  Flux-biased loops including one or more Josephson junctions are ubiquitous
elements in quantum information experiments based on superconducting circuits.
These quantum circuits can be tuned to implement a variety of Hamiltonians,
with applications ranging from decoherence-protected qubits to quantum limited
converters and amplifiers. The extrema of the Hamiltonian of these circuits are
of special interest because they govern their low-energy dynamics. However, the
theory of superconducting quantum circuits so far lacks a systematic method to
find these extrema and compute the series expansion of the Hamiltonian in their
vicinity for an arbitrary nonlinear superconducting circuit. We present such a
method, which can aid the synthesis of new functionalities in quantum devices.
