Multistability in the Duffing--Holmes oscillator: basins of antiperiodic, periodic, and chaotic attractors and their relative volumes

Abstract

The periodically driven Duffing--Holmes oscillator is strongly multistable: for the same forcing amplitude and frequency, periodic, chaotic, and antiperiodic attractors coexist, each with its own basin of attraction. Antiperiodic orbits are periodic orbits invariant under a shift of half a driving period combined with a sign reversal of the coordinates, a symmetry of the equations of motion. Guided by a continuation map of the parameter plane, which locates the regimes but is blind to their coexistence, we explore the space of initial conditions along two one-parameter cuts, at fixed forcing frequency and at fixed forcing amplitude. Stochastic sampling of the full space yields an unbiased estimate of the relative volume of each basin as a function of the control parameter, on the same parameter grids on which single-trajectory bifurcation diagrams are computed by continuation. Basin maps on two-dimensional sections, complemented by maps of the order of the attractor reached from each initial condition, resolve the geometry of the coexisting basins where the competition between attractors is strongest. The relative volumes reveal windows of global antiperiodic dominance, in which the antiperiodic orbit attracts essentially every initial condition, separated by narrow intervals in which two or three qualitatively different attractors share the space in comparable proportions, as well as extended ranges of persistent coexistence; at large forcing amplitude, a single antiperiodic orbit of order one captures the whole space. Compared with the bifurcation diagrams, the abrupt changes of basin volume are the ensemble-level counterpart of the multistability-induced discontinuities of the sweeps, and the gradual shrinking of the antiperiodic basin explains why the branch followed by a sweep depends on the initial state and on the drive phase.