  When a dynamical system contains several different modes of oscillations it
may behave in a variety of ways: If the modes oscillate at their own individual
frequencies, it exhibits quasiperiodic behavior; when the modes lock to one
another it becomes synchronized, or, as a third possibility, complex chaotic
behavior may emerge. With two modes present, like an internal oscillation
coupled to a periodic external signal, one obtains a highly structured phase
diagram that exhibits these possibilities. In essence, the details are related
to the difference between rational and irrational numbers. Natural system can
display fragments of this phase diagram, thereby offering insights into their
dynamical mechanisms.
