Complexity and chaos in piecewise-smooth dynamical systems

The paper describes the main bifurcations and routes to chaos exhibited by piecewise-smooth dynamical systems. Emphasis is given to the study of complex behavior in multidimensional piecewise-smooth systems with border-collision bifurcations. The considered phenomena include usual period-doubling and quasiperiodic routes to chaos as well as sequences of different kinds of border-collision bifurcations. We also present a family of three-dimensional piecewise-smooth maps that can be used for analyzing typical nonlinear phenomena in nonsmooth systems.