Robust control of period doubling bifurcations and implications for control of chaos

The stabilization of period doubling bifurcations for discrete-time nonlinear systems is investigated. Both static and dynamic feedback controllers have been studied. It is shown that generically such bifurcations can be stabilized using smooth feedback, even if the linearized system is uncontrollable at criticality. A connection is determined between control of the amplitude of a period doubled orbit and the elimination of a period doubling cascade to chaos. For illustration, the results are applied to the Henon system