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Über dieses Buch

This book is aimed at mathematicians, scientists, and engineers, studying models that involve a discontinuity, or studying the theory of nonsmooth systems for its own sake. It is divided in two complementary courses: piecewise smooth flows and maps, respectively. Starting from well known theoretical results, the authors bring the reader into the latest challenges in the field, going through stability analysis, bifurcation, singularities, decomposition theorems and an introduction to kneading theory. Both courses contain many examples which illustrate the theoretical concepts that are introduced.



Chapter 1. Piecewise-smooth Flows

This course is about the geometry of piecewise-smooth dynamical systems. The solutions of a system of ordinary differential equations, such as
$$\dot{x} \, = \, f(x),$$
where \( x = (x_1, x_2, . . . , x_n) \) is some n-dimensional vector or variable, and f is an n-dimensional vector field, can be pictured as trajectories (or orbits) in space (for example, \( \mathbb{R}^{n}\) or some subset of it).
Paul Glendinning, Mike R. Jeffrey

Chapter 2. Piecewise-smooth Maps

This course is about piecewise-smooth maps. If the phase space (typically \( \mathbb{R}^{n}\)) is partitioned into N disjoint open regions such that the union of the closures of these regions is the whole space, then a piecewise-smooth map is a map on this partition which is defined by a different smooth function on each region. Note that a piecewise-smooth map may be discontinuous across boundaries, or it may be continuous but the Jacobian matrix is discontinuous. Other classes exist, but these two form the basis for most studies. The decision about how to define dynamics on the boundaries of the regions can be a bit awkward and will involve us in some little technical issues later.
Paul Glendinning, Mike R. Jeffrey


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