Non-linear lattice models: complex dynamics, pattern formation and aspects of chaos

In the present paper we intend to discuss the question of pattern formation and complex dynamics developed in some lattice models. Dynamical systems are characterized by their coherent and chaotic natures. In this paper, we report non-linear lattice models which are good candidates to illustrate the fascinating complex dynamics occurring in some physical systems. One of the proposed models is devoted to the formation of localized structures in a two-dimensional lattice. Instability process of localized structures is examined in detail. The instability process is mostly related to the phase transformation in crystalline solids. On the basis of a two-dimensional lattice model involving non-linear and competing interactions the formation and dynamics of elastic domains and twin boundaries are investigated. The emphasis is placed especially on the instability mechanisms of a strain band and modulated strain structure with respect to the transverse perturbations then producing localized structures on the lattice. The long-time evolution of localized modes is also studied using asymptotic perturbative methods and an asymptotic equation is then deduced in the framework in the quasicontinuum approximation. Next, we propose a non-linear analysis in the vicinity of a critical point of the phonon-dispersion branch. From the non-linear analysis, we deduce, in the semidiscrete approach by using a multiscale technique, an amplitude equation of the Ginzburg–Landau type. The mechanism of self-generated non-linear localized modes in the two-dimensional lattice beyond the instability process or near the critical region of the phonon dispersion is numerically investigated. The numerical simulations exhibit non-trivial localized patterns. The second study deals with the soliton motion on a damped driven one-dimensional lattice. The lattice model is made of a one-dimensional chain equipped with rotatory molecules. The problem of soliton dynamics under the influence of discreteness effects, damping and time-dependent applied field allows us to show a transition to chaos of the soliton movement. Numerical simulations performed directly on the discrete system show a particularly rich dynamics of the lattice model. The last model we want to propose for illustration of the dynamics of non-linear systems is a one-dimensional lattice model with long-range interactions. In this part of the paper we are interested in the non-local description of the discrete system in the framework of the quasicontinuum approach. The idea is to consider a one-to-one correspondence between functions of discrete arguments and a class of analytic functions. The procedure allows us to use the same representation of the Lagrangian for all cases: discrete and analytic. The equation of motion deduced in the quasicontinuum representation is similar to that of a non-local model of elasticity including a local non-linear term. A perturbative asymptotic multiscale technique leads to an asymptotic equation for the long-time evolution of the non-linear wave. Numerical simulations are performed on the discrete system for different ranges of non-local action showing the propagation of discrete kink in the lattice.