CONTROL-BASED CONTINUATION OF PERIODIC ORBITS WITH A TIME-DELAYED DIFFERENCE SCHEME

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 periodic orbits that have at most one unstable Floquet multiplier sufficiently close to the unit circle. 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.