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Study of the stability of the fractional Stokes system from nonlinear optics around the zero equlibrium state

Mihai Ivan

TL;DR

The paper investigates the stability of a Caputo-fractional Stokes system modeling a single Stokes pulse in nonlinear optics, formulating the dynamics on $so(3)^{*}$ and identifying four physically inequivalent fractional types. It analyzes asymptotic stability of the zero equilibrium using Matignon's criterion and develops linear-control strategies to stabilize unstable equilibria, providing type-specific gain conditions and $q$-dependent stability results. A Fractional Euler method is derived and shown to effectively integrate the controlled systems, with numerical demonstrations (e.g., Type-4 at $q=0.48$) validating the theoretical stability and control results. The work contributes a rigorous framework for chaos control and stabilization in fractional optical systems, with implications for nonlinear optics and fractional dynamics.

Abstract

The main purpose of this paper is to study the fractional-order system with Caputo derivative associated to single Stokes pulse. The dynamic behavior for this fractional model (called the fractional Stokes system) is investigated, including: the asymptotic stability around zero equilibrium state, the stabilization problem using appropriate linear controls and the numerical integration based on fractional Euler method.

Study of the stability of the fractional Stokes system from nonlinear optics around the zero equlibrium state

TL;DR

The paper investigates the stability of a Caputo-fractional Stokes system modeling a single Stokes pulse in nonlinear optics, formulating the dynamics on and identifying four physically inequivalent fractional types. It analyzes asymptotic stability of the zero equilibrium using Matignon's criterion and develops linear-control strategies to stabilize unstable equilibria, providing type-specific gain conditions and -dependent stability results. A Fractional Euler method is derived and shown to effectively integrate the controlled systems, with numerical demonstrations (e.g., Type-4 at ) validating the theoretical stability and control results. The work contributes a rigorous framework for chaos control and stabilization in fractional optical systems, with implications for nonlinear optics and fractional dynamics.

Abstract

The main purpose of this paper is to study the fractional-order system with Caputo derivative associated to single Stokes pulse. The dynamic behavior for this fractional model (called the fractional Stokes system) is investigated, including: the asymptotic stability around zero equilibrium state, the stabilization problem using appropriate linear controls and the numerical integration based on fractional Euler method.

Paper Structure

This paper contains 5 sections, 4 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Some theoretical aspects on the single pulse Stokes system
  3. The fractional Stokes system
  4. Asymptotic stability of the fractional Stokes system $(3.5)$
  5. Numerical integration of the fractional Stokes system with controls $(4.1)$

Key Result

Lemma 3.1

(giva22)Let $~x_{e}~$ be an equilibrium state of the fractional model $~(3.1)~$ and $~\lambda_{i},~i=\overline{1,n}~$ the eigenvalues of $~J(x_{e}).$ If $~\lambda_{i} < 0,$ for all $~i=\overline{1,n},~$ then $~x_{e}~$ is asymptotically stable $~(\forall)~q\in (0,1).$

Theorems & Definitions (4)

  • Lemma 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3