[Paper Review] Fractional Maps and Fractional Attractors. Part II: Fractional Difference $\alpha$-Families of Maps

Mark Edelman|arXiv (Cornell University)|Apr 19, 2014

Fractional Differential Equations Solutions18 references4 citations

TL;DR

This paper introduces fractional difference Caputo α-families of maps—Universal, Standard, and Logistic—using the Caputo fractional difference operator to model discrete dynamical systems with falling factorial-law memory, asymptotically equivalent to power-law memory. The key contribution is the derivation of explicit iterative maps that generalize classical maps to fractional memory, revealing similar bifurcation structures and attractor dynamics as continuous fractional maps, with qualitative and quantitative similarities in chaos, convergence, and non-uniqueness of solutions across α ∈ (0,2].

ABSTRACT

In this paper we extend the notion of an $\alpha$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic $\alpha$-Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.

Motivation & Objective

• To extend fractional map theory to discrete systems using fractional difference equations with the Caputo operator.
• To model discrete nonlinear systems with asymptotically power-law memory through falling factorial-law memory kernels.
• To establish fractional difference analogs of the Universal, Standard, and Logistic maps for studying general properties of systems with long-term memory.
• To compare the dynamical behavior of fractional difference maps with continuous fractional maps, focusing on bifurcations, attractors, and memory effects.
• To provide a foundation for analyzing general properties of discrete nonlinear systems with memory using α- and K-parameterized families.

Proposed method

• Derives fractional difference Caputo α-families using the left-sided Caputo fractional difference operator with N-th order derivatives.
• Applies the Caputo definition to integro-differential equations with periodic delta-kick forcing, leading to Volterra integral equations.
• Transforms the integral equations into explicit iterative maps using falling factorial functions t(α) = Γ(t+1)/Γ(t+1−α), which asymptotically behave as power laws.
• Derives the Caputo Universal αFM as a general form, from which Standard and Logistic αFMs are obtained by substituting specific nonlinear functions GK(x).
• Expresses the maps in terms of memory-weighted sums involving Gamma functions, with weights (n−k+1)α−s−1 for s = 0,1,…,N−1.
• Reformulates the maps as 2D maps with memory by introducing conjugate variables (e.g., pn = ∆xn−1), enabling phase-space analysis and bifurcation diagrams.

Experimental results

Research questions

• RQ1 How can fractional difference equations with the Caputo operator be used to define discrete maps with long-term memory?
• RQ2 What are the explicit forms of the fractional difference Caputo Universal, Standard, and Logistic α-families of maps?
• RQ3 How do the dynamical properties (bifurcations, attractors, convergence) of fractional difference maps compare to those of continuous fractional maps?
• RQ4 What role does the memory parameter α play in shaping the bifurcation structure and chaotic behavior of these maps?
• RQ5 How do the falling factorial-law memory weights differ from ideal power-law memory, and what is their impact on trajectory non-uniqueness and attractor overlap?

Key findings

• The fractional difference Caputo Standard αFM reduces to the standard map at α = 2 and to a circle map with zero phase at α = 0, with intermediate values showing memory-dependent dynamics.
• For 0 < α < 1, the Caputo Standard αFM is given by xn = x0 − (K / Γ(α)) Σ_{k=0}^{n−1} sin(xk)(n−k)^{1−α} mod 2π, showing power-law memory effects.
• The 1 < α < 2 fractional difference Standard αFM is expressed as xn+1 = x0 + ∆x0(n+1) − (K / Γ(α)) Σ_{s=0}^{n−1} [Γ(n−s+α−1)/Γ(n−s)] sin(xs+1) mod 2π.
• The α = 1 fractional difference Logistic αFM is identical to the classical 1D Logistic map, confirming consistency with integer-order dynamics.
• Bifurcation diagrams (Figs. 1–3) show that both Caputo and fractional difference maps exhibit similar structures, including period-doubling cascades, chaos, and stable fixed points, with Kc values shifting with α.
• Trajectory analysis (Figs. 4–5) reveals cascade of bifurcations and intermittent behaviors in both continuous and discrete fractional maps, with qualitative similarity across α, though differences in memory weight distribution are significant for α ∈ (0,1), especially as α → 0+.