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Complex motions and chaos in nonlinear systems.

Akhmet, Marat
Fen, Mehmet Onur
In this paper, the authors analyze the extension of chaotic dynamics to a particular transformation of a discrete dynamical system. The main result consists in showing that starting from a chaotic map (input), the state variable obtained by adding a linear map and a continuous function of the chaotic state (output) is chaotic as well. These results are based on Devaney's definition of chaos and, for this purpose, this definition is extended to collections of sequences. Several examples are presented to show chaotic dynamics for output systems and an extension of period doubling cascades in coupled systems. The paper then analyzes the existence of homoclinic and heteroclinic orbits for the input and output systems as well as chaos control techniques for the output system in terms of the input system. The last part of the study explores in detail an application to the dynamics of a bacterial infection (gonorrhea) in two distinct heterosexual populations by means of a bidimensional map. Chaotic motion of the system is thus proven by employing the main results of the paper.