view

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

bifurcations in a planar system of differential delay equations modeling neural activity abstract a planar system of differential delay equations modeling neural activity is investigated the stationary points and their saddle node bifurcations are estimated by an analysis of the associated characteristic equation hopf bifurcations are demonstrated at the intersection points of the saddle node and hopf bifurcation curves in an appropriate parameter plane the existence of bogdanov takens singularities is shown the properties of the bogdanov takens singularities are studied by applying the center manifold and normal form theory a numerical example illustrates the obtained results

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

dynamics of fitzhugh nagumo excitable systems with delayed coupling small lattices of n nearest neighbor coupled excitable fitzhugh nagumo systems with time delayed coupling are studied and compared with systems of fitzhugh nagumo oscillators with the same delayed coupling bifurcations of equilibria in an n 2 case are studied analytically and it is then numerically confirmed that the same bifurcations are relevant for the dynamics in the case ng2 bifurcations found include inverse and direct hopf and fold limit cycle bifurcations typical dynamics for different small time lags and coupling intensities could be excitable with a single globally stable equilibrium asymptotic oscillatory with symmetric limit cycle bistable with stable equilibrium and a symmetric limit cycle and again coherent oscillatory but nonsymmetric and phase shifted for an intermediate range of time lags inverse sub critical hopf and fold limit cycle bifurcations lead to the phenomenon of oscillator death the phenomenon does not occur in the case of fitzhugh nagumo oscillators with the same type of coupling

super lie groups global topology and local structure a general mathematical framework for the super lie groups of supersymmetric theories is presented the definition of super lie group is given in terms of supermanifolds and two theorems analogous to theorems in classical lie group theory are proved the relationship of the super lie groups defined here to the formal groups of berezin and kac and the graded lie groups of kostant is analyzed

baker campbell hausdorff relations for the orthosymplectic group osp 1 2 baker campbell hausdorff relations are presented for the connected supergroup associated with the orthosymplectic algebra osp 1 2

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

hopf bifurcation and stability of periodic solutions for van der pol equation with time delay abstract in this paper the van der pol equation with a time delay is considered where the time delay is regarded as a parameter it is found that hopf bifurcation occurs when this delay passes through a sequence of critical value a formula for determining the direction of the hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem

on the theory of one parameter subgroups of supergroups the one parameter subgroups of supergroups are defined and their primary properties are delineated parametrization schemes analogous to those used in lie group theory are investigated in particular exponentiation is introduced and related to the one parameter subgroups canonical coordinates of three kinds are investigated the third of which has no analog in conventional lie theory we illustrate our results with the examples of the supersymmetric quantum mechanical superalgebra sqm 2 and the simple superalgebra osp 1 2

the haar integral for lie supergroups each supermanifold can be considered as equivalent to a certain family of real manifolds the haar integral of a lie supergroup is defined using this equivalence a useful general formula is derived together with explicit construction methods of the simple lie supergroups of types spl n m el osp n 2r el and b n el and the extended poincare group

robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances in recent years the global stability of recurrent neural networks rnns has been investigated extensively it is well known that time delays and external disturbances can derail the stability of rnns in this paper we analyze the robustness of global stability of rnns subject to time delays and random disturbances given a globally exponentially stable neural network the problem to be addressed here is how much time delay and noise the rnn can withstand to be globally exponentially stable in the presence of delay and noise the upper bounds of the time delay and noise intensity are characterized by using transcendental equations for the rnns to sustain global exponential stability moreover we prove theoretically that for any globally exponentially stable rnns if additive noises and time delays are smaller than the derived lower bounds arrived at here then the perturbed rnns are guaranteed to also be globally exponentially stable three numerical examples are provided to substantiate the theoretical results

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

periodic solutions for a two neuron network with delays in this paper we use the coincidence degree theory to establish new results on the existence of w periodic solutions for a two neuron network model

central difference scheme and chaos abstract the existence of chaotic orbits is proved for area preserving diffeomorphisms of the plane defined by polynomials of degree two derived from the discretized logistic equation

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

stability and bifurcation analysis on a discrete time neural network using techniques developed by kuznetsov to discrete time systems we study the stability of the equilibrium 0 0 and neimark sacker bifurcation also called hopf bifurcation for map of a discrete time neural network system the obtained results are less restrictive and improve upon the existing ones on neimark sacker bifurcation of discrete time neural network with special classes of transfer functions the theoretical analyses are verified by numerical simulations our results have potential applications in neural networks

stability switches and hopf bifurcations of an isolated population model with delay dependent parameters the dynamical behaviors of an isolated population model involving delay dependent parameters are investigated it is shown that the positive equilibrium switches from being stable to unstable and then back to stable as the delay increases and the hopf bifurcation occurs finite times between the two critical values of stability changes which can be analytically determined moreover the bifurcating periodic solutions are expressed analytically in an approximate form by the perturbation approach and floquet technique the direction and stability of the bifurcating periodic solutions are also determined finally the validity of the results is shown by the consistency with the numerical simulations

computation of synchronized periodic solution in a bam network with two delays a bidirectional associative memory bam neural network with four neurons and two discrete delays is considered to represent an analytical method namely perturbation incremental scheme pis the expressions for the periodic solutions derived from hopf bifurcation are given by using the pis the result shows that the pis has higher accuracy than the center manifold reduction cmr with normal form for the values of time delay not far away from the hopf bifurcation point in terms of the pis the necessary and sufficient conditions of synchronized periodic solution arising from a hopf bifurcation are obtained and the synchronized periodic solution is expressed in an analytical form it can be seen that theoretical analysis is in good agreement with numerical simulation it implies that the provided method is valid and the obtained result is correct to the best of our knowledge the paper is the first one to introduce the pis to study the periodic solution derived from hopf bifurcation for a 4 d delayed system quantitatively

supergroups have no extra topology the global topology of supergroups is studied and it is found that they are equivalent as far as homotopy is concerned to the product of the underlying ordinary lie groups

enlargeable graded lie algebras of supersymmetry a criterion to enlarge infinite dimensional lie algebras to analytic lie groups is used here for the extension of graded lie algebras of physical interest to super lie groups