Mathematische Schriften Kassel
https://kobra.uni-kassel.de:443/handle/123456789/2010081834090
2021-06-13T15:36:45ZNo Chaos in Dixon's System
https://kobra.uni-kassel.de:443/handle/123456789/11470
The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.
2020-01-01T00:00:00ZSeiler, Werner M.Seiß, MatthiasThe so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist which can easily create in numerical computations the impression of chaotic behaviour.Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads
https://kobra.uni-kassel.de:443/handle/123456789/11318
We study a rate-independent system with non-convex energy and in the
case of a time-discontinuous loading. We prove existence of
the rate-dependent viscous regularization by time-incremental problems, while
the existence of the so called parameterized BV-solutions is obtained via
vanishing viscosity in a suitable parameterized setting. In addition, we prove
that the solution set is compact.
2019-09-25T00:00:00ZKnees, DorotheeZanini, ChiaraWe study a rate-independent system with non-convex energy and in the
case of a time-discontinuous loading. We prove existence of
the rate-dependent viscous regularization by time-incremental problems, while
the existence of the so called parameterized BV-solutions is obtained via
vanishing viscosity in a suitable parameterized setting. In addition, we prove
that the solution set is compact.On the existence of symmetric minimizers
https://kobra.uni-kassel.de:443/handle/123456789/2018012354238
In this note we revisit a less known symmetrization method for functions with respect to a topological group, which we call G-averaging. We note that, although quite non-technical in nature, this method yields G-invariant minimizers of functionals satisfying some relaxed convexity properties. We give an abstract theorem and show how it can be applied to the p-Laplace and polyharmonic Poisson problem in order to construct symmetric solutions. We also pose some open problems and explore further possibilities where the method of G-averaging could be applied to.
2018-01-23T00:00:00ZStylianou, AthanasiosIn this note we revisit a less known symmetrization method for functions with respect to a topological group, which we call G-averaging. We note that, although quite non-technical in nature, this method yields G-invariant minimizers of functionals satisfying some relaxed convexity properties. We give an abstract theorem and show how it can be applied to the p-Laplace and polyharmonic Poisson problem in order to construct symmetric solutions. We also pose some open problems and explore further possibilities where the method of G-averaging could be applied to.Convergence analysis of time-discretization schemes for rate-independent systems
https://kobra.uni-kassel.de:443/handle/123456789/2017122154061
It is well known that rate-independent systems involving nonconvex energy
functionals in general do not allow for time-continuous solutions even if the
given data are smooth. In the last years, several solution concepts were
proposed that include discontinuities in the notion of solution, among them
the class of global energetic solutions and the class of BV-solutions.
In general, these solution concepts are not equivalent and numerical schemes
are needed that reliably approximate that type of solutions one is interested
in. In this paper we analyze the convergence of solutions of three
time-discretization
schemes, namely an approach based on local minimization, a relaxed version
of it and an alternate minimization scheme. For all three cases we show that under
suitable conditions on the discretization parameters discrete solutions
converge to limit functions that belong to the class of BV-solutions. The
proofs rely on a reparametrization argument. We illustrate the different
schemes with a toy example.
2017-12-21T00:00:00ZKnees, DorotheeIt is well known that rate-independent systems involving nonconvex energy
functionals in general do not allow for time-continuous solutions even if the
given data are smooth. In the last years, several solution concepts were
proposed that include discontinuities in the notion of solution, among them
the class of global energetic solutions and the class of BV-solutions.
In general, these solution concepts are not equivalent and numerical schemes
are needed that reliably approximate that type of solutions one is interested
in. In this paper we analyze the convergence of solutions of three
time-discretization
schemes, namely an approach based on local minimization, a relaxed version
of it and an alternate minimization scheme. For all three cases we show that under
suitable conditions on the discretization parameters discrete solutions
converge to limit functions that belong to the class of BV-solutions. The
proofs rely on a reparametrization argument. We illustrate the different
schemes with a toy example.