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Increasing delay as a strategy to prove stability

Ziyad AlSharawi, Jose S. Cánovas

TL;DR

The paper develops an expansion strategy for discrete-time scalar systems, generating a sequence of higher-delay maps $F_m$ by forward substitutions from $x_{n+1}=F_0(x_n,\ldots,x_{n-k+1})$. It shows that local stability of a fixed point for the original system is equivalent to the existence of a finite $m$ with $\|\nabla F_m\|_1<1$, and in the linear case provides necessary and sufficient conditions via polynomials $p_m$ and $q_m$, with $p_m(x)=p_0(x)q_m(x)$ and $Spec(J_m)=Spec(J_0)\cup\{\text{zeros of } q_m\}$. The authors further develop an embedding-based framework to obtain global stability for nonlinear systems when suitable monotonicity holds, using a monotone map $G$ in an augmented space and ensuring the absence of pseudo-fixed points. Applications to Ricker- and Clark-type models illustrate both local improvements in stability regions and global attractivity results beyond classical Jury or Schur criteria. Overall, the work provides a cohesive theory that links delay expansion, spectral criteria, and embedding techniques to advance stability analysis of discrete-time dynamics.

Abstract

We consider difference equations of the form $x_{n+1}=F_0(x_n,\ldots,x_{n-k+1}),$ and increase the delay through a process of successive substitutions to obtain a sequence of systems $y_{n+1}=F_j(x_{n-j},\ldots,x_{n-k-j+1}),\; j=0,1,\ldots$. We call this process \emph{the expansion strategy} and use it to establish novel results that enable us to prove stability. When the map $F_0$ is sufficiently smooth and has a hyperbolic fixed point, we show the fixed point is locally asymptotically stable if and only if $\|\nabla F_j\|_1<1$ for some finite number $j$. Our local stability results complement recent results obtained on Schur stability, and they can provide an alternative to the highly acclaimed Jury's algorithm. Also, we show the effectiveness of the expansion strategy in obtaining global stability results. Global stability results are obtained by integrating the expansion strategy with the embedding technique. Finally, we give illustrative examples to show the results' practical applicability across various discrete-time dynamical systems.

Increasing delay as a strategy to prove stability

TL;DR

The paper develops an expansion strategy for discrete-time scalar systems, generating a sequence of higher-delay maps $F_m$ by forward substitutions from $x_{n+1}=F_0(x_n,\ldots,x_{n-k+1})$. It shows that local stability of a fixed point for the original system is equivalent to the existence of a finite $m$ with $\|\nabla F_m\|_1<1$, and in the linear case provides necessary and sufficient conditions via polynomials $p_m$ and $q_m$, with $p_m(x)=p_0(x)q_m(x)$ and $Spec(J_m)=Spec(J_0)\cup\{\text{zeros of } q_m\}$. The authors further develop an embedding-based framework to obtain global stability for nonlinear systems when suitable monotonicity holds, using a monotone map $G$ in an augmented space and ensuring the absence of pseudo-fixed points. Applications to Ricker- and Clark-type models illustrate both local improvements in stability regions and global attractivity results beyond classical Jury or Schur criteria. Overall, the work provides a cohesive theory that links delay expansion, spectral criteria, and embedding techniques to advance stability analysis of discrete-time dynamics.

Abstract

We consider difference equations of the form and increase the delay through a process of successive substitutions to obtain a sequence of systems . We call this process \emph{the expansion strategy} and use it to establish novel results that enable us to prove stability. When the map is sufficiently smooth and has a hyperbolic fixed point, we show the fixed point is locally asymptotically stable if and only if for some finite number . Our local stability results complement recent results obtained on Schur stability, and they can provide an alternative to the highly acclaimed Jury's algorithm. Also, we show the effectiveness of the expansion strategy in obtaining global stability results. Global stability results are obtained by integrating the expansion strategy with the embedding technique. Finally, we give illustrative examples to show the results' practical applicability across various discrete-time dynamical systems.
Paper Structure (10 sections, 17 theorems, 91 equations, 4 figures, 2 tables)

This paper contains 10 sections, 17 theorems, 91 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The expansion strategy
  3. The linear case
  4. Stability and constraints on coefficients
  5. The equilibria of the expanded systems
  6. Expansion and stability in nonlinear systems
  7. Applications
  8. Ricker type models
  9. Clark's type models
  10. Conclusion and discussion

Key Result

Proposition 2.1

A solution of Eq. Eq-F0 can be made a solution of Eq. Eq-TheGeneralExpansion, but the converse is not necessarily true. In particular, a fixed point of $F_0$ is a fixed point of $F_{m}$ for all $m\geq 0,$ but the converse is not necessarily true.

Figures (4)

  • Figure 1: This figure illustrates the eigenvalues of a $2\times 2$ matrix as they appear within the unit disk $\mathbb{D}$, and as captured by our expansion technique. The colored regions in Part (i) of this diagram illustrate how non-real eigenvalues are accounted for in our expansion method. The inner gray-colored region belongs to $\|V_0\|_1<1$. The cyan-shaded region corresponds to $\|V_1\|_1<1$. In contrast, the area within the green curves reflects the non-real eigenvalues captured by $\|V_2\|_1<1.$ Part (ii) of the figure is similar but represents the case of real eigenvalues.
  • Figure 2: Part (i) of this figure shows the stability regions in the $(-a_2,a_0)$-plane obtained from the inequalities in \ref{['In-V0']} to \ref{['In-V3']}. In particular, the grey shaded region belongs to $\|V_0\|_1<1,$ the cyan shaded region belongs to $\|V_1\|_1<1,$ and the third shaded region belongs to $\|V_2\|_1<1.$ Note that there is an overlap between the regions. The connected red curve represents the boundary of the local stability region obtained by the conditions in \ref{['In-LocalStability-Ricker']}. Part (ii) of the figure is similar but reflects the regions in the $(h,b)$-plane for the particular case $f(t)=e^{b-t}$. The region that belongs to $\|V_2\|_1<1$ in Part (ii) was left unshaded to avoid overcrowding.
  • Figure 3: Part (i) of this figure shows the stability we obtain based on our expansion strategy and the parameter values. It is done according to $\|V_k\|_1<1.$ The lower part of the region shrinks as we increase $k$ ($k=2,5,8,15$). Part (ii) of the figure shows the curves in which we obtain eigenvalues on the unit disk for the case $k=8$. In this case, the upper curve forms the lower boundary of the region (see Proposition 3 in El-Lo-Li2008). A comparison between the two graphs shows the effectiveness of our approach.
  • Figure 4: This figure shows the feasible region of the inequalities in \ref{['In1-Clark']}

Theorems & Definitions (39)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Example 3.3
  • ...and 29 more