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Dynamics of a class of time-period strongly 2-cooperative system: integer-valued Lyapunov function and embedding property of limit sets

Mengmeng Gao, Dun Zhou

Abstract

We construct an integer-valued Lyapunov function $σ(\cdot)$ for generalized negative cyclic feedback system; and prove that $σ(\cdot)$ on any $ω$-limit set which generated by Poincaré mapping of bounded solution of such strongly $2$-cooperative system is constant. Therefore, the $ω$-limit can be continuously embedded into a compact subset of a two-dimensional plane. Finally, a dissipative condition is given to ensure that all orbits of such system are bounded.

Dynamics of a class of time-period strongly 2-cooperative system: integer-valued Lyapunov function and embedding property of limit sets

Abstract

We construct an integer-valued Lyapunov function for generalized negative cyclic feedback system; and prove that on any -limit set which generated by Poincaré mapping of bounded solution of such strongly -cooperative system is constant. Therefore, the -limit can be continuously embedded into a compact subset of a two-dimensional plane. Finally, a dissipative condition is given to ensure that all orbits of such system are bounded.
Paper Structure (13 sections, 15 theorems, 139 equations)

This paper contains 13 sections, 15 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic concept in dynamical systems
  4. $k$-positive and $k$-cooperative systems
  5. A property of the natural number set
  6. Integer-valued Lyapunov function for a class of strongly 2-positive system
  7. Embedding property of limit sets
  8. Proof of Theorem \ref{['changzhixing-th']}
  9. Proof of Theorem \ref{['main']}
  10. Discussions
  11. Transform \ref{['system 1.1']}+\ref{['assume system 1.1']} into \ref{['system 1.1']}+\ref{['assume simple system 1.1']}
  12. Dissipative conditions
  13. $1$-cooperative and $2$-cooperative systems

Key Result

Lemma 2.1

Let $X$ be a $Baire$ space, and $(X,T)$ be a discrete dynamical system, then $T$ is transitive if and only if $T^{-1}$ is transitive.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1
  • ...and 28 more