Stabilization of period doubling bifurcation and implications for control of chaos

The stabilization of period doubling bifurcations for discrete-time nonlinear systems is investigated. It is shown that generically such bifurcations can be stabilized using smooth feedback, even if the linearized system is uncontrollable at criticality. In the course of the analysis, expressions are derived for bifurcation stability coefficients of general n-dimensional systems undergoing period doubling bifurcation. 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 attractor