On spatial lyapunov exponents and spatial chaos

In this paper, we investigate the periodic orbits, spatial Lyapunov exponents, and stability of spatially periodic orbits of the general 2D Logistic system x_m+1,n + ax_m,n+1 = f(μ, (1 + a)x_mn), where a is a real constant and μ is a parameter. The existence of spatial chaos in the sense of Li and Yorke is proved using the Marotto theorem. These results extend the corresponding results in the 1D Logistic system x_m+1,n0 = f(μ, x_m,n0), where no is a fixed integer. These results also improve some existing results of the 2D coupled map lattice (CML) model x_m+1,n = (1 - ε) f(x_mn) + ε/2[f(x_m,n-1) + f(x_m,n+1)], where ε > 0 is the coupling constant.