Analytical treatment of the delayed feedback controlled Lorenz system close to a subcritical Hopf bifurcation

Abstract

We develop an analytical approach for the delayed feedback control of the Lorenz system close to a subcritical Hopf bifurcation. The periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique. We utilize a modification based on an unstable delayed feedback controller. The analytical approach employs the centre manifold theory, the near identity transformation, and averaging. We derive the characteristic equation for the Floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain. The analytical results are supported by numerical analysis of the original system of nonlinear differential-difference equations.
Keywords: chaos, dynamical systems, delayed feedback control, Lorenz system, subcritical Hopf bifurcation, centre manifold theory, near identity transformation, averaging
PACS: 05.45.Gg, 02.30.Yy, 02.30.Ks