Stability of fixed points in an approximate solution of the spring-mass running model

TL;DR

This paper analyzes the stability of fixed points in a simplified spring-mass model of human running, deriving conditions for stability and identifying a bifurcation through analytical and numerical methods.

Contribution

It introduces an analytical approximation for the model's solutions and establishes conditions for the stability of fixed points, including the identification of a bifurcation.

Findings

Existence of a unique stable fixed point under certain conditions

Derivation of an apex to apex return map for the model

Identification of a transcritical bifurcation via numerical continuation

Abstract

We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on our previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we construct analytical approximations of solutions in the considered model. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.