4 - Driven motion

Summary

The environment of simple systems is often not constant in time and it can influence the system in a periodically changing manner. Think of a body driven by a motor, or of the daily or annual cycle of our natural environment. The change in the environment leads to a driving of the system. We study the effect of an external, periodic driving force on a single point mass moving along a straight line. As a consequence of driving, the dimension of the phase space increases. In order to preserve an easy, two-dimensional, visualisation of the motion, it is worth introducing the concept of maps. Our aim is to show how different types of motion can be monitored by means of maps. Limit cycles appear as fixed points, limit cycles, corresponding to hyperbolic points in maps, and we formulate their stability conditions. Hyperbolic limit cycles, corresponding to hyperbolic points in maps, and curves emanated from them, the stable and unstable manifolds, constitute the skeleton of the possible motion in maps. We show that continuous time equations of motion can only lead to maps with certain well defined properties. Finally, we formulate what types of systems are candidates for exhibiting chaotic behaviour.