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

Research output: Working paperWorking paper and Preprints

  • J Sieber
  • B Krauskopf
Original languageEnglish
StatePublished - 28 Feb 2006


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

Additional information

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

Research areas

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

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  • SK06

    Submitted manuscript, 488 KB, PDF-document


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