Theory and Applications of Complex Networks 2014View this Special Issue
Research Article | Open Access
Stability and Passivity of Spatially and Temporally Complex Dynamical Networks with Time-Varying Delays
This paper proposes a new complex dynamical network model, in which the state, input, and output variables are varied with the time and space variables. By utilizing the Lyapunov functional method combined with the inequality techniques, several criteria for passivity and global exponential stability are established. Finally, numerical simulations are given to illustrate the effectiveness of the obtained results.
Complex networks can be seen everywhere, which have become an important part of our daily life. Recently, the topology and dynamical behavior of complex dynamical networks have been extensively studied by the researchers. In particular, special attention has been focused on synchronization in complex dynamical networks, and many interesting results on synchronization were derived for various complex networks [1–10].
To our knowledge, in most existing works on the complex networks, they always assume that the node state is only dependent on the time. However, such simplification does not match the peculiarities of some real networks. Food webs are among the most well-known examples of complex networks and hold a central place in ecology to study the dynamics of animal populations. A food web can be characterized by a model of complex network, in which a node represents a species. To our knowledge, species are usually inhomogeneously distributed in a bounded habitat and the different population densities of predators and preys may cause different population movements; thus it is important and interesting to investigate their spatial density in order to better protect and control their population. In such a case, the state variable of node will represent the spatial density of the species. Moreover, the input and output variables are also dependent on the time and space in many practical situations. Therefore, it is essential to study the complex networks, in which the state, input, and output variables are varied with the time and space variables.
Recently, food web [11–24] has become a focal research topic and attracted increasing attention from many fields of scientific research. In , Pao discussed the asymptotic behavior of time-dependent solutions of a three-species reaction-diffusion system in a bounded domain under a Neumann boundary condition. Kim and Lin  considered the blowup properties of solutions for a parabolic system with homogeneous Dirichlet boundary conditions, which describes dynamic properties of three interacting species in a spatial habitat. As a natural extension of the existing network models, we propose a complex dynamical network with time-varying delays, which is described by a system of parabolic partial differential equations. In addition, we investigate the global exponential stability of the proposed network model.
Stability problems are often linked to the theory of dissipative systems, which postulate that the energy dissipated inside a dynamic system is less than that supplied from external source. Passivity is part of a broader and a general theory of dissipativeness. The main point of passivity theory is that the passive properties of systems can keep the systems internally stable. The passivity theory has found successful applications in diverse areas such as complexity , signal processing , stability [27, 28], chaos control and synchronization [29, 30], and fuzzy control . Although research on passivity has attracted so much attention, little of that had been devoted to the passivity properties of spatially and temporally complex dynamical networks until Wang et al.  obtained some sufficient conditions on passivity properties for a class of reaction-diffusion neural networks with Dirichlet boundary conditions. To the best of our knowledge, very few researchers have investigated the passivity of complex dynamical networks with spatially and temporally varying state variables. Therefore, we also study the passivity of the proposed network model, and some sufficient conditions ensuring input strict passivity and output strict passivity are obtained by constructing appropriate Lyapunov functionals and using inequality techniques.
2. Network Model and Preliminaries
In this paper, we consider a parabolic complex network consisting of nonidentical nodes with diffusive and delay coupling. The entire network is described by where , is the bounded domain with smooth boundary , , , , is the time-varying delay with , is the state of node , are the input and output of node at time and in space , respectively, is the Laplace diffusion operator on , and represents the topological structure of network and coupling strength between nodes, where is defined as follows: if there is a connection from node to node (), then ; otherwise, , and
Let . The initial value and boundary value conditions associated with the network (1) are given in the form where is bounded and continuous on . Let be the state trajectory of network (1) from the above initial value and boundary value conditions.
Next, we introduce some notations and definitions.
For any , denotes for any , , we define
Definition 1. The complex network (1) is said to be globally exponentially stable if there exist constants and such that for any two solutions of network (1) with initial functions , respectively, it holds that for all . If such is an equilibrium solution (or periodic solution), then this equilibrium solution (or periodic solution) is said to be globally exponentially stable.
Definition 2 (see ). A system with input and output where is said to be passive if there is a constant such that for all , where is a bounded compact set. If in addition, there are constants and such that for all , then the system is input-strictly passive if and output-strictly passive if .
Remark 3. In , Wang and Wu discussed the global exponential stability and passivity of a parabolic complex network. In network model (1), the coupling matrix is diffusive. Namely, , . Compared with the network model considered in , the network model considered in this paper may be more reasonable. On the other hand, we investigate the input strict passivity and output strict passivity of the complex network (1), which are totally different from the work in .
Lemma 4 (see ). Let be a cube and let be a real-valued function belonging to which vanishes on the boundary of , that is, . Then, where .
3. Stability and Passivity of Parabolic Complex Network
3.1. Stability Analysis
Theorem 5. Let . The complex network (1) is globally exponentially stable if there exist constants and such that where .
Proof. Firstly, we can get from (1) that
Define , , and then the dynamics of the difference vector is governed by the following equation: where . Define the following Lyapunov functional for the system (11): Calculating the time derivative of along the trajectory of system (11), we can get From Green’s theorem and the boundary condition, we have According to Lemma 4, we can obtain Therefore, From (9), we can get So , . Since Let , , and Then , and we can obtain Namely, This completes the proof of Theorem 5.
Assume that and are periodic continuous functions with period . Then we can get from (1) that where , , .
Theorem 6. Let . The system (22) has a unique -periodic solution if there exist constants and such that where .
Proof. Define Obviously, from Theorem 5, we can derive for , where . We can choose a positive integer such that Now define a Poincaré mapping by Since the periodicity of system (22), one has . From (25) and (26), one obtains This implies that is a contraction mapping, so there exists one unique fixed point such that Since , so is also a fixed point of , then Let be the solution of system (22) with initial conditions , then is also a solution of system (22). Obviously, This means that is exactly one -periodic solution of system (22). This completes the proof of Theorem 6.
3.2. Passivity Analysis
Theorem 7. Let . The complex network (1) is input-strictly passive if there exist constants and such that where , .
Proof. Firstly, construct a Lyapunov functional for the network (1) as follows: Calculating the time derivative of along the trajectory of system (1), we can get From Green’s theorem and the boundary condition, we have According to Lemma 4, we can obtain Therefore, It follows from (32) that By integrating (38) with respect to over the time period to , we can get From the definition of , we have and . Thus, for all , where . The proof is completed.
By a minor modification of the proof of Theorem 7, we can easily get the following.
Theorem 8. Let . The complex network (1) is output-strictly passive if there exist constants and such that where , .
Remark 9. In , two kinds of impulsive parabolic complex networks were considered, in which the node states are dependent on the time and space variables. Several global exponential stability and robust global exponential stability criteria were derived for the impulsive parabolic complex networks. Both global exponential stability and passivity are taken into consideration in this paper, and some sufficient conditions are established.
In this section, an illustrative example is provided to verify the effectiveness of the proposed theoretical results.
Consider a complex network model, in which each node is a 1-dimensional dynamical system described by Take , , , , , , , . The matrix is chosen as It is easy to verify that (9) is satisfied if and . From Theorem 5, complex network (1) with above given parameters is globally exponentially stable. Moreover, if we take and , then (32) is satisfied. According to Theorem 7, complex network (1) with above given parameters is also input-strictly passive. The simulation results are shown in Figures 1 and 2.
A parabolic complex network model has been introduced, in which the state, input, and output variables are dependent on the time and space variables. The input strict passivity, output strict passivity, and global exponential stability of the proposed network model have been discussed in this paper, and several sufficient conditions have been established. Illustrative simulations have been provided to verify the correctness and effectiveness of the obtained results. In future work, we shall study the passivity and robust passivity of parabolic complex networks with impulsive effects.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank Professor J.-L. Wang (Tianjin Polytechnic University, Tianjin, China) for his kind help.
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Copyright © 2014 Shun-Yan Ren and Yue-Hui Zhao. 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.