Turing Instability and Hopf Bifurcation in 2-D Coupled Cellular Neural Networks

The dynamics of 2-D two-grid coupled cellular neural networks (CNNs) are considered. Assuming the boundary conditions of zero-flux type, the linearized model is analyzed by using the decoupling method, which is described as matrix operations, such as the Kronecker product and Kronecker sum. Then, the local stability of the zero equilibrium related to system parameters is studied. Based on the results, the sufficient conditions that induce Turing instability are derived. Furthermore, as a special kind of Turing patterns, the occurrence conditions for Hopf bifurcations are considered. In addition, the global stability of the zero equilibrium is analyzed by constructing proper Lyapunov functions. Finally, numerical simulations are given to illustrate the theoretical results.