Bifurcations of lattice structures

Abstract

Equilibrium configurations of a model for a crystal in one spatial dimension with anharmonic and up to third-neighbour harmonic interactions are related to orbits in a two- or four-dimensional space under a nonlinear symplectic mapping. In this way a connection is given with dynamical systems with two or three degrees of freedom. Depending on the parameters the orbits are smooth or stochastic. Their characterisation in terms of their fixed points and their Ljapunov exponents is given. The mapping shows a large number of bifurcations, of which one can distinguish various types, and these are discussed.