The Recurrent Sequences as Dynamical Systems

A discrete dynamical system on a metric space M is a sequence (fn) of iterations of a function f: M! M. Precisely, f0 is the identity function and for n 1, fn = f f f | {z} n times The main problem: given an initial state a 2 M what happens with orbit of a; x0 = a; xn = f (n)(x0) = f(xn1) for n 1? Dynamical Systems Theory deals with the global study of orbits. This theory was initiated by H. Poincaré in connection with his research on Celestial Mechanics. Invariant sets. Attractors In what follows f: M! M is a function acting on a metric space M. If a 2 M and fn(a) = a for some n 2 N; then a is called a periodic point of f (of period n). The smallest such n is the main period of a. The orbit of a periodic point is a \u85nite set. It can consist of only one point if a is a \u85xed point of f, that is if f(a) = a. For the identity function of R all the points are \u85xed points. For the function f(x) = x there is only one xed point, 0, all the other being periodic of period 2. The xed points and the periodic orbits are example of invariant sets. A subset A of M is called invariant under f (or for f) if f(A) = A and is called positively invariant if f(A) A. In both cases, any orbit starting in A stays in A. Figure 1: The step diagram. When M is an interval, the dynamic can be revealed by the step diagram. The intersection of the graph with the \u85rst bisector gives the xed points. The sequence (fn(a)) of the iterations of x0 = a can be seen in the following way: draw a vertical line through a, It will cross the graph at (a; f(a)). The horizontal line through this point will intersect the rst bisector at (f(a); f(a)). The vertical line through this point will cross the graph at (f(a); f2(a)) and so on. On the picture, the arrows show the order while the steps show the monotony of the function