ABSTRACT
Let us start this book by a short initiation to chaos through two archetypes inspired by the work of Ian Stewart [129].
Recurrent sequences, also called discrete dynamical systems, of the form
u0 ∈ R, un+1 = f(un), (1.1)
with f continuous, have been well studied since the early years of mathematical analysis. They are widely used to resolve equations using a Newtonian method, or when approximating the solutions to differential equations using finite difference equations to approximate derivatives. The context study was the seek for convergence, which is for instance guarantee when using monotonic functions or contractions. In the middle of the last century, Coppel established a link between this desire of convergence and the existence of a cycle in iterations [57]. More precisely, his theorem states that, considering Eq. (1.1) with
Machines
a function f : I −→ I continuous on the line segment I, the absence of any 2-cycle implies the convergence of the discrete dynamical system.
