Lattice Dynamical Systems

The following notes are based on my lectures given at CIME Session on ”Dynamical Systems” from June 19 to June 26, 2000 in Cetraro (Cosenza). In Section 1, we study spatially discrete nonlinear diffusion equations and discuss various phenomenon; in Section 2, we introduce the idea of spatial chaos by studying Nagumo equation; in Section 3, 4 and 5, we present a general theory for pattern formation and spatial chaos in lattice dynamics systems; finally in Section 6 and 7, we study the special case of synchronisation phenomena of lattice systems.