Coupled Oscillators and Quasiperiodicity

Besides the plane and the cylinder, another important two-dimensional phase space is the torus. It is the natural phase space for systems of the form 

For instance, a simple model of coupled oscillators is given by 

An intuitive way to think about (1) is to imagine two friends jogging on a circular track. Here 0,(z), 02( ) represent their positions on the track, and ft), (O are proportional to their preferred running speeds. If they were uncoupled, then each would run at his or her preferred speed and the faster one would periodically overtake the slower one (as in Example 4.2.1). But these sue. friends—they want to run around together. So they need to compromise, with each adjusting his or her speed as necessary. If their preferred speeds are too different, phase-locking will be impossible and they may want to find new running partners. 

Even the seemingly trivial case of uncoupled oscillators (/f, /f, = 0) holds some surprises. Then (1) reduces to 0, = tu, 02 = (u,. The corresponding trajectories on the square are straight lines with constant slope dd2ld6 =(a-Ja),. There are two qualitatively different cases, depending on whether the slope is a rational or an irrational number. 

Suppose for now that there are two fixed points, defined implicitly by O) -02 

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