Chaotic Features of a Class of Discrete Maps without Fixed Points

Abstract

Chaotic dynamical systems without fixed points are promising in the generation of pseudo-random sequences because orbits will not converge to a fixed point. In this class of systems there are no final fixed points, so the basin of attraction of a fixed point does not exist. The absence of fixed points makes it difficult to analyze the dynamics of the systems because a fixed point gives us a lot of information about the dynamics of the system. In addition, if there is an amplitude control parameter for the generated chaotic signals, the tolerance and adaptability are higher and better in the application process. In view of the aforementioned, in this paper, a novel class of discrete maps is presented and described by a kind of piecewise linear (PWL) maps. Necessary and sufficient conditions are given to guarantee that this class of discrete maps does not have any fixed point. Furthermore, we introduce families of these PWL discrete maps without fixed points that present positive Lyapunov exponents and have chaotic dynamics with amplitude control. From these families, we select one particular map, which is analyzed theoretically and proved to be chaotic in the sense of Devaney.