Theorems and application of local activity of CNN with five state variables and one port.

Gang Xiong,Thomas Yang,Li Xie,Xisong Dong

doi:10.1155/2012/674243

Abstract

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.

Highlights

Coupled nonlinear dynamical systems have been widely studied in recent years
These theorems can be implemented by a computer program for calculating the bifurcation diagram of the general corresponding cellular neural network (CNN) to determine emergence of complex dynamic patterns of the corresponding CNN
The local activity of CNN has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells

Summary

Introduction

Coupled nonlinear dynamical systems have been widely studied in recent years. The research on emergence and complexity has gained much attention during the past decades, the determination, prediction, and control of the complex patterns generated from highdimensional coupled nonlinear systems are still far from perfect. Chua presents the main theorem of local activity at a cell equilibrium point [1, 2], it is difficult to “test” directly the complex patterns of the high-dimensional coupled nonlinear systems, since the theorem contains no recipe for finding whether a variable exists or not.

Local Activity Theory of CNN

Conclusions