Control of degenerate Hopf bifurcations in three-dimensional maps

A feedback control method is proposed to create a degenerate Hopf bifurcation in three-dimensional maps at a desired parameter point. The particularity of this bifurcation is that the system admits a stable fixed point inside a stable Hopf circle, between which an unstable Hopf circle resides. The interest of this solution structure is that the asymptotic behavior of the system can be switched between stationary and quasi-periodic motions by only tuning the initial state conditions. A set of critical and stability conditions for the degenerate Hopf bifurcation are discussed. The washout-filter-based controller with a polynomial control law is utilized. The control gains are derived from the theory of Chenciner's degenerate Hopf bifurcation with the aid of the center manifold reduction and the normal form evolution.