Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps

Abstract

Dissipative dynamical 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. The transition is caused by interaction and overlap of mode-locked resonances and occurs at a critical line where the map loses invertibility. At this line the mode-locked intervals trace up a complete devil's staircase whose complementary set is a Cantor set with fractal dimension $D\ensuremath{\sim}0.87$. Numerical results indicate that the dimension is universal for maps with cubic inflection points. Below criticality the staircase is incomplete, leaving room for quasiperiodic behavior. The Lebesgue measure of the quasiperiodic orbits seems to be given by an exponent $\ensuremath{\beta}\ensuremath{\sim}0.35$ which can be related to $D$ through the scaling relation $D=1\ensuremath{-}\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$. The exponent $\ensuremath{\nu}$ characterizes the cutoff of narrow plateaus near the transition. A variety of other exponents describing the transition to chaos is defined and estimated numerically.