Oberseminar Nonlinear Dynamics
|Oct 14, 2014
|Oct 21, 2014
||Prof. Elena Tobisch
(Johannes Kepler University Linz, Austria)
|Nonlinear resonances in bounded domains
|Evolutionary dispersive nonlinear PDEs with periodic boundary
conditions are frequently met among the famous equations of mathematical physics, e.g. Boussinesq
eq., Hasegawa-Mima eq., Kadomtsev-Petviashvili eq., Schroedinger eq. and others. We regard a PDE
of this class in two space dimensions and with small nonlinearity. This allows reducing the study of entire
Fourier space of solutions to the study of resonantly interacting Fourier modes. As resonance condition
in this case is equivalent to a Diophantine equation of 6 to 8 variables in high powers, according to the
10th Hilbert problem there exists no general algorithm for solving it. However, the use of a special form
of the Diophantine equations appearing in physics in this context allows solving them by the method of
q-class decomposition, . Once the solutions of resonance conditions are found, their structure can be
studied; its most interesting property turned out to be the decomposition of the Fourier space into non-intersecting
subspaces with independent time evolution. Application of these results to the problem of fluid mechanics
is briefly described (water waves and atmospheric planetary waves).
. Kartashova E. Nonlinear resonance analysis: Theory, computation, applications (Cambridge University Press, 2010).
|Oct 28, 2014
|Nov 4, 2014
|Nov 11, 2014
|Nov 18, 2014
|Nov 25, 2014
||Workshop "COLLECTIVE DYNAMICS IN COUPLED OSCILLATOR SYSTEMS" at WIAS
|Dec 2, 2014
|Dec 9, 2014
||Prof. Hayato Chiba
(Kyushu University, Japan)
|A spectral theory of linear operators on a Gelfand triplet and
its application to dynamics of coupled oscillators
|The Kuramoto model is a system of ODEs (coupled oscillators)
to describe synchronization phenomena. In this talk,
an infinite dimensional Kuramoto model is considered, and Kuramoto's
conjecture on a bifurcation diagram of the system will be proved.
A linear operator obtained from the linearization of the Kuramoto
model has the continuous spectrum on the imaginary axis, so that
the dynamics of the system cannot be revealed via the usual spectrum theory.
To handle such continuous spectra, a new spectral theory of linear operators
based on Gelfand triplets is developed.
In particular, a generalized eigenvalue is defined. It is proved that
a generalized eigenvalue determines the stability and bifurcation of solutions.
|Dec 16, 2014
||Prof. Claes Uggla
(Karlstad University, Schweden)
|Scalar field cosmology and dynamical systems
|In this talk I’ll discuss Einstein’s field equations and how
Killing and homothetic Killing vectors in cosmology in combination with physical first principles,
such as general covariance and scale invariance, induce a hierarchical solution space structure,
where simpler models act as building blocks for more complicated ones (note that similar
considerations are equally applicable when it comes to modified gravity theories). To illustrate
the consequences of these quite general aspects, I will consider several examples that will furthermore
shed light on a variety of heuristic concepts such as attractor and tracker solutions. I’ll focus on flat
FLRW cosmology with a source that consists of: a scalar field representation of a modified Chaplygin
gas; monomial scalar fields and perfect fluids; inverse power-law scalar fields and perfect fluids.
I’ll show how physical principles can be used to obtain regular dynamical systems on compact state
spaces with a hierarchy of invariant subsets and how local and global dynamical systems techniques
then subsequently make it possible to obtain a global understanding of the associated solution spaces
and their properties.
|Jan 6, 2015
|Jan 13, 2015
||Dr. Oleksandr Burylko
(National Academy of Sciences of Ukraine)
|Weak chimeras in minimal networks of coupled phase oscillators
|We suggest a definition for a type of chimera state that appears
in networks of indistinguishable phase oscillators. Defining a "weak chimera" as a type of invariant set
showing partial frequency synchronization, we show that this means they cannot appear in phase
oscillator networks that are either globally coupled or too small. We exhibit various networks of four,
six and ten indistinguishable oscillators where weak chimeras exist with various dynamics and stabilities.
We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families
of weak chimera states in these example networks.
|Jan 20, 2015
||Dr. Sergey Tikhomirov
(MPI for Mathematics in the Science Leipzig, Germany)
|Shadowing and random walks
|We consider a linear skew product with the full shift in the
base and nonzero Lyapunov exponent in the fiber. We provide a sharp estimate for the precision of
shadowing for a typical pseudotrajectory of finite length. This result indicates that the high-dimensional
analog of Hammel-Yorke-Grebogi's conjecture concerning the interval of shadowability for a typical
pseudotrajectory is not correct. The main technique is reduction of the shadowing problem to the ruin
problem for a simple random walk.
|Jan 27, 2015
||Prof. Abderrahim Azouani
(Mohamed Premier University, National School of Applied Sciences,
Al Hoceima, Morocco)
|Feedback control of nonlinear dissipative dynamical systems using
general interpolant observables and continuous data assimilation
|In this work we propose a new feedback control
for controlling general dissipative evolution equations using any of the determining systems
of parameters (modes, nodes, volume elements, etc...) without requiring the presence of
separation in spatial scales, i.e. without assuming the existence of an inertial manifold.
For more reaching applications, we present a continuous data assimilation algorithm based
on our feedback control ideas in the context of the incompressible two-dimensional
|Feb 3, 2015
|Feb 10, 2015
Time and Place
Talks usually take place on Tuesday at 3:15 p.m.
at the Weierstraß Institute
Ehrhard-Schmid-Hörsaal, Mohrenstr. 39, 10117 Berlin.
Tea and coffee will be served at 2:45 p.m. on the ground floor.
Guests are always welcome !