Dynamic Cell Mapping Algorithm for Computing Basins of Attraction in Planar Filippov Systems

Discontinuities are a common feature of physical models in engineering and biological systems, e.g. stick-slip due to friction, electrical relays or gene regulatory networks. The computation of basins of attraction of such nonsmooth systems is challenging and requires special treatments, especially regarding numerical integration. In this paper we present a numerical routine for computing basins of attraction (BA) in nonsmooth systems with sliding, (so-called Filippov systems). In particular we extend the Simple Cell Mapping (SCM) method to cope with the presence of sliding solutions by exploiting an event-driven numerical integration routine specially written for Filippov systems. Our algorithm encompasses a dynamic construction of the cell-state-space so that, a lower number of integration steps are required. Moreover, we incorporate an adaptive strategy of the simulation time to render more efficient the computation of basins of attraction. We illustrate the effectiveness of our algorithm by computing basins of attraction of a sliding control problem and a dry-friction oscillator.