Bifurcations and persistence of equilibria in high-dimensional lattice dynamical systems

We discuss the similarities and differences of the bifurcations and persistence of equilibria in two kinds of high-dimensional lattice dynamical systems. If the nonlinear term has finite distinct zeros, there exists a uniform critical value below which all the equilibria at the anti-integrable limit persist without new-born equilibria. If the nonlinear term is periodic and has infinitely many zeros, then no uniform critical value exists and there always occur bifurcations as the coupling coefficient varies. We also investigate in detail the bifurcations of spatially homogeneous equilibria with different nonlinearities. The analogues of pitchfork bifurcation, saddle-node bifurcation and period-doubling bifurcation in one-dimensional maps were found. As an application, we discuss the structural stability of some high-dimensional Henon-like maps.