Many applications are modeled by autonomous systems of differential equations that contain parameters. As these parameters change, the stylized phase portraits of the differential equations may also change; parameter values where these changes occur are called bifurcation values. In this chapter we discuss how bifurcations occur. To frame the discussion we introduce two systems of differential equations — the Volterra-Lotka equations modeling the population evolution of two species (Section ??) and the CSTR equations modeling a single exothermic (or heat producing) chemical reaction (Section ??). In these models, we use scaling to identify the essential parameters, and we illustrate changes that can take place in phase portraits as a parameter is varied. The information concerning changes in phase portraits is summarized in bifurcation diagrams, which are introduced by simple examples in Section ??. Bifurcation diagrams are used in Section ?? to summarize the results of numerical explorations on the CSTR.

In Chapter ?? we showed that stylized phase portraits of planar Morse-Smale systems can often be drawn by a combination of analysis and computer. In this chapter we observe that bifurcations occur at parameter values where phase portraits of systems of differential equations are not Morse-Smale. There are three ways that an autonomous planar system of differential equations can fail to be Morse-Smale.

  • There is a nonhyperbolic equilibrium.
  • There is a periodic solution that is not a limit cycle.
  • There is a trajectory that connects a saddle to itself or that connects two different saddles.

Typically, in a system of differential equations that depends on a single parameter , there are isolated values in where the differential equation fails to be Morse-Smale and at these bifurcation values the phase portraits do actually change.

Section ?? discusses the typical bifurcations associated with nonhyperbolic equilibria: saddle-node bifurcations (where two equilibria are created) and Hopf bifurcations (where limit cycles are created). These bifurcations are called local bifurcations as the changes in the phase portraits occur in a small neighborhood of the nonhyperbolic equilibrium.

We also describe typical ways in which the remaining two failures in Morse-Smale occur. Typically, when periodic solutions are not limit cycles, two periodic solutions collide and disappear (Section ??). Other global bifurcations occur when there are saddle-saddle connections. Homoclinic bifurcations occur when there is a connection from a saddle point to itself — typically such bifurcations occur when a limit cycle disappears (Section ??). Heteroclinic bifurcations occur when a trajectory connects two different saddles (Section ??).

Additional details concerning saddle-node and Hopf bifurcations are given in Sections ?? and ??.