Introduction to the Modeling and Analysis of Complex Systems

8.2. BIFURCATIONS IN 1-D CONTINUOUS-TIME MODELS 135clear that the system’s state is converging to the stable equilibrium point while it is repellingfrom the unstable one. If you do the same for several different values of r, youobtain Fig. 8.2, which shows how to interpret this diagram.Figure 8.2: How to interpret a bifurcation diagram. Each vertical slice of the diagramdepicts a phase space of the system for a particular parameter value.There are other types of bifurcations. A transcritical bifurcation is a bifurcation whereone equilibrium point “passes through” another, exchanging their stabilities. For example:dxdt = rx − x2 (8.8)This dynamical system always has the following two equilibrium pointsx eq = 0, r, (8.9)with the exception that they collide when r = 0, which is when they swap their stabilities.Its bifurcation diagram is shown in Fig. 8.3.Another one is a pitchfork bifurcation, where an equilibrium point splits into three. Twoof these (the outermost two) have the same stability as the original equilibrium point,while the one between them has a stability opposite to the original one’s stability. Thereare two types of pitchfork bifurcations. A supercritical pitchfork bifurcation makes a stable