The Challenging Problem of Industrial Applications of Multicore-Generated Iterates of Nonlinear Mappings

Abstract

The study of nonlinear dynamics is relatively recent with respect to the long historical development of early mathematics since the Egyptian and the Greek civilization, even if one includes in this field of research the pioneer works of Gaston Julia and Pierre Fatou related to one-dimensional maps with a complex variable, nearly a century ago. In France, Igor Gumosky and Christian Mira began their mathematical researches in 1958; in Japan, the Hayashi’ School (with disciples such as Yoshisuke Ueda and Hiroshi Kawakami), a few years later, was motivated by applications to electric and electronic circuits. In Ukraine, Alexander Sharkovsky found the intriguing Sharkovsky’s order, giving the periods of periodic orbits of such nonlinear maps in 1962, although these results were only published in 1964. In 1983, Leon O. Chua invented a famous electronic circuit that generates chaos, built with only two capacitors, one inductor and one nonlinear negative resistance. Since then, thousands of papers have been published on the general topic of chaos. However, the pace of mathematics is slow, because any progress is based on strictly rigorous proof. Therefore, numerous problems still remain unsolved. For example, the long-term dynamics of the Henon map, the first example of a strange attractor for mappings, remain unknown close to the classical parameter values from a strictly mathematical point of view, 40 years after its original publication. In spite of this lack of rigorous mathematical proofs, nowadays, engineers are actively working on applications of chaos for several purposes: global optimization, genetic algorithms, CPRNG (Chaotic Pseudorandom Number Generators), cryptography, and so on. They use nonlinear maps for practical applications without the need of sophisticated theorems. In this chapter, after giving some prototypical examples of the industrial applications of iterations of nonlinear maps, we focus on the exploration of topologies of coupled nonlinear maps that have a very rich potential of complex behavior. Very long computations on modern multicore machines are used: they generate up to one hundred trillion iterates in order to assess such topologies. We show the emergence of randomness from chaos and discuss the promising future of chaos theory for cryptographic security.