Disorder of particle chains as a dynamical problem of transition to chaos: Analogy to simulation induced chaos

Abstract

The problem of the equilibria of particle chains with nearest and next-to-nearest neighbor interaction has been reduced to the dynamical system, given by 4D or 2D web-maps. It is shown that at the same time these maps can represent difference schemes for differential equations used in computational simulation. An analogy between particle disordering, dynamical chaos, and simulation induced chaos is established.The difference equation replaces an ordinary differential equation when a program for computer simulation is prepared. Such replacement generates a new dynamical system, and discretization of the initial equations leads to a high frequency perturbation which reveals in a new topology of the initial system phase space. Only fairly simple systems can be analyzed exactly what is new in their dynamics after a discretization algorithm has been applied; see [9]. To simplify the problem one may use discretized systems analogy to some physical system of particles chain in the equilibria. This analogy permits the formulation of general rules of stability or of the phase space topology changing for some class of discretized dynamical systems. where the potential energy of the chain is