Nonlinear Dynamics at the Free University Berlin

Winter 2014/15

Oberseminar Nonlinear Dynamics



Oct 14, 2014 SFB 647
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, [1]. 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).

References: [1]. Kartashova E. Nonlinear resonance analysis: Theory, computation, applications (Cambridge University Press, 2010).
Oct 28, 2014 CANCELLED
Nov 4, 2014 CANCELLED
Nov 11, 2014 SFB 647
Nov 18, 2014 CANCELLED
Dec 2, 2014 SFB 647
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 SFB 647
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 Navier-Stokes equations.
Feb 3, 2015 CANCELLED
Feb 10, 2015 SFB 647

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 !


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