Mappings and Linear Stability

Abstract

Certain topological considerations can facilitate our visualization and consequent understanding of multiply periodic motion. They lead naturally to a set of difference equations, i.e., a mapping of the dynamical trajectory onto a subspace of the system phase space. These mappings allow easy numerical visualization of the motion for problems of two degrees of freedom. Moreover, mathematical proofs concerned with the existence of various types of orbits and theoretical and numerical calculations of stochastic behavior can usually be approached most conveniently from the equations of a mapping. One the other hand, as we have seen in Chapter 2, regular motion is often conveniently described in terms of differential equations. Conversion of (Hamilton’s) differential equations into mappings, and vice versa, are common devices for calculating the motion of most nonlinear dynamical systems.