stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear schrodinger equations abstract linear stability of both sign definite positive and sign indefinite solitary waves near pitchfork bifurcations is analyzed for the generalized nonlinear schrodinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions bifurcations of linear stability eigenvalues associated with pitchfork bifurcations are analytically calculated based on these eigenvalue bifurcation formulae linear stability of solitary waves near pitchfork bifurcations is then determined it is shown that the base solution branch switches stability at the bifurcation point in addition the two bifurcated solution branches and the base branch have the opposite same stability when their power slopes have the same opposite sign furthermore the stability of these solution branches can be determined almost exclusively from their power diagram especially for positive solitary waves these stability results are also compared with the hamiltonian krein index theory and they are shown to be consistent with each other lastly various numerical examples are presented and the numerical results confirm the analytical predictions both qualitatively and quantitatively

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

large d expansion from variational perturbation theory we derive recursively the perturbation series for the ground state energy of the d dimensional anharmonic oscillator and resum it using variational perturbation theory vpt from the exponentially fast converging approximants we extract the coefficients of the large d expansion to higher orders the calculation effort is much smaller than in the standard field theoretic approach based on the hubbard stratonovich transformation

high order variational calculation for the frequency of time periodic solutions we develop a convergent variational perturbation theory for the frequency of time periodic solutions of nonlinear dynamical systems the power of the theory is illustrated by applying it to the duffing oscillator

newton conjugate gradient methods for solitary wave computations in this paper the newton conjugate gradient methods are developed for solitary wave computations these methods are based on newton iterations coupled with conjugate gradient iterations to solve the resulting linear newton correction equation when the linearization operator is self adjoint the preconditioned conjugate gradient method is proposed to solve this linear equation if the linearization operator is non self adjoint the preconditioned biconjugate gradient method is proposed to solve the linear equation the resulting methods are applied to compute both the ground states and excited states in a large number of physical systems such as the two dimensional nls equations with and without periodic potentials the fifth order kdv equation and the fifth order kp equation numerical results show that these proposed methods are faster than the other leading numerical methods often by orders of magnitude in addition these methods are very robust and always converge in all the examples being tested furthermore they are very easy to implement it is also shown that the nonlinear conjugate gradient methods are not robust and inferior to the proposed methods

two neuron dynamics and adiabatic elimination abstract we consider an effective neuron network obtained from the hopfield model by adiabatic elimination the validity of the slaving procedure is first numerically examined the self interaction of the effective neuron network may lead to non zero attractors far from the origin ui 0 the qualitative structure of the attractors for this two effective neuron model with j12j21 0 or

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 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

variational perturbation theory for markov processes we develop a convergent variational perturbation theory for conditional probability densities of markov processes the power of the theory is illustrated by applying it to the diffusion of a particle in an anharmonic potential

recursive calculation of effective potential and variational resummation we set up a method for a recursive calculation of the effective potential which is applied to a cubic potential with imaginary coupling the result is resummed using variational perturbation theory yielding an exponentially fast convergence

gap solitons and their linear stability in one dimensional periodic media an analytical theory utilizing exponential asymptotics is presented for one dimensional gap solitons that bifurcate from edges of bloch bands in the presence of a general periodic potential it is shown that two soliton families bifurcate out from every bloch band edge under self focusing or self defocusing nonlinearity and an asymptotic expression for the eigenvalues associated with the linear stability of these solitons is derived the locations of these solitons relative to the underlying potential are determined from a certain recurrence relation that contains information beyond all orders of the usual perturbation expansion in powers of the soliton amplitude moreover this same recurrence relation decides which of the two soliton families is unstable the analytical predictions for the stability eigenvalues are in excellent agreement with numerical results

hopf bifurcation of an n 1 neuron bidirectional associative memory neural network model with delays recent studies on hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two three four five or six neurons it is well known that neural networks are complex and large scale nonlinear dynamical systems so the dynamics of the delayed neural networks are very rich and complicated although discussing the dynamics of networks with a few neurons may help us to understand large scale networks there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large scale networks in this paper a general delayed bidirectional associative memory neural network model with n 1 neurons is considered by analyzing the associated characteristic equation the local stability of the trivial steady state is examined and then the existence of the hopf bifurcation at the trivial steady state is established by applying the normal form theory and the center manifold reduction explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution furthermore the paper highlights situations where the hopf bifurcations are particularly critical in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections it is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons numerical simulations are carried out to illustrate the main results

duration of transient oscillations in ring networks of unidirectionally coupled neurons abstract properties of the duration of long lasting transient oscillations in ring networks of unidirectionally coupled sigmoidal neurons are derived with a kinematical model of traveling waves in the network the duration of the transient oscillations occurring from random initial conditions increases exponentially as the number of neurons the distribution of the duration is approximated by a power law function when the number of neurons is large further transient oscillations which oscillate about one thousand cycles before ceasing are observed in a network of forty neurons in circuit experiments though the duration decreases owing to random biases

a study of network dynamics the dynamics of a discrete time neural network model are investigated first a numerical survey of network power spectra is reported for networks of varying size with random weight matrices and initial states the steepness of the logistic function and a symmetry measure of the weight matrix are taken as control parameters summary statistics are presented to give gross measures of the model s temporal activity in parameter space second a detailed study of the dynamics of a particular network is described complex dynamical behavior is observed including hopf bifurcations the ruelle takens newhouse route to chaos showing mode locking at rational winding numbers and the destruction of an invariant torus and the period doubling route to chaos

the monotone iterative method and zeros of bessel functions for nonlinear singular derivative dependent bvp in the presence of upper and lower solutions abstract in this paper we consider a class of nonlinear singular boundary value problems x y x x f x y x x y x 0 0 x 1 y 0 y 1 0 for 1 we assume that the source function f x y x y is lipschitz in x y and one sided lipschitz in y the initial approximations are an upper solution u 0 x and a lower solution v 0 x which can be ordered in one way v 0 x u 0 x or the other u 0 x v 0 x we propose an iterative scheme and establish the existence of solutions bounded by v 0 and u 0 and allow f y to take both positive and negative values the method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems

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

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

bifurcation and stabilization of oscillations in ring neural networks with inertia effects of inertia on oscillations in ring networks of unidirectionally coupled sigmoidal neurons are studied it is known that ring neural networks without inertia are bistable and the duration of transient oscillations increases exponentially with the number of neurons in this paper a kinematical description of traveling waves in the oscillations in the networks is extended to networks with inertia when the inertia is below a critical value and the state of each neuron is over damped properties of the networks are qualitatively the same as those without inertia the duration of the transient oscillations then increases with inertia and the increasing rate of the logarithm of the duration becomes more than double when the inertia exceeds a critical value and the state of each neuron becomes under damped properties of the networks qualitatively change the periodic solution is then stabilized through the pitchfork bifurcation as inertia increases more bifurcations occur so that various periodic solutions are generated and the stability of the periodic solutions changes alternately further stable oscillations generated with inertia are observed in an experiment on an analog circuit

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

delay induced instabilities in nonlinear feedback systems extending general methods developed in synergetics to construct order parameter equations we work out a systematic elimination procedure for the stable modes in nonlinear delay systems it will be shown that in the vicinity of an instability the dynamical behavior of the infinite dimensional delay system is approximately governed by a low dimensional set of order parameter equations which turn out to be of the form of ordinary differential equations i e they no longer contain memory terms the general formalism will then be compared with experimental data obtained from the nonlinear operation of a phase locked loop where the finite propagation time of the signal in the feedback loop is taken into account

stability and chaos in an inertial two neuron system abstract inertia is added to a continuous time hopfield 1984 effective neuron system we explore the effects on the stability of the fixed points of the system a two neuron system with one or two inertial terms added is shown to exhibit chaos the chaos is confirmed by lyapunov exponents power spectra and phase space plots

global robust stability of neural networks with time varying delays in this paper we study the global robust stability of neural networks with time varying delays using the lyapunov functional method and matrix inequality technique several sufficient conditions are presented to show the existence of equilibrium and global robust stability of neural networks which is easy to apply then we give a simulation to justify the obtained results

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

exponential transients in continuous time liapunov systems we consider the convergence behavior of a class of continuous time dynamical systems corresponding to so called symmetric hopfield nets studied in neural networks theory we prove that such systems may have transient times that are exponential in the system dimension i e number of neurons despite the fact that their dynamics are controlled by liapunov functions this result stands in contrast to many proposed uses of such systems in e g combinatorial optimization applications in which it is often implicitly assumed that their convergence is rapid an additional interesting observation is that our example of an exponential transient continuous time system a simulated binary counter in fact converges more slowly than any discrete time hopfield system of the same representation size this suggests that continuous time systems may be worth investigating for gains in descriptional efficiency as compared to their discrete time counterparts

families of solitons in bragg supergratings abstract we study fundamental optical gap solitons in the model of a fiber bragg grating bg which is subjected to a periodic modulation of the local reflectivity giving rise to a supergrating in addition the local refractive index is also periodically modulated with the same period it is known that the supergrating opens an infinite system of new bandgaps in the bg s spectrum we use a combination of analytical and computational methods to show that each emerging bandgap is filled with gap solitons gss including asymmetric ones and bound states of the gss in particular bifurcations of the gss created by the supergrating are studied in terms of a geometric analysis

vortex families near a spectral edge in the gross pitaevskii equation with a two dimensional periodic potential we examine numerically vortex families near band edges of the bloch wave spectrum for the gross pitaevskii equation with two dimensional periodic potentials and for the discrete nonlinear schrodinger equation we show that besides vortex families that terminate at a small distance from the band edges via fold bifurcations there exist vortex families that are continued all the way to the band edges

mean square exponential stability of stochastic inertial neural networks by introducing some parameters perturbed by white noises we propose a class of stochastic inertial neural networks in random environments to describe dynamics behaviours of this stochastic system

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

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

stochastic single neurons we study the stochastic behavior of a single self exciting model neuron with additive noise a system that has bistable stochastic dynamics we use langevin and fokker planck equations to obtain analytical expressions for the stationary distribution of activities and for the crossing rate between two stable states we adjust the parameters in these expressions to fit observed histograms of neural activity thus obtaining what we call an effective single neuron for a given network we construct an effective single neuron from an activity histogram of a representative hidden neuron in a recurrent learning network we also compare our result with an effective single neuron previously obtained analytically through the adiabatic elimination approximation

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

analytical and numerical investigations of the phase locked loop with time delay we derive the normal form for the delay induced hopf bifurcation in the first order phase locked loop with time delay by the multiple scaling method the resulting periodic orbit is confirmed by numerical simulations further detailed numerical investigations demonstrate exemplarily that this system reveals a rich dynamical behavior with phase portraits fourier analysis and lyapunov spectra it is possible to analyze the scaling properties of the control parameter in the period doubling scenario both qualitatively and quantitatively within the numerical accuracy there is evidence that the scaling constant of the time delayed phase locked loop coincides with the feigenbaum constant delta approximately 4 669 in one dimensional discrete systems

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

boundary layer theory strong coupling series and large order behavior the introduction of a lattice converts a singular boundary layer problem in the continuum into a regular perturbation problem however the continuum limit of the discrete problem is extremely nontrivial and is not completely understood this article examines two singular boundary layer problems taken from mathematical physics the instanton problem and the blasius equation and in each case examines two strategies pade resummation and variational perturbation theory to recover the solution to the continuum problem from the solution to the associated discrete problem both resummation procedures produce good and interesting results for the two cases but the results still deviate from the exact solutions to understand the discrepancy a comprehensive large order behavior analysis of the strong coupling lattice expansions for each of the two problems is done

bifurcation analysis on a turning system with large and state dependent time delay abstract stability and bifurcation analyses were performed in this study on the turning process with a state dependent and large time delay using the method of multiple scales mms the turning system tool was modeled as an oscillator with two degrees of freedom and both the cubic nonlinear stiffness and the nonlinear cutting force were considered the nonlinear cutting force was appropriately expanded in a taylor series considering the state dependency of the time delay the time delay and parameters were scaled through the proper ordering process to reflect the large delay effect on an asymptotic formulation of the mms asymptotic solutions were then obtained by the mms in the large delay regime and used to calculate the linear stability boundaries i e hopf bifurcation points and coexisting one period periodic solutions i e limit cycles of the turning system to investigate the local and global behaviors of the tool chatter bifurcation diagrams were obtained at various workpiece rotating speeds the validity of the results was examined by comparison with those obtained through the method of harmonic balance and direct numerical integration additionally using the bifurcation diagrams the effects of the state dependent time delay and nonlinear stiffness on the chatter vibration behaviors were examined

iterated defect correction for the solution of singular initial value problems we investigate the convergence properties of the iterated defect correction idec method based on the implicit euler rule for the solution of singular initial value problems with a singularity of the first kind we show that the method retains its classical order of convergence which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure

from non local gap solitary waves to bound states in periodic media solitary waves in one dimensional periodic media are discussed by employing the nonlinear schrodinger equation with a spatially periodic potential as a model this equation admits two families of gap solitons that bifurcate from the edges of bloch bands in the linear wave spectrum these fundamental solitons may be positioned only at specific locations relative to the potential otherwise they become non local owing to the presence of growing tails of exponentially small amplitude with respect to the wave peak amplitude here by matching the tails of such non local solitary waves high order locally confined gap solitons or bound states are constructed details are worked out for bound states comprising two non local solitary waves in the presence of a sinusoidal potential a countable set of bound state families characterized by the separation distance of the two solitary waves is found and each family features three distinct solution branches that bifurcate near bloch band edges at small but finite amplitude power curves associated with these solution branches are computed asymptotically for large solitary wave separation and the theoretical predictions are consistent with numerical results

stability analysis of discrete time recurrent neural networks with stochastic delay this paper is concerned with the stability analysis of discrete time recurrent neural networks rnns with time delays as random variables drawn from some probability distribution by introducing the variation probability of the time delay a common delayed discrete time rnn system is transformed into one with stochastic parameters improved conditions for the mean square stability of these systems are obtained by employing new lyapunov functions and novel techniques are used to achieve delay dependence the merit of the proposed conditions lies in its reduced conservatism which is made possible by considering not only the range of the time delays but also the variation probability distribution a numerical example is provided to show the advantages of the proposed conditions

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

high order variational perturbation theory for the free energy in this paper we introduce a generalization to the algebraic bender wu recursion relation for the eigenvalues and the eigenfunctions of the anharmonic oscillator we extend this well known formalism to the time dependent quantum statistical schrodinger equation thus obtaining the imaginary time evolution amplitude by solving a recursive set of ordinary differential equations this approach enables us to evaluate global and local quantum statistical quantities of the anharmonic oscillator to much higher orders than by evaluating feynman diagrams we probe our perturbative results by deriving a perturbative expression for the free energy which is then subject to variational perturbation theory as developed by kleinert yielding convergent results for the free energy for all values of the coupling strength