Bifurcations and transition to chaos in generalized fractional maps of the orders 0 < alpha < 1 (2303.02501v1)

Abstract: Generalized fractional maps of the orders 0 < alpha < 1 are Volterra difference equations of convolution type with kernels, which differences are absolutely summable, but the series of kernels are diverging. Commonly used in applications fractional (with the power-law kernels) and fractional difference (with the falling factorial kernels) maps of the orders 0 < alpha < 1 belong to this class of maps. We derived the algebraic equations which allow the calculation of bifurcation points of generalized fractional maps. We calculated the bifurcation points and drew bifurcation diagrams for the fractional and fractional difference logistic maps. Although the transition to chaos on individual trajectories in fractional maps may be characterized by the cascade of bifurcations type behavior and zero Lyapunov exponents, 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 df is the same as the value of the regular Feigenbaum constant delta = 4.669....