Abraham and Shaw: Dynamcis: Compound Oscillations

by Carson Reynolds

By plotting the cartesian product of the states spaces of two oscillators, we get an invariant manifold, a torus. This is different from the attractive limit cycles previously explored, which follow the surface of a torus, but are not tori themselves.

Next we look at coupled oscillators. Christiaan Huyghens noticed that when two clocks are mounted on the same wall the synchronize. This is known as entrainment. If two oscillators are coupled and the phase portrait is perturbed then the theory of Peixoto says that we get a braid: a limit set separated by a space of alternating attractors and replellors. Piexoto’s theory provides a basis for frequency entrainment, but not phase entrainment.

Then we move to the ring model for forced oscillators. We develop a three dimensional state space from the cartesian product of a 1-d and 2-d space. If the system is uncoupled, then we have an invariant manifold (torus) but one that is attractive, but not an attractor. The distinction is that attractive surfaces are not necessarily transitive (trajectories wander on it). If an attractive surface is transitive, then it is an attractor. If we re-attach the spring, then we get a braid as described by Peixoto theory.

We return our attention to Braids. We see that braids consist of a paired periodic saddle and attractor. We learn that phase entrainment (unlike freq entrainment) isn’t structurally stable. We learn that there are three periodic trajectories: a braided saddle and attractor, along with a central repellor.

If we very both amplitude and phase, we sketch out a response diagram. As we move back and forward over the parameters we see annihilation catastrophes and bifrucation behavior.

Next we examine Stoker’s experiments on electrical oscillations. We see something that looks very much like a coupled oscillator with frequency entrainment. We close on the response diagram developed by him.

This (for reasons not explained by the author) represents the close of classical dynamics. Modern dynamics, moves out of these analytical methods to more geometric techniques pioneered by Poincare.