Noise Rise in Josephson Parametric Amplifiers -- Dynamical Systems Approach Revisited

The ubiquitous noise rise in Josephson parametric amplifiers was first observed over 10 years ago. Numerous explanations of this phenomenon came forth, including a recent one, which showed that the effect could be accounted for as a universal feature of systems operated near certain types of instabilities including period doubling, symmetry breaking, and degenerate saddle node bifurcations. In this paper the analysis will be carried out for the Hopf bifurcation, a case which has significant differences from those studied previously.