TY - JOUR

T1 - Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

AU - Brown, G.

AU - Postlethwaite, C. M.

AU - Silber, M.

N1 - Funding Information:
The authors thank Luis Mier-y-Terán-Romero for extensive discussions of center manifold reduction for delay differential equations and for assistance with dde-biftool [47] . They also thank Sue Ann Campbell for useful discussion of bifurcation analysis of delay differential equations and for sharing some of her Maple™code. G.B. acknowledges support from NSF-RTG Grant ( DMS-0636574 ) and the ARCS Foundation of Chicago. She is grateful for the hospitality of the University of Auckland Mathematics Department during her visit as an NSF-EAPSI fellow. C.M.P. acknowledges support from the University of Auckland Research Committee , and M.S. from the National Science Foundation ( DMS-0709232 ).

PY - 2011/4/15

Y1 - 2011/4/15

N2 - We show that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [B. Fiedler, V. Flunkert, M. Georgi, P. Hvel, E. Schll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett. 98(11) (2007) 114101], who demonstrated that such a feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. The Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, 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: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, the Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the feedback phase angle satisfies a certain restriction.

AB - We show that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [B. Fiedler, V. Flunkert, M. Georgi, P. Hvel, E. Schll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett. 98(11) (2007) 114101], who demonstrated that such a feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. The Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, 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: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, the Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the feedback phase angle satisfies a certain restriction.

KW - Degenerate Hopf bifurcation

KW - Delay differential equations

KW - Pyragas control

KW - Subcritical Hopf bifurcation

KW - Time-delayed feedback

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U2 - 10.1016/j.physd.2010.12.011

DO - 10.1016/j.physd.2010.12.011

M3 - Article

AN - SCOPUS:79952439092

SN - 0167-2789

VL - 240

SP - 859

EP - 871

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 9-10

ER -