Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Olga Voytolovska
Universität zu Köln
Discontinuity induced Boundary Equilibrium Bifurcations in Filippov systems
A rich variety of applications in physics and mechanics can be modeled with non-smooth dynamical
systems. Nowadays there is no complete theory for bifurcations of non-smooth systems such as for
classical smooth systems. These difficulties arise due to the existence of a discontinuity manifold, separating
disjoint regions of the state space. The interaction of invariant sets, e.g. limit cycles and equilibria,
with the separating manifold may lead to bifurcations that are not possible in smooth systems. Here
we study bifurcations of equilibria in autonomous piecewise smooth discontinuous dynamical systems,
called Filippov systems. These are boundary equilibria bifurcations (BEB) that occur as a result of the
collision of a pseudo- and a standard-equilibrium branch on the discontinuity manifold. The main focus is
on analyzing of 3-dimensional Filippov systems. Based on examples we also illustrate a co-dimension 2
BEB with a non-hyperbolic equilibrium and a system where the existence and absence of a sliding area
depend solely on parameter values. The work concludes with an outlook of open problems, especially
concerning a possible construction of a center manifold for non-smooth Filippov systems.
Daniel Weiss
University of Tübingen
Existence of Invariant Cones for Piecewise Linear Systems
Recently the concept of center manifolds of smooth dynamical systems generated by a pair of complex
conjugated eigenvalues with vanishing real part has been carried over to piecewise nonlinear systems. In
this process so called invariant cones of corresponding piecewise linear systems play an important role,
invariant cones consisting of periodic orbits or orbits spiraling in respectively out.
In this talk we derive conditions of existence of invariant cones and study their stability introducing the
monodromy matrix of an invariant cone. We apply the results to general 3 dimensional piecewise linear
systems and discuss a 6 dimensional brake system.
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