Résumé: In this work, we propose a novel 3D chaotic map obtained by coupling the piecewise and logistic maps. Showing excellent properties, like a high randomness, a high complexity and a very long period, this map has enabled us to implement and investigate a new chaotic pseudo-random number generator (CPRNG). The produced pseudo-random numbers exhibit a uniform distribution and successfully pass the NIST SP 800-22 randomness tests suite. In addition, an application in the field of color image encryption is proposed where the encryption key is strongly correlated with the plain image and is then used to perform the confusion and diffusion stages. Furthermore, the ability to expand the size of our map has an impact on the complexity of the system and increases the size of the key space, making our cryptosystems more efficient and safer. We also give some statistical tests and computer simulations which confirm that the proposed algorithm has a high level of security.

Résumé: A discrete time dynamical systems represented by the iteration of nonlinear piecewise functions is introduced in this paper. The dynamical behaviors, multiple basins with fractal boundary, attractors, route to chaos via bifurcations are investigated. Moreover, we point out an fascinating form of complex basin structure in the presence of multistability and coexistence of several attractors.

Résumé: We study the behavior under iteration of a three parameters family of piece- wise linear maps of the plane. Our purpose is to study a particular kind of bifurcation for this kind of maps. We wish to show that this family possesses interesting prop- erties. Coexistence of several attractors, and characteristics of intermingled basins of different attractors are obtained.

Résumé: In this paper, we study the bifurcation space and the phase plane of the Bogdanov map. Specific bifurcation structures can be observed in the parameter space, related for instance to embedded boxes structure and configurations of bifurcation curves of periodic points near the cusp. This model is a diffeomorphism. The dynamics is extremely rich, involving periodicity, quasiperiodicity and chaos. The method of the study is a numerical iteration to an attractor in which the guesses are inspired by the theory. Bifurcation diagrams obtained in different parameter planes are given and a sketch showing the cusp bifurcations, for the versal unfolding of Bogdanov map, is given. The phase plane is also studied, different attractors are shown, their evolution giving rise to chaotic attractors is explained. Basins of attraction are considered and fuzzy boundaries of basins are put in evidence. The study of such kind of diffeomorphisms can give an interesting contribution to nonlinear systems.