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Analysis of the maps with variable fractional order

Prashant M. Gade, Sachin Bhalekar, Janardhan Chevala

TL;DR

This work investigates the stability of linear Caputo-type variable-order fractional difference maps with periodically varying order $\\alpha(t)$. By applying Z-transform techniques, it derives exact stability regions for period-2 maps—revealing a left boundary at $1-2^{\\min\\{\\alpha_1,\\alpha_2\\}}$—and analyzes period-3 maps, showing the left boundary closely matches $1-2^{(\\alpha_1+\\alpha_2+\\alpha_3)/3}$. Numerical experiments corroborate the analytic boundaries and reveal asymptotically periodic behavior within stable regions. For higher periods, the study finds that the mean order $\\langle\\alpha\\rangle$ provides a good stability proxy for odd $T$, while even $T$ do not exhibit a simple pattern, highlighting open questions about memory-changing dynamics in VO maps.

Abstract

Fractional order differential and difference equations are used to model systems with memory. Variable order fractional equations are proposed to model systems where the memory changes in time. We investigate stability conditions for linear variable order difference equations where the order is periodic function with period $T$. We give a general procedure for arbitrary $T$ and for $T=2$ and $T=3$, we give exact results. For $T=2$, we find that the lower order determines the stability of the equations. For odd $T$, numerical simulations indicate that we can approximately determine the stability of equations from the mean value of the variables.

Analysis of the maps with variable fractional order

TL;DR

This work investigates the stability of linear Caputo-type variable-order fractional difference maps with periodically varying order . By applying Z-transform techniques, it derives exact stability regions for period-2 maps—revealing a left boundary at —and analyzes period-3 maps, showing the left boundary closely matches . Numerical experiments corroborate the analytic boundaries and reveal asymptotically periodic behavior within stable regions. For higher periods, the study finds that the mean order provides a good stability proxy for odd , while even do not exhibit a simple pattern, highlighting open questions about memory-changing dynamics in VO maps.

Abstract

Fractional order differential and difference equations are used to model systems with memory. Variable order fractional equations are proposed to model systems where the memory changes in time. We investigate stability conditions for linear variable order difference equations where the order is periodic function with period . We give a general procedure for arbitrary and for and , we give exact results. For , we find that the lower order determines the stability of the equations. For odd , numerical simulations indicate that we can approximately determine the stability of equations from the mean value of the variables.

Paper Structure

This paper contains 10 sections, 2 theorems, 30 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maps with period-2 fractional order
  4. The Stable Region
  5. Corroboration of Analytic Results using Numerical Simulations
  6. Maps with period-3 fractional order
  7. The Characteristic Equation and Stable Region
  8. Validation of analytic results with Numerical Experiments
  9. Maps with higher periodic orders
  10. Conclusion

Key Result

Theorem 2.1

fulai2011existence The difference equation where $0<\alpha \leq 1$, $t \in \mathbb{N}_{1-\alpha}$, is equivalent to where $t \in \mathbb{N}_{0}$.

Figures (8)

  • Figure 1: Solutions of system (\ref{['3']}) with $\alpha_1=0.2$ and $\alpha_2=0.6$ for various values of $r$ inside and outside the stable region.
  • Figure 2: Oscillations of the system (\ref{['3']}) with $\alpha_1=0.3$ and $\alpha_2=0.9$ for various values of $r$ giving stable and unstable oscillations.
  • Figure 3: Analytical value left bound versus $1-2^{\frac{\alpha_1+\alpha_2+\alpha_3}{3}}$ for 1 lakh random triples $(\alpha_1,\alpha_2,\alpha_3)$ with $\alpha_j\in (0,1)$, $j=1,2,3$.
  • Figure 4: Behavior of the system (\ref{['3']}) with $\alpha_1=0.2$, $\alpha_2=0.4$ and $\alpha_3=0.6$ for different values of $r$ inside and outside the stable region.
  • Figure 5: Solutions of system (\ref{['3']}) with $\alpha_1=0.1$, $\alpha_2=0.5$ and $\alpha_3=0.9$ for different values of $r$.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.1
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 7 more