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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 417678, 10 pages
Multiple Nonlinear Oscillations in a -Symmetrical Coupled System of Identical Cells with Delays
College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410076, China
Received 31 January 2013; Accepted 8 May 2013
Academic Editor: Zhiming Guo
Copyright © 2013 Haijun Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A coupled system of nine identical cells with delays and -symmetry is considered. The individual cells are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. By analyzing the corresponding characteristic equations, the linear stability of the equilibrium is given. We also investigate the simultaneous occurrence of multiple periodic solutions and spatiotemporal patterns of the bifurcating periodic oscillations by using the equivariant bifurcation theory of delay differential equations combined with representation theory of Lie groups. Numerical simulations are carried out to illustrate our theoretical results.
Over the past decades, symmetry has become a topic of considerable attention in the research of nonlinear dynamical systems [1–11]. In general, the symmetry reflects a certain spatial invariant of the dynamical systems. The work of Golubitsky et al.  shows that systems with symmetry can lead to multiple patterns of oscillation, which are predictable based on the theory of equivariant bifurcations. It is well known that the introduction of time delays into some systems is more reasonable and realistic . Wu and coworkers [3, 13, 14] extended the theory of equivariant Hopf bifurcation to delay differential equations.
An artificial neural network is an information processing device that is inspired by the way biological nervous systems, such as the brain, process information simultaneously. It has many applications in different areas including pattern recognition, associative memory, signal processing, and combinatorial optimization. There has been an increasing interest in the investigation of neural networks (see, e.g., [4–6, 9–11, 15]) since Hopfield  constructed a simplified neural network model. Ring networks have been found in a variety of neural structures such as cerebellum  and even in chemistry and electrical engineering. In the field of neural networks, rings are studied to gain insight into the mechanisms underlying the behavior of recurrent networks [15, 18]. The dynamical behavior of ring networks has been investigated in more detail. For example, in order to understand which patterns occur in a particular system, Huang and Wu  studied the following ring neural network of three identical neurons with delayed feedback: where , represents the state of the th neuron at time , represents the nonlinear self-feedback function, is the connection function between neurons, and is the time delay. Afterward, some researchers have been studying many ring networks with -symmetry (see [5–8]). However, previous work just has considered the individual network but not investigated the interactions between multiple networks.
In fact, a wide variety of natural and artificial systems possess a hierarchic structure or functioning and can naturally be modeled by coupled subnetwork. For example, the brain may be conceived as a dynamic network of coupled neurons. In order to describe the complicated interaction between billions of neurons in large neural network, the neurons are often lumped into highly connected subnetworks . Coupled networks of nonlinear dynamical systems can exhibit rich dynamics, such as synchronization, symmetric bifurcation, and chaos. The spatio-temporal dynamics of systems of several coupled nonlinear oscillators is presently receiving great attention and a significant body of research has been carried out [9–11, 20, 21]. It must be pointed out that the hierarchical network of neuronal oscillators with -symmetry investigated in [20, 21] is described by a system of ordinary differential equations (ODEs), and the effect of time delays is not considered.
Motivated by the above ideas, in this paper, we consider the two-level hierarchical system which is composed of three coupled modules of interacting nonlinear neuron oscillators with time delays, modeled by the following system of delay differential equations (DDEs): where , , and represents the connection function between different modules. The individual elements are represented by a scalar equation, consisting of a linear decay term and a nonlinear time-delayed self-feedback. The architecture of the model is given in Figure 1.
It is easy to see that all cells are identical, all couplings within each group are identical, and all groups are identically coupled to each other in this model. Therefore, these lead to a -symmetry of the associated system. the model is a natural extension of system (1) and is a particularly simple example of a symmetric system exhibiting a hierarchical structure with two levels: a “macro” level concerning the interactions between the groups and a “micro” level concerning the interactions within the groups. On the other hand, system (2) can be regarded as a special example of the general Hopfield neural networks with delays .
Although model (2) is a little simple, it would be of great significance for applications to have a detailed analysis and then to understand possible mechanisms behind the observed behaviour. In this paper, our main purpose is to reveal how the time delay can affect the stability of system (2), the simultaneous occurrence of multiple periodic solutions, and spatio-temporal patterns of the bifurcating periodic oscillations depending on the -symmetry.
The rest of the paper is organized as follows. In Section 2, the associated characteristic equation is analyzed and the linear stability of the equilibrium is given. In Section 3, we discuss the existence of multiple branches of periodic oscillations and their spatio-temporal patterns with the help of symmetric bifurcation theory of delay differential equations coupled with representation theory of Lie groups. An example and numerical simulations are presented to illustrate the results in Section 4. In Section 5, a brief discussion is drawn to conclude this paper.
2. Distribution of Characteristic Roots and Linear Stability
It is clear that (2) admits the trivial solution, . The linearization of (2) at this equilibrium point is given by where , and . Regarding as the parameter, let denote the infinitesimal generator of the semigroup generated by linear system (3). We first determine when has a pair of purely imaginary eigenvalues.
The characteristic matrix of (3) is where denotes the identity matrix of order , is a circle block matrix, and is a circulant matrix of order . Then we have the following lemma.
Lemma 1. The associated characteristic equation of (3) is where .
Proof. Let , and . Then Hence The conclusion is obtained.
Note that we have In this paper, for the sake of simplicity, we consider only the case where characteristic equation (5) may have a pair of purely imaginary roots of multiplicity , that is, focus on the distribution of zeros of the factor . The other cases are simpler and can be handled in a similar way, and we omit them. Therefore, throughout this paper we suppose, , , .The following result about the distribution of the characteristic roots is obtained.
Lemma 2. Assume that holds. Define Then(i)for all , all zeros of the factors , and have negative real parts,(ii)when , all roots of the characteristic equation (5) have negative real parts; when , the characteristic equation (5) has exactly roots with positive real parts; the other roots have negative real parts; at (and only at) , has a pair of purely imaginary eigenvalues of multiplicity , and all other eigenvalues of are not integer multiples of ,(iii)for each fixed , there exist and -smooth mapping such that and for all . Moreover, ,(iv)the generalized eigenspace consists of eigenvector of associated with . Moreover, where
Proof. For , let . If is a zero of , then , from which it follows that
Thus, we claim that all the zeros of have negative real parts provided that . Noticing that holds and applying the above discussions to the factors , and , we obtain conclusion (i).
In what follows, we consider the distribution of zeros of the factor . For , becomes . Let be a root of , then Thus, Therefore, has a pair of purely imaginary roots if and only if Clearly, for each fixed , . Since there exist and -smooth mapping such that and for all . Differentiating with respect to , we have Therefore, So far, the proof of (ii) and (iii) is complete (for more details see XI.2 in ). It remains to verify (iv).
Note that ; it follows from the proof of Lemma 1 and the above discussions that the eigenspace of associated with is spanned by , , , and . Hence, the space has the real basis . On the other hand, the eigenspace associated with is of dimension , and the multiplicity of the characteristic root is also . According to the folk theorem in functional differential equations (see ), must coincide with the eigenspace of associated with . Therefore, conclusion (iv) is correct. This completes the proof.
Theorem 3. If the assumption is satisfied, then the equilibrium of system (2) is asymptotically stable for ; the equilibrium is unstable for .
3. Multiple Patterns of Oscillation
It follows from Lemma 2 that has a pair of purely imaginary eigenvalues of multiplicity and all other eigenvalues of are not integer multiples of at . Thus, the Hopf bifurcation may provide some asynchronous periodic solutions at each .
The symmetry of a system is important in determining the patterns of oscillation. In order to explore the symmetry of system and analyze the spatio-temporal patterns of the bifurcated periodic solutions, we need the following definition.
Let denote the dihedral group of order , which is generated by cyclic group together with the flip of order . Define the action of on by where is the generator of and denotes the flip.
Lemma 4. System (2) is -equivalent.
Proof. Let mapping be where Then Similarly, we can prove that Therefore, is -equivalent. This completes the proof.
Lemma 5. Let act on by Then is an absolutely irreducible representation of , and the restricted action of on is isomorphic to the action of on .
Proof. It is straightforward to verify the absolute irreducibility of the representation of on by the definition (see ). Note that Define by where the matrix The nonsingularity of the matrix implies that is a linear isomorphism. It is easy to see that Therefore, a straightforward calculation shows that This concludes the proof.
Let and let be the Banach space of all continuous -periodic functions . Then for the circle group , acts on by Denote by the subspace of consisting of all -periodic solutions of system (3) with . Then where Therefore, for , we have the following properties.
Proof. For and , note that we have Therefore Similarly, we can prove that
It is clear that if is a periodic solution of system (2), then so is for every . The spatial-temporal symmetry of a bifurcation of periodic solutions can be completely characterized by the isotropy group , and it is easy to verify that the isotropy group of is , which is conjugate to . The maximal isotropy subgroups of are listed in Table 1. For each subgroup , the -fixed-point set is a subspace of . According to Lemma 6, it is easy to verify that .
Together with Lemma 2, Lemmas 4–6 allow us to apply the equivariant Hopf bifurcation theorem for delay differential equations due to Wu  to obtain the following result on the spatio-temporal patterns of the bifurcated periodic solutions.
Theorem 7. Assume that is satisfied. Then for system (2), near each there exist distinct branches of periodic solutions bifurcated from the equilibrium . More precisely, for each isotropy group and a chosen basis of , there exist , and a -smooth mapping , where , such that for each , is an -periodic solution of system (2) with , and Furthermore, for , , every -periodic solution of system (2) with , for and must be of the above type.
4. An Example and Numerical Simulations
As a simple example, we consider the following specific Hopfield model with -symmetry: where .
Clearly, the origin is an equilibrium of system (42). It is easy to compute and verify that the hypothesis holds. According to (10), we obtain From Theorem 3, the origin is asymptotically stable if . It follows from Theorem 7 that the equivariant Hopf bifurcation occurs at and there exist distinct branches of periodic solutions bifurcated from the origin. To illustrate the analytical results found, we give some numerical simulations. Figure 2 shows that the origin of system (42) is stable when . Figure 3 shows that four periodic orbits occur simultaneously when . Unfortunately, we cannot verify the existence of all other bifurcated periodic orbits since they may be unstable.
In this paper, we have studied a coupled system of nine identical cells with delays and -symmetry. By choosing the time delay as a bifurcation parameter and analyzing the corresponding characteristic equation, we have shown that under some assumption, the equilibrium of the model loses its stability and periodic solutions via Hopf bifurcation occur when passes through a critical value. This implies that the time delay can be regarded as a source of instability and oscillatory response of the networks and is able to alter the dynamics of system (2) significantly. Moreover, the spatio-temporal patterns of bifurcating periodic solutions are explored clearly by employing the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. From Theorem 7, we have obtained the conclusion that the small-scale network with a special structure may have a large number of periodic oscillations. Therefore, it is natural that the large-scale network possesses complicated dynamics generally.
Further investigations such as the stability, direction and global existence of the periodic solutions bifurcating from the local Hopf bifurcations are essential in order to fully understand the periodic phenomenon of the system. We can compute the normal forms directly by using the method due to Faria and Magalháes [24, 25]. However, this is a complex and prolix task. In addition, we would like to point out that codimension two mode interactions may take place if the assumption is not satisfied. For example, if , , and , then system (2) undergoes a fold-Hopf bifurcation when passes through a critical value. Therefore, it is possible to study secondary bifurcations and more complex behaviours in this coupled network. We leave them for our future work.
This work is partially supported by National Natural Science Foundation of China (nos. 11101435 and 11101053), the Scientific Research Fund of Hunan Provincial Education Department of China (no. 12C0028), and the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities (no. 12FEFM07).
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