Sharp conditions for the existence and multiplicity of smooth periodic solutions to a hybrid dynamical system for human running

In this paper, we provide necessary and sufficient conditions for the existence and multiplicity of periodic smooth solutions to a hybrid system of differential equations that describes animal running at constant angular velocity during stance. We show that solutions to this system exist if and only if the angular velocity during stance is lower than the angular velocity of the corresponding linear harmonic oscillator and satisfies a non-resonance condition. The proof of our main theorem relies on a combination of tools from oscillation theory for ODEs, intersection theory for real plane conics and the analysis of Poincaré maps. Furthermore, we demonstrate the validity of our result with numerical simulations for combinations of parameters corresponding to periodic, resonant, and unbounded solutions. Finally, we show that our model fits accurately publicly available data on human running on different conditions, including comfortable and fast running on a treadmill and overground.