Bifurcations in 2D Spatiotemporal Maps

In this work, we give theoretical and numerical analyses for local bifurcations of 2D spatiotemporal discrete systems of the form x(m+1, n+1) = f (x(m, n,) x(m+1,n)), where f is a real nonlinear function, m and n are two independent integer variables, representing respectively a spatial coordinate and the time. On the basis of the spectral theory, we derive the conditions under which the local bifurcations such as flip and fold occur at the fixed points for some parameter values. As a case-study, a quite complex system, 2D spatiotemporal dynamic given by two coupled logistic maps, named 2D logistic coupled maps (2D-LCM) is considered. The proposed map provides a reliable experimental and theoretical basis for identifying some cases of local bifurcations.