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Asymptotic behavior of solutions to some classes of multi-order fractional cooperative systems

L. V. Thinh, H. T. Tuan

Abstract

This paper is devoted to the study of the asymptotic behavior of solutions to multi-order fractional cooperative systems. First, we demonstrate the boundedness of solutions to fractional-order systems under certain conditions imposed on the vector field. We then prove the global attractivity and the convergence rate of solutions to such systems (in the case when the orders of fractional derivatives are equal, the convergence rate of solutions is sharp and optimal). To our knowledge, these kinds of results are new contributions to the qualitative theory of multi-order fractional positive systems and they seem to have been unknown before in the literature. As a consequence of this result, we obtain the convergence of solutions toward a non-trivial equilibrium point in an ecosystem model (a particular class of fractional-order Kolmogorov systems). Finally, some numerical examples are also provided to illustrate the obtained theoretical results.

Asymptotic behavior of solutions to some classes of multi-order fractional cooperative systems

Abstract

This paper is devoted to the study of the asymptotic behavior of solutions to multi-order fractional cooperative systems. First, we demonstrate the boundedness of solutions to fractional-order systems under certain conditions imposed on the vector field. We then prove the global attractivity and the convergence rate of solutions to such systems (in the case when the orders of fractional derivatives are equal, the convergence rate of solutions is sharp and optimal). To our knowledge, these kinds of results are new contributions to the qualitative theory of multi-order fractional positive systems and they seem to have been unknown before in the literature. As a consequence of this result, we obtain the convergence of solutions toward a non-trivial equilibrium point in an ecosystem model (a particular class of fractional-order Kolmogorov systems). Finally, some numerical examples are also provided to illustrate the obtained theoretical results.

Paper Structure

This paper contains 8 sections, 8 theorems, 75 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation
  4. Preliminaries
  5. Asymptotic behavior of solutions to fractional-order cooperative systems
  6. Boundedness and positivity of solutions to cooperative systems
  7. Global attractivity and convergence rate of solutions to cooperative systems
  8. Numerical examples

Key Result

Proposition 2.7

Mason Suppose that $f: \mathbb{R}^d \longrightarrow \mathbb{R}^d$ is continuous and is continuously differentiable on $\mathbb{R}^d\backslash \left\{0\right\}$. Moreover, this function is homogeneous. Then, there exists a positive constant $K$ such that $\|f(x)-f(y)\|\leq K\|x-y\|$, $\forall x,y\in

Figures (3)

  • Figure 1: Orbits of the solution to the system \ref{['Eq main1']} with the initial condition $\omega=( 0.7,0.2)^{\rm T}$.
  • Figure 2: Orbits of the solution to the system \ref{['Eq main2']} with the initial condition $\omega=( 0.5, 0.3, 0.8)^{\rm T}.$
  • Figure 3: Orbits of the solution to the system \ref{['Eq main3']} with the initial condition $\omega=(0.2, 2.3)^{\rm T}.$

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • ...and 19 more