Recent works have introduced an effective technique to analyze nonlinear dynamics of a class LM of circuits containing ideal flux- or charge-controlled memristors and linear lossless elements (i.e. ideal capacitors and inductors). The technique, named Flux-Charge Analysis Method (FCAM), is based on analyzing the circuits in the flux-charge domain instead of the traditional voltage-current domain. Goal of this paper is to extend the FCAM to a larger class N of circuits containing also nonlinear capacitors and inductors. Nonlinear circuits with memristors and nonlinear lossless elements are widely used to several real nanoscale devices including the well-known Josephson junction. After deriving the constitutive relation in the flux-charge domain of each two-terminal element in N, the work focuses on a relevant subclass of N for which a state equation description can be obtained. State Equations (SE) formulation provides the fundamental basis for studying the chief features of the nonlinear dynamics: presence of invariant manifolds in autonomous circuits; coexistence of infinitely many different reduced-order dynamics on the manifolds; bifurcations due to changing of initial conditions for a fixed set of parameters, a.k.a. bifurcations without parameters.