Mathematical Modelling of Natural Phenomena

Table of Contents - 2012 - Volume 7, Issue 02 (Solitary waves)  

Research Article

Low-Dimensional Description of Pulses under the Action of Global Feedback Control

Y. Kanevsky c1 and A. A. Nepomnyashchy

Department of Mathematics, Technion – Israel Institute of Technology Haifa, 32000, Israel

Abstract

The influence of a global delayed feedback control which acts on a system governed by a subcritical complex Ginzburg-Landau equation is considered. The method based on a variational principle is applied for the derivation of low-dimensional evolution models. In the framework of those models, one-pulse and two-pulse solutions are found, and their linear stability analysis is carried out. The application of the finite-dimensional model allows to reveal the existence of chaotic oscillatory regimes and regimes with double-period and quadruple-period oscillations. The diagram of regimes resembles those found in the damped-driven nonlinear Schrödinger equation. The obtained results are compared with the results of direct numerical simulations of the original problem.

(Online publication February 29 2012)

Key Words:

  • Ginzburg-Landau equation;
  • delayed feedback control;
  • finite-dimensional models;
  • solitary waves

Mathematics Subject Classification:

  • 35B36;
  • 93B52

Correspondence:

c1 Corresponding author. E-mail: yuliyakanevsky@gmail.com