RESUMEN
Pyragas time-delayed feedback is a control scheme designed to stabilize unstable periodic orbits, which occur naturally in many nonlinear dynamical systems. It has been successfully implemented in a number of applications, including lasers and chemical systems. The control scheme targets a specific unstable periodic orbit by adding a feedback term with a delay chosen as the period of the unstable periodic orbit. However, in an experimental or industrial environment, obtaining the exact period or setting the delay equal to the exact period of the target periodic orbit may be difficult. This could be due to a number of factors, such as incomplete information on the system or the delay being set by inaccurate equipment. In this paper, we evaluate the effect of Pyragas control on the prototypical generic subcritical Hopf normal form when the delay is close to but not equal to the period of the target periodic orbit. Specifically, we consider two cases: first, a constant, and second, a linear approximation of the period. We compare these two cases to the case where the delay is set exactly to the target period, which serves as the benchmark case. For this comparison, we construct bifurcation diagrams and determine any regions where a stable periodic orbit close to the target is stabilized by the control scheme. In this way, we find that at least a linear approximation of the period is required for successful stabilization by Pyragas control.
RESUMEN
Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.
