By Ralph Abraham, Christopher D. Shaw
Dynamics: The Geometry of habit (Studies in Nonlinearity)
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Extra resources for Dynamics: The Geometry of Behavior (Studies in Nonlinearity)
The classical examples of dissipative dynamical systems-primarily those included in Rayleigh's text of 1877 - are described in this chapter. Some of the analytical descriptions, as systems of ordinary differential equations of the first order written classically, are included here without discussion. Explanations of these expressions, not essential for understanding the rest of this text, may be found in the Appendix.
3. For the sake of definiteness, let's choose a particular potential function, and visualize it as a potential surface. This one, for example, has two valleys, with a saddle ridge in between. An alternative representation of a function exemplary potential this way. is its contour map. 4. At regular intervals along the vertical axis, draw horizontal cutting planes. Mark the potential surface with a red curve where each cutting plane cuts through the surface. These are called the level curves of the surface.
Further, in dimensions greater than two, other limit sets may turn up. For example, a torus can occur as a limit set in a three-dimensional system. The solenoid, described in the preceding section, is a case in point. 11. Here, the trajectory through the point marked A is wound, like a loose solenoid, around the torus. As time goes on, it winds around tighter and tighter. It approaches its limit set, the torus, asymptotically. There are many more limit sets. Some of the more exotic ones will be shown in Part Two.