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Asymptotically periodic and bifurcation points in fractional difference maps

Mark Edelman

TL;DR

The paper addresses the challenge that fractional difference maps with memory often lack conventional periodic solutions, focusing on asymptotically periodic and bifurcation points to construct asymptotic diagrams. It develops analytic expressions for the slowly converging coefficients $S_{p,l}$ that govern $l$-cycles, using a combinatorial identity to obtain explicit formulas for $S_{p,2n}$ and $S_{p,2n+1}$ in terms of $\Gamma$, $\zeta$, and trigonometric functions. These contributions yield a framework for fast, accurate computation of the coefficients, facilitating robust analysis of asymptotic bifurcations in maps with falling-factorial memory ($0<\alpha<1$) and extending to multidimensional cases. The work has practical impact on efficiently mapping asymptotic dynamics and supports ongoing explorations of universality, such as Feigenbaum-type behavior, in fractional systems.

Abstract

The first step in investigating fractional difference maps, which do not have periodic points except fixed points, is to find asymptotically periodic points and bifurcation points and draw asymptotic bifurcation diagrams. Recently derived equations that allow calculations of asymptotically periodic and bifurcation points contain coefficients defined as slowly converging infinite sums. In this paper we derive analytic expressions for coefficients of the equations that allow calculations of asymptotically periodic and bifurcation points in fractional difference maps.

Asymptotically periodic and bifurcation points in fractional difference maps

TL;DR

The paper addresses the challenge that fractional difference maps with memory often lack conventional periodic solutions, focusing on asymptotically periodic and bifurcation points to construct asymptotic diagrams. It develops analytic expressions for the slowly converging coefficients that govern -cycles, using a combinatorial identity to obtain explicit formulas for and in terms of , , and trigonometric functions. These contributions yield a framework for fast, accurate computation of the coefficients, facilitating robust analysis of asymptotic bifurcations in maps with falling-factorial memory () and extending to multidimensional cases. The work has practical impact on efficiently mapping asymptotic dynamics and supports ongoing explorations of universality, such as Feigenbaum-type behavior, in fractional systems.

Abstract

The first step in investigating fractional difference maps, which do not have periodic points except fixed points, is to find asymptotically periodic points and bifurcation points and draw asymptotic bifurcation diagrams. Recently derived equations that allow calculations of asymptotically periodic and bifurcation points contain coefficients defined as slowly converging infinite sums. In this paper we derive analytic expressions for coefficients of the equations that allow calculations of asymptotically periodic and bifurcation points in fractional difference maps.
Paper Structure (5 sections, 1 theorem, 24 equations)

This paper contains 5 sections, 1 theorem, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sums $S_{p,l}$ for $l$-cycles of fractional difference maps
  4. Numerical simulations and discussion
  5. Conclusion

Key Result

Theorem 2.1

The $T=2^{n-1}$ -- $T=2^{n}$ bifurcation points, $2^{n-1}$ values of $x_{2^{n-1}bif,i}$ with $0<i \le 2^{n-1}$ and the value of the nonlinear parameter $K_{2^{n-1}bif}$, of a fractional generalization of a nonlinear one-dimensional map $x_{n+1}=F_K(x_n)$ written as the Volterra difference equations where $G^0(x)=h^\alpha G_K(x)/\Gamma(\alpha)$, $x_0$ is the initial condition, $h$ is the time step

Theorems & Definitions (1)

  • Theorem 2.1