Discrete one-dimensional piecewise chaotic systems without fixed points

This paper presents a new method of generating chaotic maps that do not have fixed points. The method uses an appropriate transformation of functions defined on the unit square in such a way that the newly created projection has no fixed points. In this way, countless new chaotic mappings can be created, both piecewise linear and nonlinear, that belong to the family of systems with hidden attractors. The new family of maps is presented in three examples showing phase diagrams, bifurcation diagrams, and space parameters of the Lyapunov exponent. The discussed examples of mappings were created by appropriate logistic and tent mapping transformations, and their combination. In addition, this paper analyzes the applications of the new families of chaotic mappings in chaotic cryptography. Furthermore, an algorithm for generating pseudorandom bit values (PRBG) is also presented based on the examples of proposed mappings.