From almost periodic to chaotic: The fundamental map

In this paper we present a single map that displays an extraordinary range of dynamics that lies between the Bernoulli maps and almost periodic maps. Of central importance is the fact that this map illustrates how almost-periodic dynamics evolves into Bernoulli dynamics. Due to its continuous spectrum of dynamics between Bernoulli and almost periodic, we call this map the fundamental map in contrast to the better known standard map. The range of dynamics found in this map suggests that as order gives way to chaos, both the geometry of the orbit and the sequence of coordinates of the points of the orbit evolve from order to disorder. An interesting point brought out by this map is that the spatial and temporal properties of orbits near each end of the scale bear some striking similarities. Additionally, we show how to derive a Poincare map from the fundamental map and derive the associated ODE, an equation for an electronic circuit.