ABSTRACT
Differential equations are an important class of dynamical systems; in this concluding chapter. It considers second order equations with linear damping but non-linear restoring term, driven by a periodic applied force. These restrictions have the benefit of permitting easy reduction to a two-dimensional map with constant area-reduction, while encompassing some of the most important examples, although they preclude consideration of the Lorenz and Rossler equations. One of the hallmarks of chaos, which is measured by Lyapunov exponents, is sensitive dependence of solutions on initial conditions — the Butterflyeffect. The chapter discusses the definition of Lyapunov exponents in due course. This will lead to a definition of chaotic orbits for periodically driven systems. The differential equations under consideration are three-dimensional, therefore proper definition of Lyapunov exponents should involve three contraction/expansion factors for an evolving three-dimensional ellipsoid. Moreover, the principal axis associated with the non-trivial exponents are in the phase plane.
