dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions we establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions also called linear threshold or semilinear transfer functions as hahnloser sarpeshkar mahowald douglas and seung 2000 showed networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit to obtain this behavior the network must be multistable and nondivergent and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification the first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron the second condition is restricted to symmetric networks but can also take into account the stabilizing effect of nonlocal inhibitory interactions we demonstrate the application of the conditions on a simple example and the orientation selectivity model of ben yishai lev bar or and sompolinsky 1995 we show that the conditions can be used to identify in their model regions of maximal orientation selective amplification and symmetry breaking

permitted and forbidden sets in discrete time linear threshold recurrent neural networks the concepts of permitted and forbidden sets enable a new perspective of the memory in neural networks such concepts exhibit interesting dynamics in recurrent neural networks this paper studies the basic theories of permitted and forbidden sets of the linear threshold discrete time recurrent neural networks the linear threshold transfer function has been regarded as an adequate transfer function for recurrent neural networks networks with this transfer function form a class of hybrid analog and digital networks which are especially useful for perceptual computations networks in discrete time can directly provide algorithms for efficient implementation in digital hardware the main contribution of this paper is to establish foundations of permitted and forbidden sets necessary and sufficient conditions for the linear threshold discrete time recurrent neural networks are obtained for complete convergence existence of permitted and forbidden sets as well as conditionally multiattractivity respectively simulation studies explore some possible interesting practical applications

multistability of second order competitive neural networks with nondecreasing saturated activation functions in this paper second order interactions are introduced into competitive neural networks nns and the multistability is discussed for second order competitive nns socnns with nondecreasing saturated activation functions firstly based on decomposition of state space cauchy convergence principle and inequality technique some sufficient conditions ensuring the local exponential stability of 2n equilibrium points are derived secondly some conditions are obtained for ascertaining equilibrium points to be locally exponentially stable and to be located in any designated region thirdly the theory is extended to more general saturated activation functions with 2r corner points and a sufficient criterion is given under which the socnns can have r 1 n locally exponentially stable equilibrium points even if there is no second order interactions the obtained results are less restrictive than those in some recent works finally three examples with their simulations are presented to verify the theoretical analysis

global exponential stability for complex valued recurrent neural networks with asynchronous time delays in this paper we investigate the global exponential stability for complex valued recurrent neural networks with asynchronous time delays by decomposing complex valued networks to real and imaginary parts and construct an equivalent real valued system the network model is described by a continuous time equation there are two main differences of this paper with previous works 1 time delays can be asynchronous i e delays between different nodes are different which make our model more general and 2 we prove the exponential convergence directly while the existence and uniqueness of the equilibrium point is just a direct consequence of the exponential convergence using three generalized norms we present some sufficient conditions for the uniqueness and global exponential stability of the equilibrium point for delayed complex valued neural networks these conditions in our results are less restrictive because of our consideration of the excitatory and inhibitory effects between neurons so previous works of other researchers can be extended finally some numerical simulations are given to demonstrate the correctness of our obtained results

growth and poles of meromorphic solutions of some difference equations abstract this paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth s r f we find the lower bound of the lower order or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations

on the schrodinger equation for the gaussian potential a exp λr2 the energy eigenvalues of the attractive radial gaussian potential for various eigenstates are very accurately determined within the framework of the hypervirial pade scheme

multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions in this paper we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions it is proved that under some conditions such n neuron neural networks can have at least 5 n equilibrium points 3 n of which are locally stable and the others are unstable based on the contraction mapping theorem and the theory of strict diagonal dominance matrix the investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous mexican hat type activation function or discontinuous two level activation functions an illustrative example with computer simulations is presented to verify the theoretical analysis

spherical anharmonic oscillator in self similar approximation the method of self similar approximations is applied for calculating the eigenvalues of the three dimensional spherical anharmonic oscillator the advantage of this method is in its simplicity and high accuracy for the case considered the authors show that based only on two terms of perturbation theory they find the spectrum with an error not worse than of the order 10 3 for the whole range of anharmonicity parameters from zero up to infinity and for any energy levels the comparison with other known analytical methods proves that their method is more simple and accurate

recurrent neural network based multivariable adaptive control for a class of nonlinear dynamic systems with time varying delay at the beginning an approximate nonlinear autoregressive moving average narma model is employed to represent a class of multivariable nonlinear dynamic systems with time varying delay it is known that the disadvantages of robust control for the narma model are as follows 1 suitable control parameters for larger time delay are more sensitive to achieving desirable performance 2 it only deals with bounded uncertainty and 3 the nominal narma model must be learned in advance due to the dynamic feature of the narma model a recurrent neural network rnn is online applied to learn it however the system performance becomes deteriorated due to the poor learning of the larger variation of system vector functions in this situation a simple network is employed to compensate the upper bound of the residue caused by the linear parameterization of the approximation error of rnn an e modification learning law with a projection for weight matrix is applied to guarantee its boundedness without persistent excitation under suitable conditions the semiglobally ultimately bounded tracking with the boundedness of estimated weight matrix is obtained by the proposed rnn based multivariable adaptive control finally simulations are presented to verify the effectiveness and robustness of the proposed control

stability analysis of continuous time and discrete time quaternion valued neural networks with linear threshold neurons this paper addresses the problem of stability for continuous time and discrete time quaternion valued neural networks qvnns with linear threshold neurons applying the semidiscretization technique to the continuous time qvnns the discrete time analogs are obtained which preserve the dynamical characteristics of their continuous time counterparts via the plural decomposition method of quaternion homeomorphic mapping theorem as well as lyapunov theorem some sufficient conditions on the existence uniqueness and global asymptotical stability of the equilibrium point are derived for the continuous time qvnns and their discrete time analogs respectively furthermore a uniform sufficient condition on the existence uniqueness and global asymptotical stability of the equilibrium point is obtained for both continuous time qvnns and their discrete time version finally two numerical examples are provided to substantiate the effectiveness of the proposed results

multistability of neural networks with time varying delays and concave convex characteristics in this paper stability of multiple equilibria of neural networks with time varying delays and concave convex characteristics is formulated and studied some sufficient conditions are obtained to ensure that an n neuron neural network with concave convex characteristics can have a fixed point located in the appointed region by means of an appropriate partition of the n dimensional state space when nonlinear activation functions of an n neuron neural network are concave or convex in 2k 2m 1 intervals this neural network can have 2k 2m 1 n equilibrium points this result can be applied to the multiobjective optimal control and associative memory in particular several succinct criteria are given to ascertain multistability of cellular neural networks these stability conditions are the improvement and extension of the existing stability results in the literature a numerical example is given to illustrate the theoretical findings via computer simulations

multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions abstract in this paper the multistability and multiperiodicity issues are discussed for 2 n dimensional delayed bidirectional associative memory neural networks with r level discontinuous activation functions sufficient conditions are established to ensure the existence of r n locally exponentially stable equilibria as an extension of multistability conditions are established to ensure the existence of r n locally exponentially stable periodic solutions when external inputs are periodic finally two examples are given to illustrate the effectiveness of the obtained results

the hvt technique and the treatment of two basic inequalities the quantum mechanical hypervirial theorems hvt technique is used in treating two basic inequalities relating the ground state mean square radius of the orbit of a particle in a central potential and its kinetic energy respectively to the spacing of the two lowest energy levels e for quite a wide class of those potentials the parameters of which also lead to a sufficiently small dimensionless quantity s the difference between the two sides of each inequality is of order s3 and higher while e is of order s and higher and thus it is expected in general to be quite small

the perturbed three dimensional oscillator a renormalized version of inner product and hypervirial theories is used to calculate the energy eigenvalues for four states of a perturbed three dimensional oscillator and to obtain the rayleigh schrodinger energy perturbation series

stability of complex valued impulsive system with delay abstract since many systems such as quantum system complex valued neural network and lorenz system are complex valued differential system in this paper the stability of complex valued impulsive system with delay is addressed by taking advantage of lyapunov function in complex field some new criteria for global exponential stability are established which not only generalize some known results in literature but also greatly reduce the complexity of analysis and computation as an application a new impulsive controller for complex valued lu system with delay is designed

a note on the schrodinger equation for the potential a exp x2 l l 1 x2 the bound states energies and eigenfunctions of the schrodinger equation with a radial gaussian potential are obtained using a perturbational and also a variational treatment on a conveniently chosen basis of transformed jacobi functions the accuracy of the results is fairly good

estimate of exponential convergence rate and exponential stability for neural networks estimates of exponential convergence rate and exponential stability are studied for a class of neural networks which includes hopfield neural networks and cellular neural networks both local and global exponential convergence are discussed theorems for estimation of the exponential convergence rate are established and the bounds on the rate of convergence are given the domains of attraction in the case of local exponential convergence are obtained simple conditions are presented for checking exponential stability of the neural networks

the jacobi eigenfunctions and the quantum mechanical hypervirial theorems method for bound state problems the jacobi eigenfunctions are suitably adjusted by means of the hypervirial theorems hvt scheme in order to obtain in many cases a very satisfactory analytic approximation of the s state eigenfunctions of the radial gaussian potential the results obtained show that the hvt scheme is a powerful method for obtaining analytically not only approximate eigenvalues but also the corresponding eigenfunctions

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

monostability and multistability for almost periodic solutions of fractional order neural networks with unsaturating piecewise linear activation functions since the unsaturating activation function is unbounded more complex dynamics may exist in neural networks with this kind of activation function in this article monostability and multistability results of almost periodic solutions are developed for fractional order neural networks with unsaturating piecewise linear activation functions some globally mittag leffler attractive sets are given and the existence of globally mittag leffler stable almost periodic solution is demonstrated by using ascoli arzela theorem in particular some sufficient conditions are provided to ascertain the multistability of almost periodic solutions based on locally positively invariant set it shows that there exists an almost periodic solution in each positively invariant set and all trajectories converge to this periodic trajectory in that rectangular area two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria

the energy levels of the schrodinger equation for various types of potentials using a renormalized method a perturbative method is used to compute energy levels for many different forms of potentials results of high accuracy are produced for various eigenstates

global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays by employing the lyapunov krasovskii functional and linear matrix inequality lmi approach the problem of global asymptotical stability is studied for recurrent neural networks with both discrete time varying delays and distributed time varying delays some sufficient conditions are given for checking the global asymptotical stability of recurrent neural networks with mixed time varying delay the proposed lmi result is computationally efficient as it can be solved numerically using standard commercial software two examples are given to show the usefulness of the results

local stability analysis of discrete time continuous state complex valued recurrent neural networks with inner state feedback recurrent neural networks rnns are well known for their capability to minimize suitable cost functions without the need for a training phase this is possible because they can be lyapunov stable although the global stability analysis has attracted a lot of interest local stability is desirable for specific applications in this brief we investigate the local asymptotical stability of two classes of discrete time continuous state complex valued rnns with parallel update and inner state feedback we show that many already known results are special cases of the results obtained here we also generalize some known results from the real valued case to the complex valued one finally we investigate the stability in the presence of time variant activation functions complex valued activation functions in this brief are separable with respect to the real and imaginary parts

nevanlinna theory for the q difference operator and meromorphic solutions of q difference equations it is shown that if f is a meromorphic function of order zero and q in mathbb c then begin equation label abstid m bigg r frac f qz f z bigg o t r f tag ddag end equation for all r on a set of logarithmic density 1 the remainder of the paper consists of applications of identity eqref abstid to the study of value distribution of zero order meromorphic functions and in particular zero order meromorphic solutions of q difference equations the results obtained include q shift analogues of the second main theorem of nevanlinna theory picard s theorem and clunie and mohon ko lemmas

multistability and new attraction basins of almost periodic solutions of delayed neural networks in this paper we investigate multistability of almost periodic solutions of recurrently connected neural networks with delays simply called delayed neural networks we will reveal that under some conditions the space rn can be divided into 2n subsets and in each subset the delayed n neuron neural network has a locally stable almost periodic solution furthermore we also investigate the attraction basins of these almost periodic solutions we reveal that the attraction basin of almost periodic trajectory is larger than the subset where the corresponding almost periodic trajectory is located in addition several numerical simulations are presented to corroborate the theoretical results

boundedness and complete stability of complex valued neural networks with time delay in this paper the boundedness and complete stability of complex valued neural networks cvnns with time delay are studied some conditions to guarantee the boundedness of the cvnns are derived using local inhibition moreover under the boundedness conditions a compact set that globally attracts all the trajectories of the network is also given additionally several conditions in terms of real valued linear matrix inequalities lmis for complete stability of the cvnns are established via the energy minimization method and the approach that converts the complex valued lmis to real valued ones examples with simulation results are given to show the effectiveness of the theoretical analysis

generalized halanay inequalities and their applications to neural networks with unbounded time varying delays in this brief we discuss some variants of generalized halanay inequalities that are useful in the discussion of dissipativity and stability of delayed neural networks integro differential systems and volterra functional differential equations we provide some generalizations of the halanay inequality which is more accurate than the existing results as applications we discuss invariant set dissipative synchronization and global asymptotic stability for the hopfield neural networks with infinite delays we also prove that the dynamical systems with unbounded time varying delays are globally asymptotically stable

multistability of discrete time recurrent neural networks with unsaturating piecewise linear activation functions this paper studies the multistability of a class of discrete time recurrent neural networks with unsaturating piecewise linear activation functions it addresses the nondivergence global attractivity and complete stability of the networks using the local inhibition conditions for nondivergence are derived which not only guarantee nondivergence but also allow for the existence of multiequilibrium points under these nondivergence conditions global attractive compact sets are obtained complete stability is studied via constructing novel energy functions and using the well known cauchy convergence principle examples and simulation results are used to illustrate the theory

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

convergence analysis of a discrete time recurrent neural network to perform quadratic real optimization with bound constraints presents a model of a discrete time recurrent neural network designed to perform quadratic real optimization with bound constraints the network iteratively improves the estimate of the solution always maintaining it inside of the feasible region several neuron updating rules which assure global convergence of the net to the desired minimum have been obtained some of them also assure exponential convergence and maximize a lower bound for the convergence degree simulation results are presented to show the net performance

dynamic stability conditions for lotka volterra recurrent neural networks with delays the lotka volterra model of neural networks derived from the membrane dynamics of competing neurons have found successful applications in many winner take all types of problems this paper studies the dynamic stability properties of general lotka volterra recurrent neural networks with delays conditions for nondivergence of the neural networks are derived these conditions are based on local inhibition of networks thereby allowing these networks to possess a multistability property multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process under these nondivergence conditions a compact set that globally attracts all the trajectories of a network can be computed explicitly if the connection weight matrix of a network is symmetric in some sense and the delays of the network are in l2 space we can prove that the network will have the property of complete stability

attractivity analysis of memristor based cellular neural networks with time varying delays this paper presents new theoretical results on the invariance and attractivity of memristor based cellular neural networks mcnns with time varying delays first sufficient conditions to assure the boundedness and global attractivity of the networks are derived using state space decomposition and some analytic techniques it is shown that the number of equilibria located in the saturation regions of the piecewise linear activation functions of an n neuron mcnn with time varying delays increases significantly from 2n to 22n2 n 22n2 times compared with that without a memristor in addition sufficient conditions for the invariance and local or global attractivity of equilibria or attractive sets in any designated region are derived finally two illustrative examples are given to elaborate the characteristics of the results in detail

convergence analysis of a deterministic discrete time system of oja s pca learning algorithm the convergence of oja s principal component analysis pca learning algorithms is a difficult topic for direct study and analysis traditionally the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time dct systems such a method will require the learning rate to converge to zero which is not a reasonable requirement to impose in many practical applications recently deterministic discrete time ddt systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms unlike dct systems ddt systems allow learning rates to be constant which can be a nonzero this paper will provide some important results relating to the convergence of a ddt system of oja s pca learning algorithm it has the following contributions 1 a number of invariant sets are obtained based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever thus the nondivergence of the trajectories is guaranteed 2 the convergence of the ddt system is analyzed rigorously it is proven in the paper that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix in addition exponential convergence rate are obtained providing useful guidelines for the selection of fast convergence learning rate 3 since the trajectories may diverge the careful choice of initial vectors is an important issue this paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence 4 simulation results will be furnished to illustrate the theoretical results achieved

1 n expansion for gaussian potential the method of large n expansion has been applied to an attractive radial gaussian potential to obtain its bound state energy levels it has been shown that the shifted 1 n expansion technique provides a better approximation than the method of unshifted 1 n expansion and yields energy values which are in good agreement with the accurate numerical and analytic results

absolute stability conditions for discrete time recurrent neural networks an analysis of the absolute stability for a general class of discrete time recurrent neural networks rnn s is presented a discrete time model of rnn s is represented by a set of nonlinear difference equations some sufficient conditions for the absolute stability are derived using ostrowski s theorem and the similarity transformation approach for a given rnn model these conditions are determined by the synaptic weight matrix of the network the results reported in this paper need fewer constraints on the weight matrix and the model than in previously published studies

renormalization group solutions for yukawa potential the self similar renormalization group is used to obtain expressions for the spectrum of the hamiltonian with the yukawa potential the critical screening parameter above which there are no bound states is also obtained by this method the approach presented illustrates that one can achieve good accuracy without involving extensive numerical calculations but invoking instead the renormalization group techniques

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

analysis of complete stability for discrete time cellular neural networks with piecewise linear output functions this letter discusses the complete stability of discrete time cellular neural networks with piecewise linear output functions under the assumption of certain symmetry on the feedback matrix a sufficient condition of complete stability is derived by finite trajectory length because the symmetric conditions are not robust the complete stability of networks may be lost under sufficiently small perturbations the robust conditions of complete stability are also given for discrete time cellular neural networks with multiple equilibrium points and a unique equilibrium point these complete stability results are robust and available

the hvt technique and the uncertainty relation for central potentials the quantum mechanical hypervirial theorems hvt technique is used to treat the so called uncertainty relation for quite a general class of central potential wells including the reduced poeschl teller and the gaussian one it is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for

multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions multistability is a property necessary in neural networks in order to enable certain applications e g decision making where monostable networks can be computationally restrictive this article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions it deals fully with the three basic properties of a multistable network boundedness global attractivity and complete convergence this article makes the following contributions conditions based on local inhibition are derived that guarantee boundedness of some multistable networks conditions are established for global attractivity bounds on global attractive sets are obtained complete convergence conditions for the network are developed using novel energy like functions and simulation examples are employed to illustrate the theory thus developed

foundations of implementing the competitive layer model by lotka volterra recurrent neural networks the competitive layer model clm can be described by an optimization problem the problem can be further formulated by an energy function called the clm energy function in the subspace of nonnegative orthant the set of minimum points of the clm energy function forms the set of solutions of the clm problem solving the clm problem means to find out such solutions recurrent neural networks rnns can be used to implement the clm to solve the clm problem the key point is to make the set of minimum points of the clm energy function just correspond to the set of stable attractors of the recurrent neural networks this paper proposes to use lotka volterra rnns lv rnns to implement the clm the contribution of this paper is to establish foundations of implementing the clm by lv rnns the contribution mainly contains three parts the first part is on the clm energy function necessary and sufficient conditions for minimum points of the clm energy function are established by detailed study the second part is on the convergence of the proposed model of the lv rnns it is proven that interesting trajectories are convergent the third part is the most important it proves that the set of stable attractors of the proposed lv rnn just equals the set of minimum points of the clm energy function in the nonnegative orthant thus the lv rnns can be used to solve the problem of the clm it is believed that by establishing such basic rigorous theories more and interesting applications of the clm can be found

on the nevanlinna characteristic of f qz and its applications in this paper we investigate the relation of the nevanlinna characteristic functions t r f qz and t r f z for a zero order meromorphic function f and a non zero constant q it is shown that t r f qz 1 o 1 t r f z for all r on a set of lower logarithmic density 1 this estimate is sharp in the sense that for any q c such that q 1 and 0 there exists a meromorphic function h of order such that t r h qz q o 1 t r h z as r outside of an exceptional set of finite linear measure as applications we give some results on zero order meromorphic solutions of q difference equations and on value distribution and uniqueness of certain types of q difference polynomials