A resonancelike phenomenon in a class of bistable oscillators driven by a periodic dichotomous noise process is described. The phenomenon manifests itself as fractal trajectory patterns in the phase plane of the Poincaré map of the flow. The nature of these discrete stochastic trajectories depends on damping and the bias and sharpness of the periodic noise process. In the underdamped regime the map shows self-similar spirals and other intrinsically two-dimensional patterns, whereas in the overdamped regime a one-dimensional description is possible. This one-dimensional map dynamics depends on the contraction factors ${\lambda }_{0,1}$ for the two map branches. For strong damping ${\lambda }_0$+${\lambda }_1$<1, and the dynamics produces an invariant measure, on a Cantor set, whose scaling properties depend on the bias. For weaker damping ${\lambda }_0$+${\lambda }_1$>1, and the map allows periodic orbits that induce multifractal-invariant measures. Infinite lattice families of such periodic orbits exist in the ${\lambda }_{0\mathrm{-}}$${\lambda }_1$ plane, and we examine the properties of one such lattice family. In experimental realizations, fluctuations in the periodicity of the dichotomous noise process produce parametric noise in the Poincaré map, and the dynamics of such a noisy map is studied with respect to scaling properties and self-similarity in the invariant measure.

• Received 4 September 1991

DOI:https://doi.org/10.1103/PhysRevA.45.3412