Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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.
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