We saw in the previous chapter some criteria for a differential equation for two unknown functions of time to have a periodic solution. But we only proved that there were stable periodic solutions, and we did not discuss the speed of movement. This chapter discusses speed issues.

In particular, this chapter discusses a model system that swings rapidly between two or more approximate equilibrium positions. In this chapter, the point is not so much the model, but the way in which the rapid swings between different approximate equilibria occur. Indeed, the phenomenon of rapid switching between approximately constant states occurs in many different contexts, so it is important to realize that there are reasonably general properties of certain kinds of differential equations that guarantee this switching behavior. In particular, with the model discussed here, focus on the fact that two systems are interacting; one is fundamentally slow moving while the other can move much faster. The resulting dynamics of the interacting system sometimes mirror the motion of the slow system and sometimes the fast one.

The simple model discussed here describes the switching behavior for a muscle that controls a valve in the heart. I learned of the model and learned the analysis from Beltrami's book Mathematics for Dynamical Modeling. The model is much too simple to be biologically reasonable. However, we shall see that it exhibits some remarkably lifelike behavior.