Non-smooth dynamical systems
The author deals with nonsmooth phenomena in dynamical systems theory. The canonical example under consideration is a pendulum-type equation with dry (sometimes also called Coulomb's) friction, expressed by the term \(\text{sgn }x'\), namely \[ x''+ \text{sgn }x'+ x= p(t). \] The influence of the forcing term \(p\) on the structure of the related phase plane is investigated for \(p\) being bounded, periodic, almost-periodic, etc. In fact, the above model is only a simplification (the first approximation) of the pendulum-type equation \[ x''+ \text{sgn }x'+ \sin x= p(t), \] whose study is much more difficult. Nevertheless, it is quite sufficient for demonstrating the complexity of the related nonsmooth phenomena for which the standard methods fail. The methods rely mostly on some basic theorems from set-valued analysis, when using the approximation techniques, or so. The impresssion of various topics and approaches can be made from the names of nine chapters in the book: 1. Introduction; 2. Some general theory of differential inclusions; 3. Bounded, unbounded, periodic, and almost-periodic solutions; 4. Lyapunov exponents of nonsmooth dynamical systems; 5. On the application of Conley index theory to nonsmooth dynamical systems; 6. On the application of KAM theory to nonsmooth dynamical systems; 7. Planar nonsmooth dynamical systems; 8. Melnikov's mehod for nonsmooth dynamical systems; 9. Further topics and notes. The material in the book is well arranged and self-consistent. Therefore, it can be used as a suitable introduction to the study of nonsmooth dynamics.
bounded periodic and almost-periodic solutions
scientific article; zbMATH DE number 1517962