Analyzing stability and bifurcations in a new discrete system derived from a fractional order continuous chaotic system

The main purpose of this paper is to analyze the stability of fixed points in a new fractional discrete system induced by a continuous differential system of fractional order and to explore its bifurcations. From the fractional order differential Arneodo’s system, we derive a new discrete chaotic dynamical system. Asymptotic stability of the first fixed point of this new system, is carefully investigated in three dimensions or variables. This lead to the splitting of the parameter space related to two parameters into ten regions, each of them is split again versus a third parameter into three to five subregions. In eight of these regions, we were able to analytically prove the asymptotic stability of the first fixed point. Our theoretical results are validated by numerical simulations. In addition we give some clues about bifurcation versus the parameter linked to the fractal dimension of Arneodo’s model. We highlight some chaotic behavior of the discrete model.