In the above discussion, the difference of temperature plays the role of a control parameter. We shall now discuss simple dynamical systems with a control parameter. In each cases, when the control parameter is increased beyond some critical value, some qualitative change occur in the evolution. The first example is the van der Pol equation.
The control parameter is a. For a negative, the system is ``trapped'' in an attractive fixed point. For a positive but not too large a limit cycle appears. The change when a becomes positive is called a Hopf bifurcation.
Figure 1: Some phase curves for the Van der Pol equation with a=1.