Control-based continuation of periodic orbits with a time-delayed difference scheme

J Sieber, B Krauskopf

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Abstract

This paper presents a method that is able to continue periodic orbits in systems where only output of the evolution over a given time period is available, which is the typical situation in an experiment. The starting point of our paper is an analysis of time-delayed feedback control, a method to stabilize periodic orbits experimentally that is popular among physicists. We show that the well-known topological limitations of this method can be overcome by an embedding into a pseudo-arclength continuation and prove that embedded time-delayed feedback control is able to stabilize weakly unstable periodic orbits. In the second part we introduce preconditioning into the time-delayed feedback control. In this way we extract a nonlinear system of equations from time profiles, which we solve using Newton iterations. We demonstrate the feasibility of our method by continuing periodic orbits in a laser model through folds, and by computing the family of canard orbits of the classical stiff Van der Pol system with constant forcing
Original languageEnglish
Publication statusPublished - 28 Feb 2006

Bibliographical note

Sponsorship: The research of J.S. is supported by EPSRC grant GR/R72020/01

Keywords

  • canard
  • time-delayed feedback
  • periodic orbit
  • continuation method

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