This article is an introduction and motivational contribution for the special issue “Nuclear Reactors and related nonlinear systems: Aspects of efficient modeling and stability analysis”. The authors aim to demonstrate the performance of the systematic bifurcation analysis for the stability analysis of nonlinear dynamic systems such as nuclear and thermal-hydraulic systems. In a first part, the motivation of this approach is explained in detail (part 1: Theory) and the authors provide some mathematical basics, which are necessary in order to understand the results of 3 examples of the application of bifurcation theory which will be discussed in more detail in part 2 (Applications, Progress in Nuclear Energy 113 (2019) 263-280). In this context, we would also like to point to the importance of modern methods of model order reduction (MOR) and the relationship between mathematically optimized reduced order models (ROMs) and simplified dynamical models. To represent the benefits of the bifurcation theory for our purposes compactly, already published results of earlier works were selected and partly new interpreted and revised on the advanced knowledge level. The conclusions are the result of work in this field in recent years and should be a basis of discussion for the future work of the community.