Modern theoretical methods for analysing the stability behaviour of Boiling Water Reactors (BWRs) are relatively reliable. The analysis is performed by comprehensive validated system codes comprising 3D core models and one-dimensional thermal-hydraulic parallel channel models in the frequency (linearized models) or time domain. Nevertheless the spontaneous emergence of stable or unstable periodic orbits as solutions of the coupled nonlinear differential equations determining the stability properties of the coupled thermal-hydraulic and neutron kinetic (highly) nonlinear BWR system is a surprising phenomenon, and it is worth thinking about the mathematical background controlling such behaviour. In particular the coexistence of different types of solutions, such as the coexistence of unstable limit cycles and stable fixed points, are states of stability, not all nuclear engineers are familiar with. Hence the part I of this paper is devoted to the mathematical background of linear and nonlinear stability analysis and introduces a novel efficient approach to treat the nonlinear BWR stability behaviour with both system codes and so-called (advanced) reduced order models (ROMs). The efficiency of this approach, called the RAM-ROM method, will be demonstrated by some results of stability analyses for different power plants.