The evolution of nonlinear periodicity, illustrated by the sine circle map


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The sine circle map illustrates periodic patterns discussed in the main article (“The Periodic Structure of the Natural Record, and Nonlinear Dynamics”). These patterns are generated by a recursion algorithm that describes the iterations (ticking) of a nonlinear clock (for example, a clock driven by a nonlinear pendulum). The increments of arc swept out by a clock hand or pointer per unit time (per tick) in such a device, as shown schematically in Figure 1, are not linearly proportional to time; that is, they do not fall at equally spaced tick marks around the clock face. The circle map has served as a model for many complex systems of coupled oscillation in the literature; these systems include experimental studies of the damped driven pendulum, charge density waves in semiconductors, Josephson junctions (coupled superconductors), and Rayleigh-Bénard convection “for example, Jensen et al., 1984, 1985”.