Bifurcation

This chapter considers some structural behavior of mathematical models expressed as systems of ordinary differential equations. If changes in a parameter of the system result in a structurally different phase plane, then the system may undergo the phenomenon of bifurcation. In transcritical bifurcation, at least one critical point of the system changes the stability at a critical value of the bifurcation parameter. A saddle-node bifurcation corresponds to the situation in which a saddle node and a stable node of the system approach each other and as the bifurcation parameter passes through its critical value, these critical points collide and disappear. Pitchfork bifurcation corresponds to the situation in which a stable node becomes unstable when the bifurcation parameter passes through a critical value, throwing off a pair of stable critical points of the system. The chapter explores several types of solutions commonly analyzed in dynamical systems theory.