The fuel consumption of a modern combustion engine is one of the most important purchase criteria in contemporary society. Increasing oil prices and exhaust emissions taxes force the automotive industry to continuously improve the vehicle engines. The fuel consumption is closely related to the frictional losses of an engine. New material pairings or constructive modifications of the piston group can reduce such losses. Another innovative concept to lower the frictional forces is the micro-structuring of thermo-mechanically highly stressed surfaces. Within an interdisciplinary research group sponsored by the German Research Foundation, scientists at the Leibniz Universitat Hannover and Universitat Kassel have been working together to investigate this research topic. This final report presents their findings and offers scope for further discussion.

Limit cycles or, more general, periodic solutions of nonlinear dynamical systems occur in many different fields of application. Although, there is extensive literature on periodic solutions, in particular on existence theorems, the connection to physical and technical applications needs to be improved. The bifurcation behavior of periodic solutions by means of parameter variations plays an important role in transition to chaos, so numerical algorithms are necessary to compute periodic solutions and investigate their stability on a numerical basis. From the technical point of view, dynamical systems with discontinuities are of special interest. The discontinuities may occur with respect to the variables describing the configuration space manifold or/and with respect to the variables of the vector-field of the dynamical system. The multiple shooting method is employed in computing limit cycles numerically, and is modified for systems with discontinuities. The theory is supported by numerous examples, mainly from the field of nonlinear vibrations. The text addresses mathematicians interested in engineering problems as well as engineers working with nonlinear dynamics.