| This dissertation investigates the use of time-delayed feedback control to stabilize unstable periodic orbits that arise via generic subcritical and equivariant Hopf bifurcations.;An n-dimensional system of differential equations containing a generic subcritical Hopf bifurcation from a stable equilibrium is studied. Pyragas-type feedback is added in the directions tangent to the two-dimensional center manifold of the system, and the feedback gain matrix is parameterized by a modulus and a phase angle. A linear stability analysis and center manifold reduction for delay differential equations is performed, and it is shown that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated. A bifurcation analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, Pyragas-type feedback successfully stabilizes the branch of small-amplitude unstable periodic orbits in a neighborhood of the original bifurcation point, provided that the feedback parameters satisfy certain restrictions.;The methods of Pyragas feedback control are then generalized to the situation where unstable periodic orbits arise in a D3 equivariant Hopf bifurcation. It is known that three branches of symmetry-breaking solutions emanate simultaneously from the trivial equilibrium, each with a particular group of spatio-temporal symmetries. The symmetries of these solutions are exploited in order to design non-invasive feedback controls that can select and stabilize the targeted solution branch, in the event that it bifurcates unstably. The feedback is applied in the directions tangent to the four-dimensional center manifold of the system. Where stabilization is possible, exact analytical expressions for the feedback gain parameters are determined. Numerical simulations, performed for a specific model of coupled oscillators, demonstrate the results. |
