Nonlinear systems

In this chapter, we try to obtain deeper information about nonlinear systems by looking at the corresponding linearized system. We discuss different types of bifurcations such as saddle-node, transcritical, and pitchfork. Then we move on to consider the stability of systems by linearization by characterizing the stability of the zero solution with respect to the signs of the eigenvalues of the corresponding linearized system. Also, we study limit cycles that occur in planar autonomous systems by using Poincaré–Bendixson criterion. We provide applications related to the Lotka–Volterra competition model, SIR model, and Euler's buckling beam problem. We end the chapter by stating and proving different theorems on stable and unstable manifolds, including the Hartman–Grobman theorem.