Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation

Nonlinear problems involving phases occur ubiquitously throughout applied mathematics andphysics, ranging from neuronal models to the search for elementary particles. The phase variables present in such models usually enter as harmonic terms and, being unbounded, pose an open challenge for studying bifurcations in these systems through standard numerical continuation techniques. Here, we propose to transform and embed the original model equations involving phases into structurally stable generalized systems that are more suitable for analysis via standard predictor–corrector numerical continuation methods. The structural stability of the generalized system is achieved by replacing each harmonic term in the original system by a supercritical Hopf bifurcation normal form subsystem. As an illustration of this general approach, specific details are provided for the ac-driven, Stewart–McCumber model of a single Josephson junction. It is found that the dynamics of the junction is underpinned by a two-parameter Hopf–Hopf bifurcation, detected in the generalized system. The Hopf–Hopf bifurcation gives birth to an invariant torus through Neimark–Sacker bifurcation of limit cycles. Continuation of the Neimark–Sacker bifurcation of limit cycles in the two-parameter space provides a complete picture of the overlapping Arnold tongues (regions of frequency-locked periodic solutions), which are in precise agreement with the widths of the Shapiro steps that can be measured along the current–voltage characteristics of the junction at various fixed values of the ac-drive amplitude.