filters, and DC to DC transformers that make use of periodic switching to transform the voltage available from some supply, such as a battery, to the voltage required by the transistor or motor that is being powered. In many cases, the explanation of the behavior of these types of examples is much more subtle than the explanation of ordinary linear regulation. In this paper we use examples from various domains to illustrate the mathematical ideas that lie at the heart of these problems.

Perhaps the most fundamental arguments as to why periodic processes are required to produce nonperiodic effects seem to be based on considerations of kinematics and force amplification. Animals and automobiles need to cover distances that far exceed the longest linear dimension in their makeup. They cannot simply reconfigure their bodies to cover the distances involved. Moreover, muscles, magnets, and expanding gases can only generate their significant force over a limited range of displacements. Having generated a force over this range, it is necessary to reconfigure before being able to generate the same level of force again. Among the possible temporal patterns of reconfiguration, some are more effective than others. Having found an effective one, it can be repeated over and over, giving a cyclic process that enables the coverage of large distances by means of repeated short distance movements.

Within this overall paradigm there is a further important distinction to be made. Certain periodic processes operate with a fixed amplitude, piston engines being a good example. Other periodic processes, such as the motion of an inchworm and the swimming motion of a fish, can operate at a variety of amplitudes. In the case of variable amplitude devices it may happen that the mechanical advantage increases as the amplitude decreases. Theories dealing with nonlinear controllability provide considerable insight into the capabilities of systems of this latter type. Understanding the dynamics of their regulatory processes requires more study, and only recently has there been an appropriate mathematical formulation of a control problem in which pattern generation plays a decisive role. We touch on this in our final section.

ORDER SOMETIMES MATTERS

In choosing from various possible actions that one may take, it sometimes happens that a particular set of actions applied in one order has an overall effect that is different from that obtained when the same set of actions is applied in a different order. The order in which we make deposits and withdrawals in a checking account does not affect the end-of-the-month balance. Driving in a city laid out on a rectangular grid, we can go north for one block then east for one block and get to the same location as we would if we first went east for one block and then north for one block. On the other hand, there are situations in which order matters very much. The most obvious examples, such as opening a door and then walking though it versus walking through the door and then opening it, do not lead to very interesting, or general, mathematical models. However, the situation shown in Figure 2.1 does embody a rather general mathematical/physical principle. Because it is illustrative of several of the main points we will analyze it in detail.

The illustration depicts a pair of tanks. The top tank holds fluid in a vessel that is fitted to create a sealed chamber below it. The lower chamber is full of fluid. Fluid can be pumped into or out of the tanks. For the purposes of exposition, we suppose that there are individual agents responsible for the



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