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Josephson junction oscillators can generate chaotic signals with a wide frequency spectrum. An improved scheme of Lyapunov functions is proposed to control chaotic resonators of this type and forces them to converge to an arbitrary selected target signal. A changeable gain coefficient is introduced into the Lyapunov function, and the controllers are designed analytically. The controllers operate automatically when the output series are deviated from the target orbit synchronously. A resistive-capacitive-inductive-shunted Josephson junction in chaotic parameter region is investigated in our studies, and power consumption is estimated from the dimensionless model. It is found that the power consumption of controller is dependent on the amplitude and/or angular frequency of the external target signal to be tracked. For example, larger power costs are observed when the target signal is in larger amplitude and/or angular frequency. The numerical results are consistent with the analytical discussion. © 2014 Wiley Periodicals, Inc. Complexity 21: 370-378, 2015 [ABSTRACT FROM AUTHOR]