From U sequence to Farey sequence: A unification of one-parameter scenarios

A mechanism is presented and analyzed that accomplishes the unification of a number of experimentally observed one-parameter sequences of dynamical behavior as parts of a continuous spectrum that ranges from the U sequence to Farey sequences. Farey sequences, by which we mean one-parameter sequences whose periodic windows correspond to Farey series of rational numbers, have generally been associated with motion on a state-space torus. The mechanism analyzed here does not involve a torus.