Bifurcations Analysis for Beginners

We study a class of hybrid dynamical systems that arise from various fields of mathematical sciences. We provide a rigorous analytical framework for the con- struction of the model, including explicit solutions within orthants, analytical determination of switching times, and the derivation of a boundary-to-boundary return map governing the global dynamics.This work presents a systematic ana- lytical study of bifurcation phenomena arising in low- and moderate-dimensional dynamical systems with applications to biological regulation and switching pro- cesses. Starting from a general nonlinear system depending on a control pa- rameter, we develop a rigorous Taylor expansion framework that enables the precise identification of non-hyperbolic equilibria and the derivation of reduced normal forms. Particular attention is given to saddle-node, transcritical, and Hopf bifurcations, with explicit genericity conditions formulated in terms of higher-order derivatives. These conditions guarantee structural stability and codimension-one unfoldings, allowing biologically meaningful parameter inter- pretations.