Transitions to Bubbling of Chaotic Systems

Certain dynamical systems (e.g., synchronized chaotic oscillators) exhibit a phenomenon called bubbling, whereby small perturbations induce intermittent bursting. In this Letter we show that, as a parameter is varied through a critical value, the transition to bubbling can be {open_quote}{open_quote}hard{close_quote}{close_quote} (the bursts appear abruptly with large amplitude) or {open_quote}{open_quote}soft{close_quote}{close_quote} (the maximum burst amplitude increases continuously from zero), and that the presence or absence of symmetry in the unperturbed system has a fundamental effect on these transitions. These results are confirmed by numerical and physical experiments. {copyright} {ital 1996 The American Physical Society.}