A Detailed Look at the SLIP Model Dynamics: Bifurcations, Chaotic Behavior, and Fractal Basins of Attraction

Abstract

This paper provides a comprehensive numerical analysis of a simple 2D model of running, the spring-loaded inverted pendulum (SLIP). The model consists of a point-mass attached to a massless spring leg; the leg angle at touch-down is fixed during the motion. We employ numerical continuation methods combined with extensive simulations to find all periodic motions of this model, determine their stability, and compute the basins of attraction of the stable solutions. The result is a detailed and complete analysis of all possible SLIP model behavior, which expands upon and unifies a range of prior studies. In particular, we demonstrate and explain the following effects: (i) saddle-node bifurcations, which lead to two distinct solution families for a range of energies and touch-down angles; (ii) period-doubling (PD) bifurcations which lead to chaotic behavior of the model; and (iii) fractal structures within the basins of attraction. In contrast to prior work, these effects are found in a single model with a single set of parameters while taking into account the full nonlinear dynamics of the SLIP model.