  Josephson parametric amplifiers (JPA) have become key devices in quantum
science and technology with superconducting circuits. In particular, they can
be utilized as quantum-limited amplifiers or as a source of squeezed microwave
fields. Here, we report on the detailed measurements of five flux-driven JPAs,
three of them exhibiting a hysteretic dependence of the resonant frequency
versus the applied magnetic flux. We model the measured characteristics by
numerical simulations based on the two-dimensional potential landscape of the
dc superconducting quantum interference devices (dc-SQUID), which provide the
JPA nonlinearity, for a finite screening parameter $\beta_\mathrm{L}\,{>}\,0$
and demonstrate excellent agreement between the numerical results and the
experimental data. Furthermore, we study the nondegenerate response of
different JPAs and accurately describe the experimental results with our
theory.
