Introduction to the Modeling and Analysis of Complex Systems

136 CHAPTER 8. BIFURCATIONS105x_eq051010 5 0 5 10rFigure 8.3: Bifurcation diagram of a transcritical bifurcation, obtained from Eq (8.8).equilibrium point split into three, two stable and one unstable. For example:dxdt = rx − x3 (8.10)This dynamical system has the following three equilibrium pointsx eq = 0, ± √ r, (8.11)but the last two exist only for r ≥ 0. You can show that x eq = 0 is stable for r < 0 andunstable for r > 0, while x eq = ± √ r are always stable if they exist. Its bifurcation diagramis shown in Fig. 8.4.In the meantime, a subcritical pitchfork bifurcation makes an unstable equilibrium pointsplit into three, two unstable and one stable. For example:dxdt = rx + x3 (8.12)This dynamical system has the following three equilibrium pointsx eq = 0, ± √ −r, (8.13)but the last two exist only for r ≤ 0. Its bifurcation diagram is shown in Fig. 8.5.