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Integrating the enveloping technique with the expansion strategy to establish stability

Ziyad AlSharawi, Jose S. Cánovas

TL;DR

This paper addresses establishing global stability for multidimensional maps by reducing to a one-dimensional enveloping map using the expansion strategy and enveloping technique. It introduces strong local asymptotic stability (SLAS) and shows how expansions $F_j$ with $ig\| abla F_jig\\_1<1$ enable constructing a one-dimensional $g$ that inherits LAS from $F$, facilitating GAS for the original system. The authors develop a geometric enveloping framework in two dimensions, integrate it with embedding approaches, and provide case analyses for maps with mixed monotonicity ($F(,)$, $F(,)$, etc.), including criteria for no $2$-cycles and region-based envelopes. The work yields practical, verifiable global stability criteria in 2D with explicit constructions of $g$ and $$ maps, supported by illustrative examples and a conceptual mind map linking expansion, enveloping, and embedding. Overall, the approach broadens the applicability of enveloping to higher dimensions and offers a geometric toolkit for GAS verification in discrete-time dynamical systems.

Abstract

In this paper, we focus on finding one-dimensional maps that detect global stability in multidimensional maps. We consider various local and global stability techniques in discrete-time dynamical systems and discuss their advantages and limitations. Specifically, we navigate through the embedding technique, the expansion strategy, the dominance condition technique, and the enveloping technique to establish a unifying approach to global stability. We introduce the concept of strong local asymptotic stability (SLAS), then integrate what we call the expansion strategy with the enveloping technique to develop the enveloping technique for two-dimensional maps, which allows to give novel global stability results. Our results make it possible to verify global stability geometrically for two-dimensional maps. We provide several illustrative examples to elucidate our concepts, bolster our theory, and demonstrate its application.

Integrating the enveloping technique with the expansion strategy to establish stability

TL;DR

This paper addresses establishing global stability for multidimensional maps by reducing to a one-dimensional enveloping map using the expansion strategy and enveloping technique. It introduces strong local asymptotic stability (SLAS) and shows how expansions with enable constructing a one-dimensional that inherits LAS from , facilitating GAS for the original system. The authors develop a geometric enveloping framework in two dimensions, integrate it with embedding approaches, and provide case analyses for maps with mixed monotonicity (, , etc.), including criteria for no -cycles and region-based envelopes. The work yields practical, verifiable global stability criteria in 2D with explicit constructions of and maps, supported by illustrative examples and a conceptual mind map linking expansion, enveloping, and embedding. Overall, the approach broadens the applicability of enveloping to higher dimensions and offers a geometric toolkit for GAS verification in discrete-time dynamical systems.

Abstract

In this paper, we focus on finding one-dimensional maps that detect global stability in multidimensional maps. We consider various local and global stability techniques in discrete-time dynamical systems and discuss their advantages and limitations. Specifically, we navigate through the embedding technique, the expansion strategy, the dominance condition technique, and the enveloping technique to establish a unifying approach to global stability. We introduce the concept of strong local asymptotic stability (SLAS), then integrate what we call the expansion strategy with the enveloping technique to develop the enveloping technique for two-dimensional maps, which allows to give novel global stability results. Our results make it possible to verify global stability geometrically for two-dimensional maps. We provide several illustrative examples to elucidate our concepts, bolster our theory, and demonstrate its application.

Paper Structure

This paper contains 13 sections, 24 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Local stability and expansions
  3. The expansion strategy
  4. Local stability and one-dimensional maps
  5. Global stability techniques
  6. A mind map
  7. One-dimensional maps through enveloping
  8. Enveloping in contrast to embedding
  9. The case $\mathbf{F(\uparrow,\uparrow)}$
  10. The case $\mathbf{F(\downarrow,\uparrow)}$
  11. The case $\mathbf{F(\downarrow,\downarrow)}$
  12. The case $\mathbf{F(\uparrow,\downarrow)}$
  13. Conclusion

Key Result

Theorem 2.1

Al-Ca2024 Consider Eq. Eq-F-Normalized, and let $F_j$ be the sequence of expansions introduced in Eq. Eq-Fj. If $\{\bar{y}=1\}$ is a hyperbolic equilibrium solution, then it is LAS for $F_0$ iff $\|\nabla F_m\|_1<1$ for some non-negative integer $m.$

Figures (2)

  • Figure 1: This diagram is a cognitive map showcasing the ideas presented in this paper. In a prior study Al-Ca2024, the authors combined the expansion strategy and the embedding technique to achieve stability outcomes. This paper aims to complete the diagram, integrate the expansion strategy with the enveloping technique, and achieve new stability outcomes.
  • Figure 2: This figure illustrates the two scenarios of the curves $y=F_j(x,y)$ and $x=F_j(x,y)$. Both scenarios are possible, as we illustrate in our examples. $M_1$ and $M_2$ are defined in Eqs. \ref{['Eq-M1ANDM2']}. The shaded region represents the region $\mathcal{R}$ as defined in Eq. \ref{['Eq-RegionOfInterest']}. The dashed line $y=2\bar{x}-x$ is given to emphasize the importance of the slope $-1.$

Theorems & Definitions (53)

  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.1
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 43 more