Attractors in k-Dimensional Discrete Systems of Mixed Monotonicity

• Published in: Qualitative theory of dynamical systems Vol. 23; no. Suppl 1
• Main Authors: AlSharawi, Ziyad, Cánovas, Jose S., Kallel, Sadok
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Difference and Functional Equations; Difference equations; Dimensional stability; Discrete systems; Discrete time systems; Dynamical Systems and Ergodic Theory; Mathematics; Mathematics and Statistics; 39A30; Rational difference equations; Ricker model; Global stability; Periodic solutions; 37N25; Embedding; 39A60; Local stability
• DOI: 10.1007/s12346-024-01123-8
• EISSN: 1662-3592
• ISSN: 1575-5460

We consider
k
-dimensional discrete-time systems of the form
x
n
+
1
=
F
(
x
n
,
…
,
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
R
+
2
k
, compatible with the monotonicity of
F
, and then use it to embed the
k
-dimensional system into a 2
k
-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.