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(mu, (1 + a)x(mn)), where a is a real constant and mu 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(mu, x(m,n0)) , where n(0) is a fixed integer. These results also improve some existing results of the 2D coupled map lattice (CML) model x(m+1,n) = (1 - epsilon)f(x(mn)) + (epsilon)/(2)[f(x(m,n-1)) + f(x(m,n+1))], 2 where epsilon > 0 is the coupling constant.