# Christian Kuehn

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NameChristian Kuehn |
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## Pubs By Year |
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## Pub CategoriesMathematics - Dynamical Systems (37) Nonlinear Sciences - Pattern Formation and Solitons (14) Mathematics - Classical Analysis and ODEs (14) Nonlinear Sciences - Chaotic Dynamics (12) Mathematics - Probability (11) Mathematics - Numerical Analysis (9) Mathematics - Analysis of PDEs (7) Quantitative Biology - Neurons and Cognition (6) Nonlinear Sciences - Adaptation and Self-Organizing Systems (5) Physics - Biological Physics (4) Quantitative Biology - Quantitative Methods (3) Quantitative Biology - Populations and Evolution (3) Physics - Computational Physics (2) Physics - Statistical Mechanics (2) Physics - Data Analysis; Statistics and Probability (2) Mathematical Physics (2) Mathematics - Mathematical Physics (2) Physics - Fluid Dynamics (2) Mathematics - Complex Variables (1) Computer Science - Numerical Analysis (1) Physics - Medical Physics (1) Computer Science - Graphics (1) Physics - Physics and Society (1) Mathematics - Combinatorics (1) Computer Science - Mathematical Software (1) Mathematics - Differential Geometry (1) Quantitative Biology - Molecular Networks (1) |

## Publications Authored By Christian Kuehn

We investigate a singularly perturbed, non-convex variational problem arising in materials science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan M\"uller, where it is proven that the solutions of the variational problem are periodic and exhibit a complicated multi-scale structure. In order to get more insight into the rich solution structure, we transform the corresponding Euler-Lagrange equation into a Hamiltonian system of first order ODEs and then use geometric singular perturbation theory to study its periodic solutions. Read More

We study model spaces, in the sense of Hairer, for stochastic partial differential equations involving the fractional Laplacian. We prove that the fractional Laplacian is a singular kernel suitable to apply the theory of regularity structures. Our main contribution is to study the dependence of the model space for a regularity structure on the three-parameter problem involving the spatial dimension, the polynomial order of the nonlinearity, and the exponent of the fractional Laplacian. Read More

Hysteresis operators appear in many applications such as elasto-plasticity and micromagnetics, and can be used for a wider class of systems, where rate-independent memory plays a role. A natural approximation for systems of evolution equations with hysteresis operators are fast-slow dynamical systems, which - in their used approximation form - do not involve any memory effects. Hence, viewing differential equations with hysteresis operators in the non-linearity as a limit of approximating fast-slow dynamics involves subtle limit procedures. Read More

In this work, we study the numerical approximation of local fluctuations of certain classes of parabolic stochastic partial differential equations (SPDEs). Our focus is on effects for small spatially-correlated noise on a time scale before large deviation effects have occurred. In particular, we are interested in the local directions of the noise described by a covariance operator. Read More

Particles moving inside a fluid near, and interacting with, invariant manifolds is a common phenomenon in a wide variety of applications. One elementary question is whether we can determine once a particle has entered a neighbourhood of an invariant manifold, when it leaves again. Here we approach this problem mathematically by introducing balance functions, which relate the entry and exit points of a particle by an integral variational formula. Read More

We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Read More

Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. Read More

Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socio-economic changes and climate transitions between ice-ages and warm-ages. From bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition. Read More

High-dimensional computational challenges are frequently explained via the curse of dimensionality, i.e., increasing the number of dimensions leads to exponentially growing computational complexity. Read More

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. Read More

A cross-diffusion system modeling the information herding of individuals is analyzed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Read More

We prove local existence of solutions for a class of suitably renormalised coupled SPDE-ODE systems driven by space-time white noise, where the space dimension is equal to 2 or 3. This class includes in particular the FitzHugh-Nagumo system describing the evolution of action potentials of a large population of neurons, as well as models with multidimensional gating variables. The proof relies on the theory of regularity structures recently developed by M. Read More

In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping points). Read More

**Authors:**Christian Kuehn

We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a stochastic $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand, how numerical continuation methods can be carried over to the SPDE setting. First, we compute the deterministic bifurcation diagram for the PDE, i. Read More

**Authors:**Christian Kuehn

Numerical continuation calculations for ordinary differential equations (ODEs) are, by now, an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences. Although several excellent standard software packages are available for ODEs, there are - for good reasons - no standard numerical continuation toolboxes available for partial differential equations (PDEs), which cover a broad range of different classes of PDEs automatically. A natural approach to this problem is to look for efficient gluing computation approaches, with independent components developed by researchers in numerical analysis, dynamical systems, scientific computing and mathematical modelling. Read More

We consider a scalar reaction-diffusion equation in one spatial dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. We present our analytical results on the existence, uniqueness (up to translations) and stability of a traveling wave solution connecting two stable homogeneous steady states. Moreover, we review numerical methods for the case of reaction-diffusion equations with fractional Laplacian and discuss possible extensions to our reaction-diffusion equations with Riesz-Feller operators. Read More

There have been significant recent advances in our understanding of the potential use and limitations of early-warning signs for predicting drastic changes, so called critical transitions or tipping points, in dynamical systems. A focus of mathematical modeling and analysis has been on stochastic ordinary differential equations, where generic statistical early-warning signs can be identified near bifurcation-induced tipping points. In this paper, we outline some basic steps to extend this theory to stochastic partial differential equations with a focus on analytically characterizing basic scaling laws for linear SPDEs and comparing the results to numerical simulations of fully nonlinear problems. Read More

We study the Olsen model for the peroxidase-oxidase reaction. The dynamics is analyzed using a geometric decomposition based upon multiple time scales. The Olsen model is four-dimensional, not in a standard form required by geometric singular perturbation theory and contains multiple small parameters. Read More

A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consider non-hyperbolic steady states, for example in the context of bifurcation theory. Read More

Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Read More

We quantify the effect of Gaussian white noise on fast--slow dynamical systems with one fast and two slow variables, which display mixed-mode oscillations owing to the presence of a folded-node singularity. The stochastic system can be described by a continuous-space, discrete-time Markov chain, recording the returns of sample paths to a Poincar\'e section. We provide estimates on the kernel of this Markov chain, depending on the system parameters and the noise intensity. Read More

We consider a single component reaction-diffusion equation in one dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. Our main result shows the existence, uniqueness (up to translations) and stability of a traveling wave solution connecting two stable homogeneous steady states. In particular, we provide an extension to classical results on traveling wave solutions involving local diffusion. Read More

A large variety of complex systems in ecology, climate science, biomedicine and engineering have been observed to exhibit tipping points, where the internal dynamical state of the system abruptly changes. For example, such critical transitions may result in the sudden change of ecological environments and climate conditions. Data and models suggest that detectable warning signs may precede some of these drastic events. Read More

We study the existence of stationary solutions for a nonlocal version of the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is a recent study by Berestycki et {al.} [Nonlinearity 22 (2009), {pp. Read More

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a $Q$-Wiener process and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers' law for neural fields poses substanial difficulties but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Read More

Invasion waves are a fundamental building block of theoretical ecology. In this study we aim to take the first steps to link propagation failure and fast acceleration of traveling waves to critical transitions (or tipping points). The approach is based upon a detailed numerical study of various versions of the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. Read More

This work is motivated by mathematical questions arising in differential equation models for autocatalytic reactions. In particular, this paper answers an open question posed by Guckenheimer and Scheper [SIAM J. Appl. Read More

Slow manifolds are important geometric structures in the state spaces of dynamical systems with multiple time scales. This paper introduces an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds. We present two examples of bifurcation problems where these manifolds play a key role and a third example in which saddle-type slow manifolds are part of a traveling wave profile of a partial differential equation. Read More

Hopf bifurcations in fast-slow systems of ordinary differential equations can be associated with surprising rapid growth of periodic orbits. This process is referred to as canard explosion. The key step in locating a canard explosion is to calculate the location of a special trajectory, called a maximal canard, in parameter space. Read More

This paper investigates travelling wave solutions of the FitzHugh-Nagumo equation from the viewpoint of fast-slow dynamical systems. These solutions are homoclinic orbits of a three dimensional vector field depending upon system parameters of the FitzHugh-Nagumo model and the wave speed. Champneys et al. Read More

Geometric Singular Perturbation Theory (GSPT) and Conley Index Theory are two powerful techniques to analyze dynamical systems. Conley already realized that using his index is easier for singular perturbation problems. In this paper, we will revisit Conley's results and prove that the GSPT technique of Fenichel Normal Form can be used to simplify the application of Conley index techniques even further. Read More

The FitzHugh-Nagumo equation has been investigated with a wide array of different methods in the last three decades. Recently a version of the equations with an applied current was analyzed by Champneys, Kirk, Knobloch, Oldeman and Sneyd using numerical continuation methods. They obtained a complicated bifurcation diagram in parameter space featuring a C-shaped curve of homoclinic bifurcations and a U-shaped curve of Hopf bifurcations. Read More

**Authors:**Christian Kuehn

Recent studies have shown that adaptive networks driven by simple local rules can organize into "critical" global steady states, providing another framework for self-organized criticality (SOC). We focus on the important convergence to criticality and show that noise and time-scale optimality are reached at finite values. This is in sharp contrast to the previously believed optimal zero noise and infinite time scale separation case. Read More

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). Read More

The method of generalized modeling has been applied successfully in many different contexts, particularly in ecology and systems biology. It can be used to analyze the stability and bifurcations of steady-state solutions. Although many dynamical systems in mathematical biology exhibit steady-state behaviour one also wants to understand nonlocal dynamics beyond equilibrium points. Read More

Epileptic seizures are one of the most well-known dysfunctions of the nervous system. During a seizure, a highly synchronized behavior of neural activity is observed that can cause symptoms ranging from mild sensual malfunctions to the complete loss of body control. In this paper, we aim to contribute towards a better understanding of the dynamical systems phenomena that cause seizures. Read More

**Authors:**Christian Kuehn

Bifurcations can cause dynamical systems with slowly varying parameters to transition to far-away attractors. The terms ``critical transition'' or ``tipping point'' have been used to describe this situation. Critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance. Read More

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Read More

Generalized models provide a framework for the study of evolution equations without specifying all functional forms. The generalized formulation of problems has been shown to facilitate the analytical investigation of local dynamics and has been used successfully to answer applied questions. Yet their potential to facilitate analytical computations has not been realized in the mathematical literature. Read More

We consider the effect of Gaussian white noise on fast-slow dynamical systems with one fast and two slow variables, containing a folded-node singularity. In the absence of noise, these systems are known to display mixed-mode oscillations, consisting of alternating large- and small-amplitude oscillations. We quantify the effect of noise and obtain critical noise intensities above which the small-amplitude oscillations become hidden by fluctuations. Read More

Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological and engineering systems. These mixed-mode oscillations (MMOs) are often found in systems with multiple time scales. Previous differential equation modeling and analysis of MMOs has mainly focused on local mechanisms to explain the small oscillations. Read More

The parameter space of dynamical systems arising in applications is often found to be high-dimensional and difficult to explore. We construct a fast algorithm to numerically analyze "quantitative features" of dynamical systems depending on parameters. Using a classical problem from mathematical ecology as an example, we demonstrate how to apply the algorithm to investigate the amplitude of a limit cycle depending on seven parameters. Read More

The scaling of the time delay near a "bottleneck" of a generic saddle-node bifurcation is well-known to be given by an inverse square-root law. We extend the analysis to several non-generic cases for smooth vector fields. We proceed to investigate $C^0$ vector fields. Read More

We introduce the basic concepts related to subharmonic functions and potentials, mainly for the case of the complex plane and prove the Riesz decomposition theorem. Beyond the elementary facts of the theory we deviate slightly from the usual path of exposition and introduce further concepts alongside with applications. We cover the Dirichlet problem in detail and illustrate the relations between potential theory and probability by considering harmonic measure and its relation to Brownian motion. Read More

Convergence properties of binary stationary subdivision schemes for curves have been analyzed using the techniques of z-transforms and eigenanalysis. Eigenanalysis provides a way to determine derivative continuity at specific points based on the eigenvalues of a finite matrix. None of the well-known subdivision schemes for curves have complex eigenvalues. Read More