Bifurcation analysis in a recurrent neural network model with delays

In this paper, we study the dynamical behaviors of a three-node recurrent neural network model with four discrete time delays. We study several types of bifurcation, and use the method of multiple time scales to derive the normal forms associated with Hopf-zero bifurcation, non-resonant and resonant double Hopf bifurcations. Moreover, bifurcations are classified in two-dimensional parameter space near these critical points, and numerical simulations are presented to demonstrate the applicability of the theoretical results.     

Bifurcation analysis for a discrete-time Hopfield neural network of two neurons with two delays and self-connections

In this paper, a bifurcation analysis is undertaken for a discrete-time Hopfield neural network of two neurons with two different delays and self-connections. Conditions ensuring the asymptotic stability of the null solution are found, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, Fold or Neimark-Sacker bifurcations occur, but Flip and codimension 2 (Fold–Neimark-Sacker, double Neimark-Sacker, resonance 1:1 and Flip–Neimark-Sacker) bifurcations may also be present. The direction and the stability of the Neimark-Sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory.     

Bifurcation of a three-unit neural network

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

Periodic oscillation for a three-neuron network with delays

By means of the coincidence degree theory, we give some criteria for the existence of periodic solutions of a three-neuron network model.     

Bifurcation of periodic solution in a three-unit neural network with delay

1. IntroductionDynamical characteristics of neural networks have become recelltly a subject of intenseresearch activity. J. B6lair and S. Dufou.[1] investigated a system of neural networks introduced by Hopfield[2]. Especially' they studied the three-uult network system with noself connectiondxi(t) 3dt = --xi(t) Z Ti,f; (x;(t -- T)), i = 1, 2, 3, (l)J = 1where f;(0) = 0, j = 1, 2, 3 and T, = 0, i = 1, 2, 3, give the stability properties of the nullsolution. [2] discussed the lineaJr stab…     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Bifurcation analysis of a chemostat model with two distributed delays

Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-type model with two distributed delays. For this model, He et al. [SIAM J. Math. Anal. 29 (1998) 681] showed that the positive equilibrium can be globally asymptotically stable if the mean delays are sufficiently small. In this paper, using the average time delay as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations. Computer simulations illustrate the result. The mistakes in [Chaos, Solitons & Fractals 17 (2003) 879] are pointed out and corrected.     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays

In this paper, we consider a three-dimensional delayed differential equation representing a bidirectional associate memory (BAM) neural network with three neurons and two discrete delays. By analyzing the number and stability of equilibria, the pitchfork bifurcation curve of the system is obtained. Furthermore, on the pitchfork bifurcation curve, by using the sum of two delays as the bifurcation parameter, we find that the system can undergo a Hopf bifurcation at the origin and the three-dimensional ordinary differential equation describing the flow on the center manifold is given.     

Global Hopf bifurcation analysis on a BAM neural network with delays

A BAM neural network with three neurons is considered. Sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large. Numerical simulations are presented to support the theoretical results found.     

Codimension two bifurcation in a delayed neural network with unidirectional coupling

In this paper, a delayed neural network model with unidirectional coupling is considered. Zero–Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm form at the zero–Hopf singularity and show that the model can exhibit pitchfork, Hopf bifurcation, and double Hopf bifurcation is also found to occur in this model. Some numerical simulations are given to support the analytic results.     

A discrete dynamical system as a model of a neural network with generalized Hebbian synapses

We study the dynamical behavior of a discrete time dynamical system which can serve as a model of a learning process. We determine fixed points of this system and basins of attraction of attracting points. This system was studied by Fernanda Botelho and James J. Jamison in [A learning rule with generalized Hebbian synapses, J. Math. Anal. Appl. 273 (2002) 529-547] but authors used its continuous counterpart to describe basins of attraction.     

Chaos and asymptotical stability in discrete-time recurrent neural networks with generalized input-output function

We theoretically investigate the asymptotical stability, local bifurcations and chaos of discrete-time recurrent neural networks with the form of

Bifurcation analysis of an oscillatory CNN model with two cells

The aim of this paper is to carry out the full bifurcation analysis of the two-parameter two-dimensional oscillatory cellular neural network (CNN) model (3)–(4) in Chap. 8 of the recent monograph of Chua and Roska (Cellular Neural Networks and Visual Computing, Cambridge University Press, [2002]). The main tool is an averaged divergence inequality implying that—regardless the dimension of the phase space—compact invariant sets are of zero Lebesgue measure.     

Statistical analysis of brain waves from randomly interconnected neural nets

A statistical analysis of brain waves from randomly connected neural nets is presented here based to our previous work. The most important findings from this analysis is that several neural networks have the ability to create periods of spontaneous rhythmic activity similar to those observed in animals and human thalamus, cerebral cortex and hippocampus.     

Towards a unified recurrent neural network theory: The uniformly pseudo-projection-anti-monotone net

In the past decades, various neural network models have been developed for modeling the behavior of human brain or performing problem-solving through simulating the behavior of human brain. The recurrent neural networks are the type of neural networks to model or simulate associative memory behavior of human being. A recurrent neural network (RNN) can be generally formalized as a dynamic system associated with two fundamental operators: one is the nonlinear activation operator deduced from the input-output properties of the involved neurons, and the other is the synaptic connections (a matrix) among the neurons. Through carefully examining properties of various activation functions used, we introduce a novel type of monotone operators, the uniformly pseudo-projectionanti-monotone (UPPAM) operators, to unify the various RNN models appeared in the literature. We develop a unified encoding and stability theory for the UPPAM network model when the time is discrete. The established model and theory not only unify but also jointly generalize the most known results of RNNs. The approach has lunched a visible step towards establishment of a unified mathematical theory of recurrent neural networks.     

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.     

Global exponential stability of high order recurrent neural network with time-varying delays

In this paper, global exponential stability of high order recurrent neural network with time-varying delay and bounded activation functions is investigated. Some improved conditions are obtained involving external input, connection weights, and time delays of recurrent neural network. Moreover, the location of the equilibrium point can be estimated. In addition, two examples are demonstrated to illustrate the effectiveness of the proposed criteria in comparison with some existing results.