Tomas Bohr

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 Devils 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.

Scaling for supercritical circle-maps: numerical investigation of the onset of bistability and period doubling

Abstract The scaling of mode-locked regions in supercritical circle-maps is studied. We have calculated distances from the critical line to the first period doubling and to overlaps by neighboring tongues. Around quadratic irrationals we find usual scaling behavior corroborating earlier results. For sequences converging to rationals novel behavior is found: the distances remain finite and the limiting value depends strongly on the particular sequence.

A bound for the existence of invariant circles in a class of two-dimensional dissipative maps

Abstract By generalizing results due to Mather we obtain an upper bound for the existence of smooth invariant circles in a class of dissipative two-dimensional maps. In particular the map x n+1 = x n + Ω + by n − (k/2π)sin 2πx n , y n+1 = by n − (k/2π) X sin 2πx n can have no smooth invariant circle for | k | > 2(1 + b )/(2 + b ).

The Complete Devil′s Staircase and Universality of Mode Locking Structures in Discrete Mappings

We study mode locking in certain discrete mappings. The mode locking traces up a complete devil′s staircase of fractal dimension D ~ 0.87. This exponent is universal for a large class of functions and we expect to find similar behaviour in systems with two competing frequencies.