Stability Analysis

This chapter provides a systematic introduction to stability analysis, a fundamental tool in understanding the qualitative behavior of dynamical systems. It begins with the study of one-dimensional flows, where equilibrium points and their local stability are determined through graphical and analytical techniques. The analysis is then extended to higher-dimensional systems. Linear systems, whose stability is characterized by the eigenvalues of the system matrix, and nonlinear systems, where local behavior near equilibria is examined via linearization, are studied locally and globally. Hopf bifurcation and limit cycles are also explored to illustrate how changes in system parameters can give rise to self-sustained oscillations. The phenomenon of hysteresis is examined through the quenching reaction, demonstrating how nonlinear feedback can induce multiple equilibria and abrupt transitions between stable states.