On the origin of periodicity in dynamical systems

We prove a theorem establishing a direct link between macroscopically observed periodic motions and certain subsets of intrinsically discrete orbits which are selected naturally by the dynamics from the skeleton of unstable periodic orbits (UPOs) underlying classical and quantum dynamics. As a simple illustration, an infinite set of UPOs of the quadratic (logistic) map is used to build ab initio the familiar trigonometric and hyperbolic functions and to show that they are just the first members of an infinite hierarchy of functions supported by the UPOs. Although all microscopic periodicities of the skeleton involve integer (discrete) periods only, the macroscopic functions resulting from them have real (non-discrete) periods proportional to very complicate non-integer numbers, e.g. 2π and 2πi, where i=(−1)1/2.