University of Bristol
Almost every paper on nonsmooth systems references the book by Filippov Differential Equations with Discontinuous Righthand Sides (Kluwer, 1988). Yet it is a difficult read, with an obsolete section numbering system, few diagrams, dense text and very few examples. It is tempting to ask how many people have actually read it all, or know of the results that it contains.
For the past 18 months or so, a group at the University of Bristol (the Bristol Filippov Group, BFG) has been systematically going through the text. Meeting for two hours once every week or so (with homework in between), we have endeavoured to understand all the material, and to extract those results that are relevant to the modern audience.
In this lecture, we will focus on a major result which does not appear to be widely known. This is Theorem 1 on p. 217 (originally due to Kozlova, itself a notoriously difficult paper to find), which gives necessary and sufficient conditions for a planar non smooth system of a certain class to be structurally stable in a closed bounded domain.
To derive this result, we need, within the context of non smooth systems, to fundamentally revise our notions of 1) the solution to an equation, 2) singularities, 3) separatrices, 4) structural stability, and 5) topological equivalence.
This talk will not dwell on the fine detail of the many lemmas and corollaries needed to prove the theorem. Instead we will direct attention toward the importance of revising our understanding of notions well accepted in the smooth setting and the relevant sections of the book where that detail is contained. Several examples will be given to illustrate the talk.