Quasi-periodic invariant 2-tori in a delayed BAM neural network☆
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 equationswhere xi(t) and 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 μi and νj. The cji and dij represent the connection weights among the neurons in two layers. On the I-layer, the ith neurons xi(t) receives the external input Ii and the inputs outputted by those neurons on the J-layer via the activation function fi. Conversely, on the J-layer, the jth neurons yj(t) receives the external input Jj and the inputs outputted by those neurons on the I-layer via the activation function gj. τji and γij 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 equationswhere denotes the state of the ith neuron. The stability of internal neuron processes on the I-layer and J-layer is described by . The real constants ci1 and 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 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 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 of delays and connection weight b () 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.
Section snippets
Double Hopf bifurcation
In this section we study the existence of double Hopf bifurcation by analysing the associated characteristic equation with two pairs of purely imaginary roots, which is somewhat analogous to the analysis of the existence of Hopf bifurcation in [1]. For the sake of simplicity, we let and and the system (1.2) can be rewritten as
Computation of normal form
In this section, we will derive the normal form by the method in [3], [4]. According to the center manifold theorem and normal form method, it is usually necessary to determine the expression of the central manifold and bring it into the first equation of (3.7) to obtain a system of the ordinary differential equations which is satisfied by the flow on the central manifold. But the method in [3], [4] is used to calculate the central manifold and simplify the normal form at the same time. By
Quasi-periodic invariant 2-tori
In this section, we first discuss the existence of invariant 2-tori of the truncated normal form of (3.30), then the persistence of these invariant 2-tori adding the higher-degree terms. Consider the truncated normal form of (3.30)As the amplitude equations in (4.1) are independent of the angle variables, the existence of invariant tori of (4.1) can be studied
Numerical simulations
In this section, we give numerical simulations of our main theoretical analysis. For fixed which make (K1) in Lemma 1 hold, we can obtain the critical point: . Consequently, (2.3) has two pairs of simple imaginary roots with the imaginary parts ω10 ≈ 0.32217013 and ω20 ≈ 2.56247241, respectively. For the sake of simplicity, the transformations and are applied to place the critical points at the origin
Conclusions
Although, the Hopf bifurcation of a general delayed BAM neural network model with neurons is studied in [15], investigating bifurcation with more higher codimension in a simplified neural network models is necessary in order to know more dynamical behaviors in a BAM neural network model. In this paper, we consider a four-neuron BAM neural network model with two delays and take the sum of delays and the connection weight as bifurcation parameters. We first obtain the
CRediT authorship contribution statement
Xuejing Deng: . Xuemei Li: Supervision, Writing - original draft, Visualization. Fang Wu: .
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Xuejing Deng is a graduate student majoring in applied mathematics at Hunan Normal University.
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Cited by (0)
Xuejing Deng is a graduate student majoring in applied mathematics at Hunan Normal University.
Xuemei Li, Professor, Department of Mathematics, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan Normal University Research Interest: Ordinary differential equations and dynamical systems Biography 1987 - Hunan Normal University 2008/04 2004/04, Visiting Scholar, University of Texas at Austin, USA 1999–2002 Hunan University, China, Ph. D. in Mathematics 1985–1987 Inner Mongolia University, China, Ma. Sc. In Mathematics
Fang Wu is a graduate student majoring in applied mathematics at Hunan Normal University.
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This work is supported by the NNSF(11971163) of China.