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On Fractional Generalizations of the Logistic Map and their Applications

Mark Edelman

TL;DR

The paper surveys fractional generalizations of the logistic map, focusing on Caputo-based fractional maps (FLM) and fractional difference maps (FDLM), and proposes a unified framework (GFLM) to capture power-law memory via Volterra-type kernels. It analyzes fixed points, asymptotic periodicity, stability, CBTT phenomena, and the convergence rates of trajectories, showing that asymptotic behavior preserves key logistic-map features while memory introduces rich finite-time dynamics and slow convergence. The work also documents Feigenbaum-like universality in fractional maps, derives exact and asymptotic bifurcation structures, and illustrates applications spanning biology, economics, memristors, image encryption, and even longevity modeling. Overall, the study demonstrates that memory-encoded fractional maps extend the classical logistic map with practically relevant dynamics, enabling refined modeling of real-world systems exhibiting power-law memory and cascade-type bifurcations. The results have implications for both theoretical understanding of nonlinear systems with memory and diverse applied domains where history-dependent dynamics are essential.

Abstract

The regular logistic map was introduced in 1960s, served as an example of a complex system, and was used as an instrument to demonstrate and investigate the period doubling cascade of bifurcations scenario of transition to chaos. In this paper, we review various fractional generalizations of the logistic map and their applications.

On Fractional Generalizations of the Logistic Map and their Applications

TL;DR

The paper surveys fractional generalizations of the logistic map, focusing on Caputo-based fractional maps (FLM) and fractional difference maps (FDLM), and proposes a unified framework (GFLM) to capture power-law memory via Volterra-type kernels. It analyzes fixed points, asymptotic periodicity, stability, CBTT phenomena, and the convergence rates of trajectories, showing that asymptotic behavior preserves key logistic-map features while memory introduces rich finite-time dynamics and slow convergence. The work also documents Feigenbaum-like universality in fractional maps, derives exact and asymptotic bifurcation structures, and illustrates applications spanning biology, economics, memristors, image encryption, and even longevity modeling. Overall, the study demonstrates that memory-encoded fractional maps extend the classical logistic map with practically relevant dynamics, enabling refined modeling of real-world systems exhibiting power-law memory and cascade-type bifurcations. The results have implications for both theoretical understanding of nonlinear systems with memory and diverse applied domains where history-dependent dynamics are essential.

Abstract

The regular logistic map was introduced in 1960s, served as an example of a complex system, and was used as an instrument to demonstrate and investigate the period doubling cascade of bifurcations scenario of transition to chaos. In this paper, we review various fractional generalizations of the logistic map and their applications.

Paper Structure

This paper contains 27 sections, 4 theorems, 102 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Regular Logistic Map
  3. One-dimensional generalizations
  4. Two- and three-dimensional logistic maps
  5. Defining the fractional logistic map
  6. First fractional logistic maps
  7. The fractional logistic map
  8. Fractional difference logistic map
  9. Generalized fractional logistic map
  10. Properties of the Fractional and Fractional Difference Logistic Maps
  11. Asymptotically periodic and bifurcation points
  12. Asymptotically periodic points in the GFLM of the orders $\alpha>0$
  13. Calculation of the sums $S^{m-1}_{j,l}$
  14. Poincaré plots
  15. Bifurcation points
  16. ...and 12 more sections

Key Result

theorem 1

For $\alpha \in \mathbb{R}$, $\alpha \ge 0$ the Caputo-like h-difference equation where $t\in (h\mathbb{N})_{m}$, with the initial conditions is equivalent to the map with $h$-factorial-law memory where $x_k=x(kh)$, which is called the $h$-difference Caputo universal $\alpha$-family of maps.

Figures (20)

  • Figure 1: (a) The bifurcation diagram for the logistic map $x_{n+1}=Kx_n(1-x_n)$. (b) The bifurcation diagram for the 1D standard map (the circle map with the zero driving phase) $x_{n+1}= x_n - K \sin (x_n)$, $({\rm mod} \ 2\pi )$. This figure is reprinted from ME6 with the permission of Springer Nature.
  • Figure 2: Bifurcations in the 2D Logistic Map: (a) $T=1$$\rightarrow$$T=2$ bifurcation at $K = 5$ ($K=5.05$ on the figure). (b) $T=8$$\rightarrow$$T=16$ bifurcation at $K \approx 5.5319$ ($K=5.53194$ on the figure). This figure is reprinted from ME6 with the permission from Springer Nature.
  • Figure 3: Asymptotic bifurcation K-$\alpha$ curves (Eqs. (\ref{['KcL']}) and (\ref{['Kcal']})) on which transition from a fixed point to a $T=2$ cycle occurs for fractional (upper curve) and fractional difference logistic maps ($h=1$). This figure is reprinted from Cycles with the permission of Springer Nature.
  • Figure 4: The Poincaré plot (500000 iterations) for fractional difference logistic map with $\alpha=0.75$, $K=3.2$, $h=1$, and the initial condition $x_0=0.3$. The asymptotically stable $T=2$ sink is marked by the stars and the unstable fixed point $(K-1)/K$ by the circle. This figure is reprinted from Cycles with the permission of Springer Nature.
  • Figure 5: The Poincaré plot (500000 iterations) for fractional difference logistic map with $\alpha=0.75$, $K=3.3$, $h=1$, and the initial condition $x_0=0.3$. Stable $T=4$ sink marked by the plus signs. The asymptotically unstable $T=2$ sink is marked by the stars and the unstable fixed point $(K-1)/K$ by the circle. This figure is reprinted from Cycles with the permission of Springer Nature.
  • ...and 15 more figures

Theorems & Definitions (4)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4