Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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DMV Tagung 2011 - Köln, 19. - 22. September Albert Granados, John Hogan, Tere Seara Universität Stuttgart, University of Bristol, Universitat Politècnica de Catalunya Melnikov’s method for subharmonic orbits in a piecewise-defined Hamiltonian system with impacts In this work we consider a two-dimensional piecewise smooth system, defined in two sets separated by the switching curve x = 0. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two critical saddle points located at each side of x = 0. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. When considering a non-autonomous (T -periodic) Hamiltonian perturbation of amplitude ε, using an impact Poincaré map, we rigorously prove that, for every n and m relatively prime and ε > 0 small enough, there exists a nT -periodic orbit impacting 2m times with the switching curve at every period. In addition, we also prove that, if the orbits are forced to undergo a discontinuity when they cross x = 0, which simulates a loss of energy, then all these orbits persist if the relative size of ε > 0 with respect to the magnitude of this jump is large enough. Hany A. Hosham Universität zu Köln Nonstandard bifurcation phenomena in nonsmooth system Due to the presence of discontinuities on the manifold, nonsmooth system (PWS) present a wide variety of bifurcations. In particular, if the behavior of PWS relies on the dynamics of the separation boundaries, nonstandard bifurcations may occur. It was recently shown that the existence of invariant cones C plays an important role in describing the dynamical behavior for PWS relevant to the transition law between the subsystems. The sliding dynamics along the separation manifold for PWS are formulated by using differential inclusions. We show the existence of C containing a segment of sliding orbits and study stability on these cones. Our approach is developed to investigate the existence of C induced sliding bifurcation. Different sliding bifurcation scenarios such as: invariant cones exhibiting crossing-sliding, grazing-sliding and switching-sliding bifurcation are treated. Further, catastrophic bifurcation may occur. Literatur CFPT Carmona, V., Freire, E., Ponce, E., Torres, F. (2005). Bifurcation of invariant cones in piecewise linear homogeneous systems. Int. J. Bifur. Chaos, 15, 2469-2484. K Küpper, T. (2008). Invariant cones for non-smooth systems, Mathematics and Computers in Simulation, 79 1396-1409. KH Küpper, T., Hosham, H. A. (2011). Reduction to invariant cones for non-smooth systems, special Issue of Mathematics and Computers in Simulation, 81 980-995. HKW Weiss, D., Küpper, T., Hosham, H. A. (2011). Invariant manifolds for nonsmooth systems, Physica D: Nonlinear Phenomena, to appear. 174
DMV Tagung 2011 - Köln, 19. - 22. September Tassilo Küpper Universität zu Köln Bells as impacting system “Die Kaiserglocke im Kölner Dom” Ringing bells provide a beautiful example for a dynamical system where impacts occur in a natural way. Bells considered as a dynamical system give rise to several interesting mathematical problems. Starting with the fascinating story of the famous emperor’s bell in the Cathedral of Cologne who could not be forced to ring appropriately and for that reason had been nicknamed “Die Stumme”. We provide a dynamical system approach to investigate motions of a bells inducting impacts. Literatur Köker, S. (2009). Zur Dynamik des Glockenläutens. Diplomarbeit. Universität zu Köln. 2009. Köker, S., Küpper, T. (2009/2010). Die Kaiserglocke als Dynamisches System. Jahrbuch für Glockenkunde 20. - 21. Dudtschenko K.(2011). Zur Glockendynamik. Simulation und Visualisierung. Staatsexamensarbeit. Universität zu Köln. Küpper, K., Hosham, H. A., Dudtschenko, K.(2011). The dynamics of bells as impacting system, to appear in J. Mech. Eng. Sci.. Veltmann, W. (1876). Über die Bewegung einer Glocke. Dinglers Polytechnisches Journal, 22, 481-494. Veltmann, W. (1880). Die Kölner Kaiserglocke. Enthüllungen über die Art und Weise wie der Kölner Dom zu einer mißratenen Glocke gekommen ist. Bonn. Tassilo Küpper Universität zu Köln Bifurcation of periodic orbits for non-smooth systems For smooth dynamical systems Hopf bifurcation provides a well established approach to generate periodic solutions. In recent studies this approach has been extended to non-smooth systems. In this lecture we will present an overview of recent results concerning the bifurcation of periodic solutions. In particular we will show how the dynamics of high-dimensional systems can be reduced to the investigation of low dimensional systems. For that purpose the concept of center manifolds is generalized to invariant "conelike” objects which may involve grazing/sliding bifurcation. Literatur K Küpper, T. (2008). Invariant cones for non-smooth systems, Mathematics and Computers in Simulation, 79 1396-1409. KH Küpper, T., Hosham, H. A. (2011). Reduction to invariant cones for non-smooth systems, special Issue of Mathematics and Computers in Simulation, 81 980-995. HKW Weiss, D., Küpper, T., Hosham, H. A. (2011). Invariant manifolds for nonsmooth systems, Physica D: Nonlinear Phenomena, to appear. Hosham, H. A. (2011). Conelike invariant manifolds for non-smooth systems, in preparation. Filippov, A. F. (1988). Differential equations with discontinuous right-hand sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, Netherlands. 175
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Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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