Turing Instability and Hopf Bifurcation in Cellular Neural Networks

Abstract

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.