Generalized Vertigo Maps — A New Family of Chaotic Maps with Robust Chaos but Without Fixed Points

This work proposes a generalization of the family of chaotic maps without fixed points, proposed by Jafari et al.; 2016 and termed the Vertigo maps. The original map family is parameterized by four control parameters, which can be used to scale the function used as a seed and control its domain. Several theoretical results are provided regarding the existence of the fixed points, the periodic cycles, and the Lyapunov exponents of the maps. Furthermore, two map examples are provided based on the logistic and tent seed functions, which are then studied using a series of numerical tools, like phase portraits, bifurcation diagrams, and Lyapunov exponent diagrams. Finally, an application to a Pseudo-Random Bit Generator is considered. The generator utilizes an exponential-based hash function in combination with the remainder operator.