Oberseminar Dynamische Systeme:
Mathematische Grundlagen und Anwendungen

Wintersemester 2016/2017
Montag, 17.10.16, 16 Uhr c.t., Raum MI 03.06.011:
M.Sc. Maximilian Engel (Imperial College London, UK)
Bifurcation analysis of stochastically driven limit cycles

Montag, 17.10.16, 17 Uhr c.t., Raum MI 03.06.011:
B.Sc. Andreas Schlattl (TUM)
Verkehrsflussanalyse mit rekurrenten neuronalen Netzen

Montag, 24.10.16, 16 Uhr c.t., Raum MI 03.06.011:
Dr. Gero Schnücke (Universität Würzburg)
An Arbitrary Lagrangian-Eulerian Discontinuous Galerkin method for conservative laws

Montag, 7.11.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Messoud Efendiev (TUM)
Mitochondria swellin: in vitro-1

Abstract: In this talk we present a new mathematical model of mitochondria swelling scenario, which include spatial effects. This spatial effect is of great importance in applications. Based on the experimental data we derived a new mathematical model which leads to PDE-ODE coupling. Our model considers three mitochondrial subpopulations varying in the degree of swelling. We will discuss the derived model with respect to existence and long-time behavior of solutions and obtain a complete mathematical classification of the swelling process.

Montag, 14.11.16, 16 Uhr c.t., Raum MI 03.06.011:
Dr. Elena Popova (Moscow State University)
On the stability of Nanocraft orientation while illuminated by intense laser beam

Abstract: In this talk among it others we are mainly interested in the study of stability of nanocraft position and orientation inside intense laser beam column. The nanocraft driven by intense laser beam pressure acting on its lightsail is sensitive to the torques and lateral forces reacting on surface of sail. This forces influence the orientation and lateral displacement of spacecraft. The assumptions in choosing the model: 1. flat or concave (part of the sphere, conical) circular sail; 2. configuration of nanocraft is treated as rigid body (applicability of Euler equations); 3. mirror reflection of laser beam from surface of the lightsail. We will discuss how sail shape and profile of the laser beam (Gaussian or flat) can affect stability of nanocraft position. We will also discuss the effect of nanocraft rotation around the axis along which it is moving on the stability.

Montag, 21.11.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Matthias Althoff (TUM)
Reachability Analysis of Mixed Discrete and Continuous Systems

Abstract: Functionality, autonomy, and complexity of products and production processes is steadily increasing due to growing computing resources. The advanced capabilities of new systems make it possible to automate tasks that were previously performed by humans, such as (semi-)automated operation of road vehicles, surgical robots, smart grids, flight control systems, and collaborative human-robot systems, to name only a few. It is obvious that most of those systems are either safety- or operation-critical, demanding methods that automatically verify their safety and correct operation.

In this talk, I present reachability analysis as a method to formally verify cyber-physical and embedded systems. I use hybrid automata to jointly model the discrete behavior of computing elements and the continuous behavior of the physical elements. Based on this modeling formalism, I present algorithms for reachability analysis, which automatically explore all possible states of the system for a given set of initial states, parameters, and inputs. If the set of reachable states is not in a set of forbidden states, the correct behavior of the system is proven. Note that this is not possible with classical simulation techniques, since they only generate a finite and thus incomplete set of behaviors.

The applicability of the method is demonstrated for autonomous driving, phase-locked loops, and smart grids.

Montag, 28.11.16, 16 Uhr c.t., Raum MI 03.06.011:
Prof. Dr. Sergiy Kolyada (Institute of Mathematics, NAS of Ukraine / Max-Planck-Institut für Mathematik,Bonn)
Dynamical Compactness and Dynamical Topology

Abstract: The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is called Topological Dynamics. Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area is called Dynamical Topology. Some results of this talk can be considered as a contribution to Dynamical Topology. To link the Auslander point dynamics property with topological transitivity, we introduced the notion of Dynamical Compactness with respect to a family as a new concept of chaoticity of a dynamical system (X,T) given by a compact metric space X and a continuous surjective self-map T:X --> X. In particular, we will show that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. Observe that each topologically weak mixing system is dynamically (transitive) compact. We will discuss the relationships among it and other several stronger forms of sensitivity.

Based on joint works with Wen Huang, Danylo Khilko, Alfred Peris and Guo Hua Zhang.

Donnerstag, 8.12.16, 16 Uhr c.t., Raum MI 02.12.020:
Dr. Frits Veerman (University of Edinburgh)
Breathing pulses in singularly perturbed reaction-diffusion systems

Abstract: Pulse solutions in reaction-diffusion systems can exhibit a wide range of interesting dynamics. The dynamical behaviour of such solutions in example systems has been explored using numerical simulations; however, analytical understanding is often lacking. We study the weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation, using a centre manifold expansion in function space. We present a general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm-Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.

Montag, 9.1.17, 16 Uhr c.t., Raum MI 03.06.011:
Cris Hasan (University of Auckland, Neuseeland)
Mixed-mode oscillations, slow manifolds and twin canards in chemical systems

Abstract: A mixed-mode oscillation (MMO) is a complex waveform with a pattern of alternating small- and large-amplitude oscillations. MMOs have been observed experimentally in many physical and biological applications, and most notably in chemical reactions. We are mainly interested in MMOs that appear in dynamical systems with different time scales. In particular, we consider an autocatalytic model with an explicit time-scale separation parameter. The mathematical analysis of MMOs is very geometric in nature and based on singular limits of the time-scale ratios. Near the singular limit one finds so-called slow manifolds that guide the dynamics on the slow time scale. In the considered autocatalator model, slow manifolds are surfaces that can be either attracting or repelling. Transversal intersections between attracting and repelling slow manifolds are called canard orbits. Our aim is to study a parameter regime where the time-scale ratio is relatively large. We use continuation methods based on two-point boundary value problems to investigate the underlying complex dynamics of the autocatalator in such a parameter regime. By employing these methods, we observe unexpected phenomena such as twin canard orbits and ribbons of the attracting slow manifold.

Montag, 16.1.17, 16 Uhr c.t., Raum MI 03.06.011:
Niclas Kruff M.Sc. (RWTH Aachen)
Coordinate-free computation of normal forms and reductions

Abstract: In the theory of autonomous ordinary differential equations we often have parameter depending systems and we are interested in bifurcations such as pitchfork or Hopf. In order to study those systems near a stationary point it is possible to compute Poincare-Dulac normal forms and to reduce the system using invariants of the linearization. However, in practice it turns out that those computations are not feasible in general. We provide a new approach and a new algorithm to study bifurcations without explicitly computing normal forms.

Montag, 30.1.17, 16 Uhr c.t., Raum MI 03.06.011:
Stud. math. Stefan Luchs (TUM)
Numerische Untersuchung von Hamiltonschen Schwingungsgleichungen

Montag, 13.2.17, 16 Uhr c.t., Raum MI 03.06.011:
Manon Baudel (Universite d'Orleans, Frankreich)
Spectral theory for random Poincare maps

Abstract: We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting N asymptotically stable periodic orbits. To quantify the rare transitions between periodic orbits, we construct a discrete-time, continuous-space Markov chain, called a random Poincare map. We show that this process admits exactly N eigenvalues which are exponentially close to 1, and provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighbourhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasistationary distributions of processes killed upon hitting some of these neighbourhoods. The proofs rely on Feynman-Kac-type representation formulas for eigenfunctions, Doob's h-transform, spectral theory of compact operators, and a detailed balance property satisfied by committor functions.
Joint work with Nils Berglund (Orleans).