A study of the nonlinear behavior of growing density-wave oscillations is presented in the framework of a reduced-order model. Nonlinear effects are included in both the hydraulic and neutron kinetics equations, where both were found to contribute to the observed limit cycles. In this paper, Part I, the basic concepts were developed and applied to the global oscillation mode, where only the fundamental neutron flux mode excitation is considered. Approximate analytical solutions for the limit cycle amplitude and the time evolution of the transient were derived. In Part II, the model order is increased to allow the representation of the azimuthal neutron flux harmonic and the simulation of growing regional mode oscillations. Analysis demonstrates that the regional mode, unlike the global mode, may not always reach a stable limit cycle, and if it does, the regional limit cycle amplitudes are large compared with the global mode. An extended reduced-order model has been developed for use as an accurate quantitative tool for simulating actual reactor situations, whereas the current paradigm restricts the applicability of reduced-order models to gaining qualitative insights.