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Attractors in $k$-dimensional discrete systems of mixed monotonicity

Ziyad AlSharawi, Jose S. Cánovas, Sadok Kallel

Abstract

We consider $k$-dimensional discrete-time systems of the form $x_{n+1}=F(x_n,\ldots,x_{n-k+1})$ in which the map $F$ is continuous and monotonic in each one of its arguments. We define a partial order on $\mathbb{R}^{2k}_+$, compatible with the monotonicity of $F$, and then use it to embed the $k$-dimensional system into a $2k$-dimensional system that is monotonic with respect to this poset structure. An analogous construction is given for periodic systems. Using the characteristics of the higher-dimensional monotonic system, global stability results are obtained for the original system. Our results apply to a large class of difference equations that are pertinent in a variety of contexts. As an application of the developed theory, we provide two examples that cover a wide class of difference equations, and in a subsequent paper, we provide additional applications of general interest.

Attractors in $k$-dimensional discrete systems of mixed monotonicity

Abstract

We consider -dimensional discrete-time systems of the form in which the map is continuous and monotonic in each one of its arguments. We define a partial order on , compatible with the monotonicity of , and then use it to embed the -dimensional system into a -dimensional system that is monotonic with respect to this poset structure. An analogous construction is given for periodic systems. Using the characteristics of the higher-dimensional monotonic system, global stability results are obtained for the original system. Our results apply to a large class of difference equations that are pertinent in a variety of contexts. As an application of the developed theory, we provide two examples that cover a wide class of difference equations, and in a subsequent paper, we provide additional applications of general interest.
Paper Structure (10 sections, 16 theorems, 66 equations, 3 figures)

This paper contains 10 sections, 16 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Embedding discrete-time systems
  3. Trapping box
  4. Monotonic sequences under the new partial orders
  5. Fixed points and global attractors
  6. Embedding periodic systems
  7. Applications
  8. The Ricker model with delays and stocking
  9. Rational difference equations
  10. Conclusion

Key Result

Lemma 2.3

(Monotone Convergence) Let $V$ be a closed subset of ${\mathbb R}$ (eg. $V={\mathbb R},{\mathbb R}^+$ or $V=[a,b]$), and consider on $V^k$ any monotone ordering $\tau$. Suppose $G$ is increasing (resp. decreasing) with respect to the partial order $\tau$ and is bounded above (resp. below) in $\tau$.

Figures (3)

  • Figure 1: The "Cobweb convergence" of a sequence $x_{n+1}=f(x_n)$, with $f$ decreasing. The sequence is embedded along the diagonal in the form $(x_n,x_n)$. When $f$ has no cycles of length two, the converging spiral illustrates "box convergence", which we develop for higher-dimensional maps (see §\ref{['embeddingtheory']}).
  • Figure 2: This figure shows the stability regions in the $(h,r)-$plane for several choices of the delay. The main curve is the bottom red curve representing $r=r_\infty,$ which shows the global stability region obtained by our theory. The curves from top to bottom are as follows: The curve $r=r_0$ represents the boundary of the local and global stability regions when no delay is involved in the model, i.e., $k=0.$ The curve $r=r_1$ represents the boundary of the local stability region when the delay is $k=1.$ The curve $r=r_2$ represents the boundary of the local stability region when the delay is $k=2.$ The curve $r=r_2$ can be found explicitly. The curve $r=r_3$ represents the boundary of the local stability region when the delay is $k=3.$ This curve is found numerically. The curve $r=r_\infty$ represents the boundary of the global stability region that is found based on our theory for any finite value of $k.$
  • Figure 3: This figure shows the feasible region of the inequalities in \ref{['In-ArtificialFixed1']} when $x<y.$ The functions $L$ and $q_2$ are given in Eq. \ref{['Eq-FixedPoints']} and Eq. \ref{['Eq-q2']}, respectively.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Remark 2.1
  • ...and 30 more