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

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....

arXiv comment21 pages, 2 figures. arXiv admin note: text overlap with arXiv:2209.15462

References in corpus (10)

• Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions
• Fractional $h$-difference equations arising from the calculus of variations
• Logistic map with memory from economic model
• Fractional Standard Map
• Fractional Equations of Kicked Systems and Discrete Maps
• Caputo Standard $\alpha$-Family of Maps: Fractional Difference vs. Fractional
• Universal Fractional Map and Cascade of Bifurcations Type Attractors
• Asymptotic cycles in fractional maps of arbitrary positive orders
• Fractional Dynamics with Non-Local Scaling
• Asymptotically periodic points, bifurcations, and transition to chaos in fractional difference maps

Cited by in corpus (2)

• Asymptotic cycles in fractional generalizations of multidimensional maps
• Characterizing and quantifying weak chaos in fractional dynamics