^{1}

^{2}

^{1}

^{2}

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 [

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 [

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 [

In this paper, we consider a parabolic complex network consisting of

Let

Next, we introduce some notations and definitions.

For any

The complex network (

A system with input

In [

Let

Let

Firstly, we can get from (

Define

Practically, Theorem

Assume that

Let

Define

Let

Firstly, construct a Lyapunov functional for the network (

By a minor modification of the proof of Theorem

Let

In [

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

The change processes of

The change process of

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.

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.