Abstract
Pyragas control is a widely used time-delayed feedback control for the stabilization of periodic orbits in dynamical systems. In this paper we investigate how we can use equivariance to eliminate restrictions of Pyragas control, both to select periodic orbits for stabilization by their spatio-temporal pattern and to render Pyragas control possible at all for those orbits. Another important aspect is the optimization of equivariant Pyragas control, i.e. to construct larger control regions. The ring of $n$ identical Stuart-Landau oscillators coupled diffusively in a bidirectional ring serves as our model.
Highlights
A successful method of time-delayed feedback control has been introduced in 1992 by Pyragas [10]
A network of n identical Stuart-Landau oscillators coupled in a bidirectional ring was investigated
whose symmetry group is given by Dn × S1
Summary
A successful method of time-delayed feedback control has been introduced in 1992 by Pyragas [10]. We use a modification of Pyragas control for stabilizing periodic orbits with prescribed spatio-temporal symmetries on networks. This modification has been discussed previously in [13, 9] for general equivariant systems near Hopf bifurcation. In first works on small networks consisting of two and three Stuart-Landau oscillators, such as [5, 12], it was shown that the control method (1) can stabilize unstable periodic orbit with prescribed symmetry near equivariant Hopf bifurcation. We want to apply equivariant Pyragas control (1) to a specific network type, which consists of n identical Stuart-Landau oscillators coupled in a ring.
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