Poincaré-Andronov-Melnikov analysis for non-smooth systems (Q2813067)

The content of this interesting and very useful book is well described in the preface written by its authors, which is reproduced below. NEWLINENEWLINE``Discontinuous systems describe many real processes characterized by instantaneous changes, such as electrical switching or impacts of a bouncing ball. This is the reason why many papers and books have appeared on this topic in the last few years. This book is a contribution to this direction; namely, it is devoted to the study of bifurcations of periodic solutions for general \(n\)-dimensional discontinuous systems. First, we study these systems under assumptions of transversal intersections with discontinuity/switching boundaries and sufficient conditions are derived for the persistence of single periodic solutions under nonautonomous perturbations from single solutions; or under autonomous perturbations from non-degenerate families of solutions; or from isolated solutions. Furthermore, bifurcations of periodic sliding solutions are studied from sliding periodic solutions of unperturbed discontinuous equations. Then bifurcations of forced periodic solutions are investigated for impact systems from single periodic solutions of unperturbed impact equations. We also study weakly coupled discontinuous systems. In addition, local asymptotic properties of derived perturbed periodic solutions are investigated for all studied problems. The relationship between non-smooth systems and their continuous approximations is investigated as well. Many examples of discontinuous ordinary differential equations and impact systems are given to illustrate the theoretical results. To achieve our results, we mostly use the so-called discontinuous Poincaré mapping, which maps a point to its position after one period of the periodic solution. This approach is rather technical. On the other hand, by this method we can get results for general dimensions of spatial variables and parameters as well as asymptotic results such as stability, instability and hyperbolicity of solutions. Moreover, we explain how this approach can be modified for differential inclusions. These are the aims of this book and make it unique, since no one else in any book has ever before studied bifurcations of periodic solutions in discontinuous systems in such general settings. Therefore, our results in this book are original. Some parts of this book are related to our previous works. But we are substantially improving these results, give more details in the proofs and present more examples. Needless to say, this book contains brand new parts. So the aim of this book is to collect and improve our previous results, as well as to continue with new results. Numerical computations described by figures are given with the help of the computational software \texttt{Mathematica}. This book is intended for post-graduates students, mathematicians, physicists and theoretically inclined engineers studying either oscillations of nonlinear discontinuous mechanical systems or electrical circuits by applying the modern theory of bifurcation methods in dynamical systems.''