Bifurcations and transition to chaos in generalized fractional maps of the orders 0 <  α  < 1

Abstract

In this paper, we investigate the generalized fractional maps of the orders 0<α<1. Commonly used in publications, fractional and fractional difference maps of the orders 0<α<1 belong to this class of maps. As an example, we numerically solve the equations, which define asymptotically periodic points to draw the bifurcation diagrams for the fractional difference logistic map with α=0.5. For periods more than four (T>4), these bifurcation diagrams are significantly different from the bifurcation diagrams obtained after 105 iterations on individual trajectories. We present examples of transition to chaos on individual trajectories with positive and zero Lyapunov exponents. We derive the algebraic equations, which allow the calculation of bifurcation points of generalized fractional maps. We use these equations to calculate the bifurcation points for the fractional and fractional difference logistic maps with α=0.5. The results of our numerical simulations allow us to make a conjecture that the cascade of bifurcations scenarios of transition to chaos in generalized fractional maps and regular maps are similar, and the value of the generalized fractional Feigenbaum constant δf is the same as the value of the regular Feigenbaum constant δ=4.669….