Bifurcations in Two-Dimensional Systems

In the previous chapters, we observed that varying a single control parameter led to three local bifurcations in one-dimensional systems, which allowed us to establish their normal forms and analyze their behavior. This approach is not limited to one dimension; it also applies to two-dimensional systems. However, in two-dimensional systems, changing a single control parameter results in a different type of local bifurcation, known as the Hopf bifurcation. With this in mind, we proceed as follows. First, we introduce a simple nonlinear model of a two-dimensional system of codimension one, in which periodic bifurcations may occur. Second, we present the general conditions for its occurrence in dynamical systems of dimension \(n \ge 2\) . Third, we show the normal form of the Hopf bifurcation. Lastly, we explore the classification that distinguishes between the supercritical and subcritical cases.