Abstract
We consider the problem of stabilization of unstable periodic solutions to autonomous systems by the non-invasive delayed feedback control known as Pyragas control method. The Odd Number Theorem imposes an important restriction upon the choice of the gain matrix by stating a necessary condition for stabilization. In this paper, the Odd Number Theorem is extended to equivariant systems. We assume that both the uncontrolled and controlled systems respect a group of symmetries. Two types of results are discussed. First, we consider rotationally symmetric systems for which the control stabilizes the whole orbit of relative periodic solutions that form an invariant two-dimensional torus in the phase space. Second, we consider a modification of the Pyragas control method that has been recently proposed for systems with a finite symmetry group. This control acts non-invasively on one selected periodic solution from the orbit and targets to stabilize this particular solution. Variants of the Odd Number Limitation Theorem are proposed for both above types of systems. The results are illustrated with examples that have been previously studied in the literature on Pyragas control including a system of two symmetrically coupled Stewart-Landau oscillators and a system of two coupled lasers.
Highlights
Stabilization of unstable periodic solutions is an important problem in applied nonlinear sciences
An elegant method suggested by Pyragas [10] is to introduce delayed feedback with the delay equal, or close, to the period T of the target unstable periodic solution x∗(t) to the uncontrolled system x (t) = f (t, x(t))
We extend the odd-number limitation type results considered in [7] to treat the case when control of the form (5) is applied to a system with a finite group of symmetries (Section 2); and, the case when the standard Pyragas control such as in (2) is applied to a target cycle, which is not hyperbolic, because the system is S1-symmetric (Section 3)
Summary
Stabilization of unstable periodic solutions is an important problem in applied nonlinear sciences. Stabilization of periodic orbits, Pyragas control, delayed feedback, S1-equivariance, finite symmetry group. The theorem provides necessary conditions on the control matrix K to allow stabilization of an unstable hyperbolic cycle x∗ of the autonomous system x (t) = f (x(t)).
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