High speed rotor systems mounted on gas foil bearings present bifurcations which change the quality of stability, and may compromise the operability of rotating systems, or increase noise level when response amplitude drastically increases. The paper identifies the dissipating work in the gas film to be the source of self-excited motions driving the rotor whirling close to bearing's surface. The energy flow among the components of the system is evaluated for various design sets of bump foil properties, rotor stiffness and unbalance magnitude. The paper presents a methodology to retain the dissipating work at positive values during the periodic limit cycle motions caused by unbalance. An optimization technique is embedded in the pseudo arc length continuation of limit cycles, those evaluated (when exist) utilizing an orthogonal collocation method. The optimization scheme considers the bump foil stiffness and damping as the variables for which bifurcations do not appear in a certain speed range. It is found that Neimark-Sacker bifurcations, which trigger large limit cycle motions, do not exist in the unbalanced rotors when bump foil properties follow the optimization pattern. Period doubling bifurcations are possible to occur, without driving the rotor in high response amplitude. Different design sets of rotor stiffness and unbalance magnitude are investigated for the efficiency of the method to eliminate bifurcations. The quality of the optimization pattern allows optimization in real time, and bearing properties shift values during operation, eliminating bifurcations and allowing operation in higher speed margins.