arXiv subjects

Christian Kuehn

Publications and source records attributed to Christian Kuehn.

At least 37 records · Page 2Linked to original sources

arXiv2026

Stability of Phase-Locked States in Signed Kuramoto Networks: Structure versus Adaptation

Adaptive Kuramoto models admit a variety of nontrivial phase-locked configurations, including antipodal and rotating-wave states. A central open question is whether the observed persistence of such configurations can be attributed to intrinsic properties of the associated signed interaction networks, or whether it relies essentially on adaptive coupling dynamics. To address this question, we study the stability of antipodal and rotating-wave phase configurations on fixed signed networks that preserve the same phase symmetries but are not generated by adaptive dynamics. We show that for two canonical classes of static signed networks, stability is highly constrained, with unstable modes persisting under parameter variations generically, and we characterize how adaptive coupling influences invariant sets and basins of attraction for the configurations where stability is permitted. Taken together, these results show that while static network structure imposes severe constraints on the stability of phase-locked configurations, adaptive coupling dynamics organize and delineate their robustness when stability is permitted.

math.DS↗

arXiv2026

Tracking Finite-Time Lyapunov Exponents to Robustify Neural ODEs

We investigate finite-time Lyapunov exponents (FTLEs), a measure for exponential separation of input perturbations, of deep neural networks within the framework of continuous-depth neural ODEs. We demonstrate that FTLEs are powerful organizers for input-output dynamics, allowing for better interpretability and the comparison of distinct model architectures. We establish a direct connection between Lyapunov exponents and adversarial vulnerability, and propose a novel training algorithm that improves robustness by FTLE regularization. The key idea is to suppress exponents far from zero in the early stage of the input dynamics. This approach enhances robustness and reduces computational cost compared to full-interval regularization, as it avoids a full ``double'' backpropagation.

math.DS↗

arXiv2026

Analysis of the Geometric Structure of Neural Networks and Neural ODEs via Morse Functions

Besides classical feed-forward neural networks such as multilayer perceptrons, also neural ordinary differential equations (neural ODEs) have gained particular interest in recent years. Neural ODEs can be interpreted as an infinite depth limit of feed-forward or residual neural networks. We study the input-output dynamics of finite and infinite depth neural networks with scalar output. In the finite depth case, the input is a state associated with a finite number of nodes, which maps under multiple non-linear transformations to the state of one output node. In analogy, a neural ODE maps an affine linear transformation of the input to an affine linear transformation of its time-$T$ map. We show that, depending on the specific structure of the network, the input-output map has different properties regarding the existence and regularity of critical points. These properties can be characterized via Morse functions, which are scalar functions where every critical point is non-degenerate. We prove that critical points cannot exist if the dimension of the hidden layer is monotonically decreasing or the dimension of the phase space is smaller than or equal to the input dimension. In the case that critical points exist, we classify their regularity depending on the specific architecture of the network. We show that except for a Lebesgue measure zero set in the weight space, each critical point is non-degenerate if for finite depth neural networks the underlying graph has no bottleneck, and if for neural ODEs, the affine linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and infinite depth cases. The established theorems allow us to formulate results on universal embedding and universal approximation, i.e., on the exact and approximate representation of maps by neural networks and neural ODEs.

math.DS↗

arXiv2026

Chaotic Kramers' Law: Hasselmann's Program and AMOC Tipping

In bistable dynamical systems driven by Wiener processes, the widely used Kramers' law relates the strength of the noise forcing to the average time it takes to see a noise-induced transition from one attractor to the other. We extend this law to bistable systems forced by fast chaotic dynamics, which we argue is in some cases a more realistic modeling approach than unbounded noise forcing. Transitions similar to the noise-driven case can only occur if the amplitude of the chaotic forcing is large enough. If this is the case, in our numerical example - a reduced-order model of the Atlantic Meridional Overturning Circulation (AMOC) - we observe the chaotic Kramers' law to hold even when the chaotic forcing is far from the stochastic limit. We discuss the limitations of the chaotic Kramers' law, how to address the numerical issues associated with the timescale separation, and give a possible explanation for the dynamics of recently found AMOC collapses and recoveries in complex climate models.

nlin.CD↗

arXiv2025

Existence, scaling, and spectral gap for traveling fronts in the 2D renormalized Allen--Cahn equation

We study the deterministic skeleton of the renormalized stochastic Allen--Cahn equation in spatial dimension $2$. For all sufficiently small regularization parameters $δ>0$, we construct monotone traveling wave front solutions connecting the renormalized equilibria, derive a small-$δ$ asymptotic description of their profile and speed, and identify the leading-order contributions. Linearizing about the wave and working in a naturally chosen weighted space, we show that there exists a spectral gap between the symmetry induced eigenvalue $0$ and the rest of the spectrum. The spectral gap grows linearly in the renormalization constant as $δ\downarrow 0$.

math.AP↗

arXiv2025

Simplicial Complex Emergence on Directed Hypergraphs

We study when co-evolving (or adaptive) higher-order networks defined on directed hypergraphs admit a simplicial description. Binary and triadic couplings are modelled by time-dependent weight tensors. Using representation theory of the symmetric group $S_k$, we decompose these tensors into fully symmetric, fully antisymmetric, and mixed isotypic components, and track their Frobenius norms to define three asymptotic regimes and a quantitative notion of convergence. In the symmetric (resp. antisymmetric) limit, we certify emergence and stability of simplicial complexes via a local boundary test and interior drift conditions that enforce downward-closure; in the mixed limit, we show that the minimal faithful object is a semi-simplicial set. We illustrate the theory with simulations that track the isotypic Frobenius norms and the higher-order structure. Practically, our work provides rigorous conditions under which homological tools are justified for adaptive higher-order systems.

math.CO↗

arXiv2025

Spectral theory of dense hypergraph limits

In this work, we develop a spectral theory for hypergraph limits. We prove the convergence of the spectra of adjacency and Laplacian matrices for hypergraph sequences converging in the $1$-cut metric. On the other hand, we give examples of matrix operators associated with hypergraphs whose spectra are not continuous with respect to the $1$-cut metric. Furthermore, we show that these operators are continuous with respect to other cut norms.

math.CO↗

arXiv2025

The Mean-Field Ott-Antonsen Manifold is an Unstable Manifold in the Continuum Limit

We study interacting particle systems of Kuramoto-type. Our focus is on the dynamical relation between the partial differential equation (PDE) arising in the continuum limit (CL) and the one obtained in the mean-field limit (MFL). Both equations arise when we are considering the limit of infinitely many interacting particles but the classes of PDEs are structurally different. The CL tracks particles effectively pointwise, while the MFL is an evolution for a typical particle. First, we briefly discuss the relation between solutions of the CL and the MFL showing how to generate solutions of the CL starting from solutions of the MFL. Our main result concerns a dynamical relation between important invariant manifolds of the CFL and the MFL. In particular, we give an explicit proof that the unstable manifold of the homogeneous steady state of the CL is the direct dynamical analogue of the famous Ott-Antonsen manifold for the MFL.

math.DS↗

arXiv2025

Local asymptotics for the nonlocal Swift-Hohenberg equation

The nonlocal-to-local asymptotics investigation for evolutionary problems is a central topic both in the theory of PDEs and in functional analysis. More recently, it became the main core of the mathematical analysis of phase-separation models. In this paper we focus on the Swift-Hohenberg equations which are key benchmark models in pattern formation problems and amplitude equations. We prove well-posedness of the nonlocal Swift-Hohenberg equation, and study the nonlocal-to-local asymptotics with one and two nonlocal contributions under homogeneous Neumann boundary conditions using suitable energy estimates on the nonlocal problems.

math.AP↗

arXiv2025

Phase transitions for polyadic epidemic and voter models with multiscale groups

Polyadic (or higher-order) interactions can significantly impact the dynamics of interacting particle systems. However, previous studies have often assumed group sizes to be relatively small. In this work, we examine the influence of multiscale polyadic group interactions, where some groups are small and others are very large. We consider two paradigmatic examples, an SIS-epidemic and the adaptive voter model. On the level of the mean field, we specifically discuss the impact of the multiscale polyadic interactions on equilibrium dynamics and phase transitions. For the SIS-epidemic model, we find a region of bistability that protects the disease-free state over a wide range beyond the classical epidemic threshold from a significant outbreak. For the adaptive voter model, we show that multiscale polyadic interactions can stabilize the network or increase the convergence rate to an unbiased equilibrium.

nlin.AO↗

arXiv2025

Computing Resilience Measures in Dynamical Systems

Resilience broadly describes a quality of withstanding perturbations. Measures of system resilience have gathered increasing attention across applied disciplines, yet existing metrics often lack computational accessibility and generalizability. In this work, we review the literature on resilience measures through the lens of dynamical systems theory and numerical methods. In this context, we reformulate pertinent measures into a general form and introduce a resource-efficient algorithm designed for their parallel numerical estimation. By coupling these measures with a global continuation of attractors, we enable their consistent evaluation along system parameter changes. The resulting framework is modular and easily extendable, allowing for the incorporation of new resilience measures as they arise. We demonstrate the framework on a range of illustrative dynamical systems, revealing key differences in how resilience changes across systems. This approach provides a more global perspective compared to traditional linear stability metrics used in local bifurcation analysis, which can overlook inconspicuous but significant shifts in system resilience. This work opens the door to genuinely novel lines of inquiry, such as the development of new early warning signals for critical transitions or the discovery of universal scaling behaviours. All code and computational tools are provided as an open-source contribution to the DynamicalSystems.jl software library.

math.DS↗

arXiv2025

Pattern formation and nonlinear waves close to a 1:1 resonant Turing and Turing--Hopf instability

In this paper, we analyse the dynamics of a pattern-forming system close to simultaneous Turing and Turing--Hopf instabilities, which have a 1:1 spatial resonance, that is, they have the same critical wave number. For this, we consider a system of coupled Swift--Hohenberg equations with dispersive terms and general, smooth nonlinearities. Close to the onset of instability, we derive a system of two coupled complex Ginzburg--Landau equations with a singular advection term as amplitude equations and justify the approximation by providing error estimates. We then construct space-time periodic solutions to the amplitude equations, as well as fast-travelling front solutions, which connect different space-time periodic states. This yields the existence of solutions to the pattern-forming system on a finite, but long time interval, which model the spatial transition between different patterns. The construction is based on geometric singular perturbation theory exploiting the fast travelling speed of the fronts. Finally, we construct global, spatially periodic solutions to the pattern-forming system by using centre manifold reduction, normal form theory and a variant of singular perturbation theory to handle fast oscillatory higher-order terms.

math.AP↗

arXiv2025

Phase Transitions in a Particle Model for the Self-Adaptive Response to Cancer Dynamics

In this paper, we present a probabilistic analysis of a dynamical particle model for the self-adaptive immune response to cancer, as proposed by the first author in a previous work. The model is motivated by the interplay between immune surveillance and cancer evolution. We rigorously confirm the sharp phase transition in immune system learning predicted in the original work. Additionally, we compute the expected amount of information acquired by the immune system about cancer cells over time. Our analysis relies on time-reversal techniques.

math.PR↗

arXiv2025

The Influence of the Memory Capacity of Neural DDEs on the Universal Approximation Property

Neural Ordinary Differential Equations (Neural ODEs), which are the continuous-time analog of Residual Neural Networks (ResNets), have gained significant attention in recent years. Similarly, Neural Delay Differential Equations (Neural DDEs) can be interpreted as an infinite depth limit of Densely Connected Residual Neural Networks (DenseResNets). In contrast to traditional ResNet architectures, DenseResNets are feed-forward networks that allow for shortcut connections across all layers. These additional connections introduce memory in the network architecture, as typical in many modern architectures. In this work, we explore how the memory capacity in neural DDEs influences the universal approximation property. The key parameter for studying the memory capacity is the product $K τ$ of the Lipschitz constant and the delay of the DDE. In the case of non-augmented architectures, where the network width is not larger than the input and output dimensions, neural ODEs and classical feed-forward neural networks cannot have the universal approximation property. We show that if the memory capacity $Kτ$ is sufficiently small, the dynamics of the neural DDE can be approximated by a neural ODE. Consequently, non-augmented neural DDEs with a small memory capacity also lack the universal approximation property. In contrast, if the memory capacity $Kτ$ is sufficiently large, we can establish the universal approximation property of neural DDEs for continuous functions. If the neural DDE architecture is augmented, we can expand the parameter regions in which universal approximation is possible. Overall, our results show that by increasing the memory capacity $Kτ$, the infinite-dimensional phase space of DDEs with positive delay $τ>0$ is not sufficient to guarantee a direct jump transition to universal approximation, but only after a certain memory threshold, universal approximation holds.

math.DS↗

arXiv2025

Adaptive Cucker-Smale Networks: Limiting Laplacian Time-Varying Dynamics

Differences in opinion can be seen as distances between individuals, and such differences do not always vanish over time. In this paper, we propose a modeling framework that captures the formation of opinion clusters, based on extensions of the Cucker Smale and Hegselmann Krause models to a combined adaptive (or co-evolutionary) network. Reducing our model to a singular limit of fast adaptation, we mathematically analyze the asymptotic behavior of the resulting Laplacian dynamics over various classes of temporal graphs and use these results to explain the behavior of the original proposed adaptive model for fast adaptation. In particular, our approach provides a general methodology for analyzing linear consensus models over time-varying networks that naturally arise as singular limits in many adaptive network models.

math.DS↗

arXiv2025

Critical Slowing Down in Bifurcating Stochastic Partial Differential Equations with Red Noise

The phenomenon of critical slowing down (CSD) has played a key role in the search for reliable precursors of catastrophic regime shifts. This is caused by its presence in a generic class of bifurcating dynamical systems. Simple time-series statistics such as variance or autocorrelation can be taken as proxies for the phenomenon, making their increase a useful early warning signal (EWS) for catastrophic regime shifts. However, the modelling basis justifying the use of these EWSs is usually a finite-dimensional stochastic ordinary differential equation, where a mathematical proof for the aptness is possible. Only recently has the phenomenon of CSD been proven to exist in infinite-dimensional stochastic partial differential equations (SPDEs), which are more appropriate to model real-world spatial systems. In this context, we provide an essential extension of the results for SPDEs under a specific noise forcing, often referred to as red noise. This type of time-correlated noise is omnipresent in many physical systems, such as climate and ecology. We approach the question with a mathematical proof and a numerical analysis for the linearised problem. We find that also under red noise forcing, the aptness of EWSs persists, supporting their employment in a wide range of applications. However, we also find that false or muted warnings are possible if the noise correlations are non-stationary. We thereby extend a previously known complication with respect to red noise and EWSs from finite-dimensional dynamics to the more complex and realistic setting of SPDEs.

math.PR↗

arXiv2025

Mathematical Models for Self-Adaptive Response to Cancer Dynamics

We consider two minimal mathematical models for cancer dynamics and self-adaptation. We aim to capture the interplay between the rapid progression of cancer growth and the possibility to leverage and enhance self-adaptive defense mechanisms of an organism, e.g., motivated by immunotherapy. Yet, our two models are more abstract and generic encapsulating the essence of competition between rapid cancer growth and the speed of adaptation. First, we propose a four-dimensional ordinary differential equation model on a macroscopic level. The model has three main parameter regimes and its most important features can be studied analytically. At small external stimulation of adaptation, cancer progresses to a deadly level. At intermediate external input, a bifurcation is observed leaving behind only a locally stable cancer-free steady state. Unfortunately, in a large parameter regime after the bifurcation, a transient cancer density spike is observed. Only for a combination of external input and speeding up adaptation, cancer does not reach critical levels. To study the adaptation speed-up in the initial dynamics phase, we switch to a microscopic probabilistic model, which we study numerically. The microscopic model undergoes a sharp transition under variation of the self-adaptation probability. It is shown that a combination of temporal memory and rare stochastic positive adaptation events is crucial to move the sharp transition point to a desired regime. Despite their extreme simplicity, the two models already provide remarkably deep insight into the challenges one has to overcome to leverage/enhance the self-adaptation of an organism in fighting cancer.

nlin.AO↗

arXiv2025

Probability of Transition to Turbulence in a Reduced Stochastic Model of Pipe Flow

We study the phenomenon of turbulence initiation in pipe flow under different noise structures by estimating the probability of initiating metastable transitions. We establish lower bounds on turbulence transition probabilities using linearized models with multiplicative noise near the laminar state. First, we consider the case of stochastic perturbations by Itô white noise; then, through the Stratonovich interpretation, we extend the analysis to noise types such as white and red noise in time. Our findings demonstrate the viability of detecting the onset of turbulence as rare events under diverse noise assumptions. The results also contribute to applied SPDE theory and offer valuable methodologies for understanding turbulence across application areas.

math.PR↗