Stability analysis of non-smooth dynamical systems with an application to biomechanics

This thesis discusses a two dimensional non-smooth dynamical system described by an autonomous ordinary differential equation. The right hand side of the differential equation is assumed to be discontinuous. We provide a local theory of existence, uniqueness and exponential asymptotic stability and state a formula for the basin of attraction. Our conditions are sufficient. Thetheory generalizes smooth dynamical systems theory by providing contraction conditions for two nearby trajectories at a jump. Such conditions have only previously been studied for a two dimensional nonautonomous differential equation. We provide an example of the theory developed in this thesis and show that we can determine stability of a periodic orbit without explicitly calculating it. This is the main advantage of our theory. Our conditions require to define a metric. This however, can turn out to be a difficult task, and at present, we do not have a method for finding such a metric systematically. The final part of this thesis considers an application of a nonsmooth dynamical system to biomechanics. We model an elderly person stepping over an obstacle. Our model assumes stiff legs, and suggests a gait strategy to overcome an obstacle. This work is in collaboration with Professor Wagner’s research group at Institute for Sport Science at the University of M?nster. However, we only present work developed independently in this thesis.