An Experimental Verification of Basin Metamorphoses in A Nonlinear Mechanical System

Abstract

This paper describes bifurcations and the basin boundary metamorphoses that give rise to post-fold outcome indeterminacy from a primarily experimental perspective. A gravity-loaded cart-and-track system is constrained to mimic the twin-well, single-degree-of-freedom Duffing oscillator. Of primary interest is the study of how motion, initially contained within a single well, "spills over" into the adjacent well. Although this system is globally bounded, it retains the same generic features of the single-well canonical escape equation. Using time-embedded coordinates, the technique of stochastic interrogation is used to generate the initial condition maps at three different forcing levels corresponding to three different regimes of post-fold outcomes. These three regions are characterized, respectively, by smooth basin boundaries with safe jumps to resonance, fractal basin boundaries with jumps that may or may not restabilize on to the resonant attractor, and eroded basins with unsafe jumps leading to escape from the local well. This experiment successfully replicates much of the subtle global behavior observed in numerical simulations.