Clarifying chaos 3. Chaotic and stochastic processes, chaotic resonance, and number theory

In this tutorial we continue our program of clarifying chaos by examining t he relationship between chaotic and stochastic processes. To do this, we co nstruct chaotic analogs of stochastic processes, stochastic differential eq uations, and discuss estimation and prediction models. The conclusion of th is section is that from the composition of simple nonlinear periodic dynami cal systems arise chaotic dynamical systems, and from the time-series of ch aotic solutions of finite-difference and differential equations are formed chaotic processes, the analogs of stochastic processes. Chaotic processes a re formed from chaotic dynamical systems in at least two ways. One is by th e superposition of a large class of chaotic time-series. The second is thro ugh the compression of the time-scale of a chaotic time-series. As stochast ic processes that arise from uniform random variables are not constructable , and chaotic processes are constructable, we conclude that chaotic process es are primary and that stochastic processes are idealizations of chaotic p rocesses. Also, we begin to explore the relationship between the prime numbers and th e possible role they may play in the formation of chaos.