Balancing with reflex delay

We consider the simplest possible model describing the “man-machine” system when somebody places the end of a stick on his fingertip, and tries to move the lowest point of the stick in a way that its upper position should be stable. The stability analysis of the mechanical model provides surprisingly simple and clear results in spite of the fact that modelling the time delay makes the corresponding differential equation infinite dimensional in mathematical sense. The calculated critical delay of reflexes and the frequency of the nonlinear vibrations arising as a result of a Hopf bifurcation at the limit of linear stability shows good agreement with experimental observations. By means of the above results, we also get some in-view into the work of the organ called “labyrinthus” in the inner ear which helps in self-balancing of the human body even when our eyes are closed.