Quasi-periodic invariant 2-tori in a delayed BAM neural network

Abstract

In this paper, we consider a four-neuron bi-directional associative memory (BAM, for short) neural network with two delays. We choose connection weights and the sum of delays as bifurcation parameters and derive the critical values where a double Hopf bifurcation may occur by analyzing the associated characteristic equation which is a fourth-degree polynomial exponential equation. Meanwhile, we obtain some parameter conditions on the existence of invariant 2-tori of the truncated normal form near the bifurcation point by the center manifold theorem and normal form method. Despite the fact that the higher-degree terms may destroy the invariant 2-tori of the truncated normal form, we prove that the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where the truncated normal form possesses invariant 2-tori in a sufficiently small neighborhood of the bifurcation point. Numerical examples and simulations are given to support the theoretical analysis.

Introduction

Based on the Hopfied [7], [8] neural network model (a simplified ordinary differential equation), Marcus and Westervelt [14] argued that time delays always occur in the signal transmission and proposed a neural network model with delays. Afterward, various artificial models have been established to describe neural networks with delays, including the bi-directional associative memory (BAM, for short) neural network model which is a two-layer neural network. The BAM neural network as an extension of the unidirectional autoassociator of Hopfield neural network has been applied to signal processing, pattern recognition and automatic control, modeled by the following equations

$$\{\begin{array}{c}\dot{x_i}\left(t\right)=-{\mu }_i{x}_i\left(t\right)+\sum _{j=1}^m{c}_{ji}{f}_i\left(y_j(t-{\tau }_{ji})\right)+I_i,\hfill \\ \\ \dot{y_j}\left(t\right)=-{\nu }_j{y}_j\left(t\right)+\sum _{i=1}^n{d}_{ij}{g}_j\left(x_i(t-{\gamma }_{ij})\right)+J_j,\hfill \end{array}$$

where x_i(t) and $y_j\left(t\right)(i=1,2,\dots ,n,j=1,2,\dots ,m)$ represent the states of the ith neuron on the I-layer and the jth on the J-layer at time t, respectively. The stability of internal neuron processes on the I-layer and J-layer is described by μ_i and ν_j. The c_ji and d_ij represent the connection weights among the neurons in two layers. On the I-layer, the ith neurons x_i(t) receives the external input I_i and the inputs outputted by those neurons on the J-layer via the activation function f_i. Conversely, on the J-layer, the jth neurons y_j(t) receives the external input J_j and the inputs outputted by those neurons on the I-layer via the activation function g_j. τ_ji and γ_ij correspond to the finite time delays of neural processing and delivery of signals, respectively.

The study on the simplified ones of (1.1) plays an important role since there are still no effective methods for the qualitative analysis of large-scale neural networks with delays. The stability and bifurcation of delayed BAM neural networks with three, four and five neurons respectively, have been extensively studied in [1], [2], [5], [6], [9], [13], [16], [17], [18], [19], [20]. For the three-neuron BAM neural network, Song-Han-Wei [16] discussed the stability and Hopf bifurcation in the case with a single delay, and Yan [18] investigated the pitchfork bifurcation in the case with two discrete delays. The BAM neural network with four neurons and multiple delays is considered in [1], [6], [13], [19] and references therein. The model with two delays, consisting of two neurons on the I-layer and three neurons on the J-layer, is discussed in [9], the local stability of zero solution and a family of bifurcating periodic solutions are obtained as the delays pass through a certain critical value.

In this paper, we consider a four-neuron BAM neural network model with delays in [1]. There are three neurons with respective activation functions f₂, f₃ and f₄ on the J-layer and one neuron with the activation function f₁ on the I-layer. For the sake of simplicity, we assume that the delay from the I-layer to the J-layer is τ₁, while the delay from the J-layer back to the I-layer is τ₂. The architecture of this network is illustrated in Fig. 1, and it can be modeled by the following system of delay differential equations

$$\{\begin{array}{c}\dot{x_1}\left(t\right)=-{\mu }_1{x}_1\left(t\right)+c_{21}{f}_1\left(x_2(t-{\tau }_2)\right)+c_{31}{f}_1\left(x_3(t-{\tau }_2)\right)\hfill \\ \\ +c_{41}{f}_1\left(x_4(t-{\tau }_2)\right),\hfill \\ \\ \dot{x_2}\left(t\right)=-{\mu }_2{x}_2\left(t\right)+c_{12}{f}_2\left(x_1(t-{\tau }_1)\right),\hfill \\ \\ \dot{x_3}\left(t\right)=-{\mu }_3{x}_3\left(t\right)+c_{13}{f}_3\left(x_1(t-{\tau }_1)\right),\hfill \\ \\ \dot{x_4}\left(t\right)=-{\mu }_4{x}_4\left(t\right)+c_{14}{f}_4\left(x_1(t-{\tau }_1)\right),\hfill \end{array}$$

where $x_i(i=1,2,3,4)$ denotes the state of the ith neuron. The stability of internal neuron processes on the I-layer and J-layer is described by ${\mu }_i>0(i=1,2,3,4)$. The real constants c_i1 and $c_{1i}(i=2,3,4)$ are the connection weights among the neurons on two layers. There are many results on this model, for example, Cao and Xiao [1] have obtained that the sign of $f^{{}^{″\prime }}\left(0\right)/f^{{}^{\prime }}\left(0\right),$ $(f_i=f,i=1,\dots ,4),$ determines the direction and stability of Hopf bifurcation by using the center manifold theorem and normal form method. Ge and Xu [5] introduced the perturbation-incremental scheme (PIS) to get expressions of the periodic solutions emerging from Hopf bifurcation, and showed the solution from the PIS having higher accuracy than one from the center manifold reduction. For most of results on BAM neural networks only is the delay regarded as the bifurcation parameter, but the real network modeling inevitably requires the change of the connection topology. Later, Liu[13] chose the connection weights as bifurcation parameters and obtained bifurcation diagrams of Bogdanov-Takens (B-T) and triple zero bifurcations at the origin, which is an equilibrium point of the neuron network, respectively.

Min, Zhang and Cao [15] analyzed the Hopf bifurcation of a general delayed BAM neural network model with $n+1$ neurons. In order to know more dynamical behaviors in a BAM neural network model, the study of bifurcation with more higher codimension is necessary. The goal of this paper is to study the double Hopf bifurcation and the persistence of quasi-periodic invariant 2-tori of the truncated normal form in the double Hopf bifurcation in a BAM neural network model, taking the model (1.2) as an example. Due to the complexity of the calculation of normal forms near the critical point of double Hopf bifurcation, most of the work about 2-codimensional bifurcations was focused on numerical simulations. We first obtain the critical conditions where double Hopf bifurcation occurs by discussing the distribution of eigenvalues of the associated characteristic equation which is a fourth-degree polynomial exponential equation. Then taking the sum $\tau ={\tau }_1+{\tau }_2$ of delays and connection weight b ($b:=c_{12}=c_{21}$) as bifurcation parameters, we obtain the normal form in a neighborhood of the critical value by normal form computation [3], [4]. Finally, we study the persistence of quasi-periodic invariant 2-tori in double Hopf bifurcation by a KAM theorem [12], and prove that in a sufficiently small neighborhood of the bifurcation point, the original system (equivalently, the model (1.2)) has quasi-periodic invariant 2-tori for most of the parameter set where its truncated normal form possesses invariant 2-tori. To the best of our knowledge, the paper is the first one to introduce the existence of quasi-periodic invariant 2-tori of double Hopf bifurcation in a four-neuron BAM neural network model.

The paper is organized as follows. In Section 2, the existence of the double Hopf bifurcation is discussed by analyzing the associated characteristic equation. In Section 3, the normal form of (1.2) near the double Hopf bifurcation critical point is given based on the center manifold theorem and normal form method. Finally, the persistence of quasi-periodic invariant 2-tori in double Hopf bifurcation of (1.2) is proved by a KAM theorem in Section 4. Sections 5 and 6 give some numerical simulations and conclusions. Finally, in Appendix, the proof of Lemmas 2 and 3 and some complex coefficients of the normal form are attached.