A numerical approach for the bifurcation analysis of nonsmooth delay equations

Abstract

Mathematical models based on nonsmooth dynamical systems with delay are widely used to understand complex phenomena, specially in biology, mechanics and control. Due to the infinite-dimensional nature of dynamical systems with delay, analytical studies of such models are difficult and can provide in general only limited results, in particular when some kind of nonsmooth phenomenon is involved, such as impacts, switches, impulses, etc. Consequently, numerical approximations are fundamental to gain both a quantitative and qualitative insight into the model dynamics, for instance via numerical continuation techniques. Due to the complex analytical framework and numerical challenges related to delayed nonsmooth systems, there exists so far no dedicated software package to carry out numerical continuation for such type of models. In the present work, we propose an approximation scheme for nonsmooth dynamical systems with delay that allows a numerical bifurcation analysis via continuation (path-following) methods, using existing numerical packages, such as COCO (Dankowicz and Schilder). The approximation scheme is based on the well-known fact that delay differential equations can be approximated via large systems of ODEs. The effectiveness of the proposed numerical scheme is tested on a case study given by a periodically forced impact oscillator driven by a time-delayed feedback controller.