Christian Kuehn

Bio: Christian Kuehn is an academic researcher from Technische Universität München. The author has contributed to research in topics: Dynamical systems theory & Mathematics. The author has an hindex of 25, co-authored 206 publications receiving 3233 citations. Previous affiliations of Christian Kuehn include Max Planck Society & Cornell University.

Multiple Time Scale Dynamics

Abstract: Introduction.- General Fenichel Theory.- Geometric Singular Perturbation Theory.- Normal Forms.- Direct Asymptotic Methods.- Tracking Invariant Manifolds.- The Blow-Up Method.- Singularities and Canards.- Advanced Asymptotic Methods.- Numerical Methods.- Computing Manifolds.- Scaling and Delay.- Oscillations.- Chaos in Fast-Slow Systems.- Stochastic Systems.- Topological Methods.- Spatial Dynamics.- Infinite Dimensions.- Other Topics.- Applications.

Mixed-Mode Oscillations with Multiple Time Scales

Abstract: Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

Moment Closure—A Brief Review

Abstract: Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one “moment”, a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing “higher-order moments” in terms of “lower-order moments”. In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.

Coupled dynamics on hypergraphs: Master stability of steady states and synchronization.

Abstract: In the study of dynamical systems on networks or graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that, instead of microscopic details of the individual nodes or vertices, rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here, we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As, for instance, in the theory of coupled map lattices, we study Laplace-type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of different dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.

Stochastic Differential Equations

Abstract: We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

On the theory of superconductivity

Abstract: IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.