This article presents new two-tier harmonic-balance methodologies for calculating all coexisting periodic solutions of nonlinear circuits, including multivalued branches, disconnected curves, and closed loops missed by conventional approaches. Harmonic balance provides the inner computational tier, while driven or equilibrium control functions form the outer tier according to circuit operation. Hybrid formulations combine HB-extracted oscillator models with external networks for efficient system-level analysis and optimization. Six applications demonstrate the methods: near-field wireless power transfer with a nonlinear secondary resonator; a dual-band injection-locked oscillator sensor; high-order subharmonic injection locking for phase-noise reduction; a nonreciprocal isolator with two nonlinear resonators; wireless power transfer resonator optimization; and tag-to-reader communication using two wirelessly coupled oscillators.