Towards Stabilization of Odd-Number Orbits in Experiments

Abstract

Although the counterexample provided in Chap. 3 has fulfilled its purpose to refute the alleged odd-number limitation, it is often difficult to realize in experiments, in order to stabilize odd-number orbits, and in particular subcritical Hopf orbits. One reason why the counterexample is not immediately applicable, is the special choice of the gain matrix, i.e., a feedback term, which only involves z and not the complex conjugate $\overline{z}$. This gain matrix conserves the $S^1$-symmetry of the normal form, but in order to realize this control matrix experimentally one needs to have access to two dynamical variables in the rotation plane of the orbit, process these to generate the rotation phase $\beta$, and feed the control signal back into the corresponding two dynamic degrees of freedom. This may be possible in certain situations, for instance, when stabilizing an unstable mode of a laser, where the optical phase can naturally introduce a rotation "as reported by (Fiedler et al. Phys. Rev. E 77:066207, 2008)" (see Sect. 7.1). But what happens, for example, if we have only access to one dynamical variable? We will give some answers to this question in the following "as reported by (Flunkert and Schöll. Towards stabilization of odd-number orbits in experiments, in preparation, 2011)".