Universal Fractional Map and Cascade of Bifurcations Type Attractors

Abstract

We modified the way in which the Universal Map is obtained in the regular dynamics to derive the Universal $\alpha$-Family of Maps depending on a single parameter $\alpha >0$ which is the order of the fractional derivative in the nonlinear fractional differential equation describing a system experiencing periodic kicks. We consider two particular $\alpha$-families corresponding to the Standard and Logistic Maps. For fractional $\alpha <2$ in the area of parameter values of the transition through the period doubling cascade of bifurcations from regular to chaotic motion in regular dynamics corresponding fractional systems demonstrate a new type of attractors - cascade of bifurcations type trajectories.

10 pages 8 figures

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Cited by in corpus (16)

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• General Fractional Dynamics
• Stability of Fixed Points and Chaos in Fractional Systems
• Caputo Standard $\alpha$-Family of Maps: Fractional Difference vs. Fractional
• Asymptotic cycles in fractional maps of arbitrary positive orders
• Fractional Dynamics with Non-Local Scaling
• On Fractional Eulerian Numbers and Equivalence of Maps with Long Term Power-Law Memory (Integral Volterra Equations of the Second Kind) to Gr$\ddot{u}$nvald-Letnikov Fractional Difference (Differential) Equations
• Fractional Dynamical Systems
• Predator-prey models with memory and kicks: Exact solution and discrete maps with memory
• Stability of Fixed Points in Generalized Fractional Maps of the Orders $0<\alpha <1$
• Bifurcations and transition to chaos in generalized fractional maps of the orders 0 < alpha < 1
• Caputo Fractional Standard Map: Scaling Invariance Analyses
• From Fractional Differential Equations with Hilfer Derivatives: To Discrete Maps with Memory
• Leaking from the phase space of the Riemann-Liouville fractional standard map
• Asymptotic cycles in fractional generalizations of multidimensional maps
• Characterizing and quantifying weak chaos in fractional dynamics