Asymptotically periodic points, bifurcations, and transition to chaos in fractional difference maps

Abstract

In this paper, we derive analytic expressions for coefficients of the equations that allow calculations of asymptotically periodic points in fractional difference maps. Numerical solution of these equations allows us to draw the bifurcation diagram for the fractional difference logistic map. Based on the numerically calculated bifurcation points, we make a conjecture that in fractional maps the value of the Feigenbaum constant $\delta$ is the same as in regular maps, $\delta=4.669...$.