Basin boundary metamorphoses due to changes in accessible boundary orbits in passive dynamic walking

Passive dynamic walking is a mechanical system that walks down a shallow slope without any input or control, and is a useful tool for understanding the dynamic properties of walking. This system has a wide variety of periodic solutions through bifurcations depending on the slope angle, resulting in chaotic attractors and fractal basin boundaries. In addition, basin boundary metamorphoses occur at certain slope angles, where the boundaries of the basin of attraction change abruptly, but the mechanism underlying this phenomenon remains largely unclear. A well-known dynamical system, the Hénon map, exhibits similar properties, and its basin boundary metamorphoses have been explained in terms of changes in accessible boundary orbits caused by intersections of manifolds associated with bifurcating solutions. Inspired by this framework, we propose a hypothesis for the mechanism of basin boundary metamorphoses in passive dynamic walking by introducing the concept of accessible boundary orbits and verify it numerically. Our results provide new insights into the governing dynamics of walking and contribute to a deeper understanding of nonlinear phenomena in locomotion systems.