Noise-induced bifurcations, multiscaling and on-off intermittency

We present recent results on noise-induced transitions in a nonlinear oscillator with randomly modulated frequency. The presence of stochastic perturbations drastically alters the dynamical behaviour of the oscillator: noise can wash out a global attractor but can also have a constructive role by stabilizing an unstable fixed point. The random oscillator displays a rich phenomenology but remains elementary enough to allow for exact calculations: this system is thus a useful paradigm for the study of noise-induced bifurcations and is an ideal testing ground for various mathematical techniques. We show that the phase is determined by the sign of the Lyapunov exponent ( which can be calculated non-perturbatively for white noise), and we derive the full phase diagram of the system. We also investigate the effect of time correlations of the noise on the phase diagram and show that a smooth random perturbation is less efficient than white noise. We study the critical behaviour near the transition and explain why noise-induced transitions often exhibit intermittency and multiscaling: these effects do not depend on the amplitude of the noise but rather on its power spectrum. By increasing or filtering out the low frequencies of the noise, intermittency and multiscaling can be enhanced or eliminated.