Mode-Locking and the Transition to Chaos in Dissipative Systems

Dissipative systems with two competing frequencies exhibit transitions to chaos. We have investigated the transition through a study of discrete maps of the circle onto itself, and by constructing and analyzing return maps of differential equations representing some physical systems. The transition is caused by interaction and overlap of mode-locked resonances and takes place at a critical line where the map looses invertibility. At this line the mode-locked intervals trace up a complete Devil's Staircase whose complementary set is a Cantor set with universal fractal dimension D ~ 0.87. Below criticality there is room for quasiperiodic orbits, whose measure is given by an exponent β ~ 0.34 which can be related to D through a scaling relation, just as for second order phase transitions. The Lebesgue measure serves as an order parameter for the transition to chaos. The resistively shunted Josephson junction, and charge density waves (CDWs) in r.f. electric fields are usually described by the differential equation of the damped driven pendulum. The 2d return map for this equation collapses to 1d circle map at and below the transition to chaos. The theoretical results on universal behavior, derived here and elsewhere, can thus readily be checked experimentally by studying real physical systems. Recent experiments on Josephson junctions and CDWs indicating the predicted fractal scaling of mode-locking at criticality are reviewed.