This paper presents a simple Josephson-junction circuit with two parameters (inductance and capacitance) which can be tuned to represent different energy landscapes with different physical properties. By tuning this quantum circuit through external accessible elements we can move from two to three and more energy levels depending on the parameter setting. The inductance, the capacitance as well as the external voltage (driving terms) condition the number of relevant energy levels as well as the model to be used.

In this paper, the equivalent circuit of the non-autonomous Josephson junction (JJ) is presented and the effect of the proper frequency on the phase φ is studied. We also study nonlinear resonance phenomena in the oscillations of a modified Josephson junction (MJJ). These oscillations are probed through a system of nonlinear differential equations and the multiple time scale method is employed to investigate all different types of resonance that occur. The results of primary, superharmonic and subharmonic resonances are obtained analytically. We show that the system exhibits hardening and softening behaviors, as well as hysteresis and amplitude hopping phenomena in primary and superharmonic resonances, and only the hysteresis phenomenon in subharmonic resonance. In addition, the stabilities and the steady state solutions in each type of resonances are kindly evaluated. The number of equilibrium points that evolve with time and their stabilities are also studied. Finally, the equations of motion are numerically integrated to check the correctness of analytical calculations. We further show that the dynamics of the MJJ is strongly influenced by its parameters.