arXiv subjects

Christian Kuehn

Publications and source records attributed to Christian Kuehn.

At least 19 recordsLinked to original sources

arXiv2026

On the optimal control of entry-exit phenomena in planar fast-slow dynamical systems

We study a geometric singular perturbation theory (GSPT) approach to optimal control problems involving entry-exit phenomena in planar fast-slow dynamical systems. In contrast to previous work, we assume that the control acts only on the slow dynamics and we consider minimum-time optimal control problems (i.e., we aim at minimizing the time to reach a certain target). We apply this method to a simplified fast-slow toy model, characterized by a repetition of entry-exit processes, enabling a transparent analytical exploration of the control dynamics and allowing us to derive the key results explicitly. First, we analyze each non-autonomous entry-exit phenomenon separately, and we derive the optimal way to control them. Second, by artificially generating a stable limit cycle, we determine under which conditions it is possible to reach the target. Finally, under such conditions, we combine all entry-exit processes together and we demonstrate the existence of an optimal control. Our approach mainly exploits techniques of GSPT and dynamic programming. Indeed, we show that our problem can be formulated in a way resembling the Bellman equation, even though not in its standard setting. Numerical simulations, built on such Bellman-like formulation, illustrate our analytical results. In particular, the optimal control is derived with an algorithm that converges in a finite number of steps. Lastly, we extend this novel approach to more complex planar fast-slow dynamical systems.

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arXiv2026

The Role of Bifurcations in Parameter Estimation: A UQ Analysis of the Non-spatial Klausmeier Model

We employ uncertainty quantification methods to investigate how parameter identifiability changes in the vicinity of a bifurcation point. We perform numerical experiments on the non-spatial Klausmeier vegetation model with random coefficients, which describes biomass-water interactions and exhibits a fold bifurcation. We partition the domain around the bifurcation into regions with distinct convergence behaviors. In each region, we follow a UQ workflow that includes sensitivity analysis, Bayesian inference, and Fisher information evaluation to assess parameter identifiability. The results show that proximity to the bifurcation point is decisive for parameter estimation. Reliable joint inference of both model parameters is not possible when the system exhibits bistability. Furthermore, we can reconstruct the model's bifurcation pattern by Fisher information heatmaps, presenting a practical tool for bifurcation detection. Lastly, we highlight the importance of transient data for successful parameter estimation.

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arXiv2026

A dispersive extension result for a class of nonlocal nonlinearities

We prove that a class of nonlocal nonlinearities on periodic Sobolev spaces can be expressed as a trace of the unique mild solution to a local dispersive problem consisting of a system of forced Schrödinger equations. The proof is based on a reformulation of the nonlocal nonlinearities as the resonant orbit average of a real-analytic function under the unitary group generated by the free Schrödinger operator $\mathrm{i} \partial_x^2$. We illustrate our result by providing an equivalent local reformulation for a recently obtained nonlocal amplitude equation that formally captures the dynamics of parabolic systems close to a conserved Hopf instability.

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arXiv2026

Unstable Manifolds for the Kuramoto Model: Convergence to the Ott-Antonsen Manifold

In this paper, we study the finite-dimensional, homogeneous, all-to-all coupled Kuramoto model. We begin by performing a complete spectral analysis of all equilibria of the system. Motivated by this analysis, we then derive an explicit description of the unstable manifolds associated with the family of incoherent equilibria. Subsequently, we establish the convergence of this family of unstable manifolds to the Ott-Antonsen manifold $\mathcal{M}_{\mathrm{OA}}$, with respect to the Hausdorff distance induced by the $p$-Wasserstein metric. We further carry out an analogous analysis for the corresponding counterpart of $\mathcal{M}_{\mathrm{OA}}$ in the continuum limit. Moreover, we establish the uniform-in-time convergence of trajectories of the finite-dimensional Kuramoto model on these invariant manifolds towards their corresponding mean-field limit trajectories. Our results provide a direct geometric link between finite-dimensional particle systems and their mean-field, or continuum, limits.

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arXiv2026

Synchronization for the Rough Kuramoto Model

We study the local synchronization of phases and frequencies for the Kuramoto model driven by rough noise. In particular, we prove exponential convergence towards synchronization and we give the explicit rate of convergence and quantify the size of the random basin of attraction. Furthermore, we show that the long time behavior of the system is determined by the evolution of phases' mean. Our result relies on the use of a Lyapunov function, capable of overriding the particular structure of the noise, taking in account only its intensity. Finally, we illustrate our analytical results and possible extensions with the help of numerical simulations.

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arXiv2026

Cascades on Networks with Functional Structure

We consider a version of the Watts threshold model on directed multiplex configuration model networks, and present a detailed analysis of the cascade size, single-seed cascade probability and cascade condition. We then introduce a smaller class of network models that we call "constrained multiplex networks", which is designed to represent networks with so-called "functional" or "complementary" structure. We find that the particular choice of functional structure affects the phase transitions of the cascade model in a variety of ways.

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arXiv2026

A Dynamical Systems Perspective on the Analysis of Neural Networks

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements. We also tackle three concrete challenges. First, we consider the process of information propagation through a neural network, i.e., we study the input-output map for different architectures. We explain the universal embedding property for augmented neural ODEs representing arbitrary functions of given regularity, the classification of multilayer perceptrons and neural ODEs in terms of suitable function classes, and the memory-dependence in neural delay equations. Second, we consider the training aspect of neural networks dynamically. We describe a dynamical systems perspective on gradient descent and study stability for overdetermined problems. We then extend this analysis to the overparameterized setting and describe the edge of stability phenomenon, also in the context of possible explanations for implicit bias. For stochastic gradient descent, we present stability results for the overparameterized setting via Lyapunov exponents of interpolation solutions. Third, we explain several results regarding mean-field limits of neural networks. We describe a result that extends existing techniques to heterogeneous neural networks involving graph limits via digraph measures. This shows how large classes of neural networks naturally fall within the framework of Kuramoto-type models on graphs and their large-graph limits. Finally, we point out that similar strategies to use dynamics to study explainable and reliable AI can also be applied to settings such as generative models or fundamental issues in gradient training methods, such as backpropagation or vanishing/exploding gradients.

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arXiv2026

Normal Forms for Rough Differential Equations

We address the existence of normal forms for rough ordinary differential equations. We assume suitable smoothness and the hyperbolicity of an equilibrium point. In this context, we establish local formal equivalence of the two solution flows generated by a random nonlinear RDE and its linearized version. This provides the foundation for extending normal form theory to rough differential equations.

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arXiv2026

Compatibility of Higher-Order Slow-Manifold Reduction and Continuum Limits in Adaptive Networks

Adaptive networks couple the evolution of node states to the evolution of the interactions between them. In fast-adapting phase oscillator networks, a slow-manifold reduction of a pairwise microscopic model can generate effective higher-order terms in the phase dynamics. We ask whether this higher-order structure survives the dense-graph continuum limit, and whether it matters if one first reduces and then passes to the continuum, or first passes to the continuum and then reduces. We prove well-posedness and discrete-to-continuum convergence for the unreduced and first-order reduced models, and we construct the continuum slow manifold directly in a Banach-space setting. Along admissible equal-cell step approximations, the two routes give the same first-order continuum vector field, including the same pairwise correction and triplet operator, up to controlled $O(\varepsilon^2)$ remainders. A continuum mixed-derivative criterion then shows that, for suitable coupling functions, the resulting triplet operator is genuinely nonpairwise in the smooth bounded-kernel class. Thus the higher-order term is not a finite-network artefact, but persists in the macroscopic continuum description considered here.

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arXiv2026

Fréchet Means of Periodic Orbits in Dynamical Systems: Geometry, Dynamics, and Diagnostics

In many dynamical systems applications, one seeks representative trajectories summarizing families of periodic orbits arising from parameter variation or uncertainty. While the Fréchet mean provides a natural notion of averaging in nonlinear metric spaces, its application to periodic trajectories is complicated by the interplay between geometric shape and temporal dynamics. Inspired by ideas from shape analysis, we develop a framework for computing Fréchet means of periodic orbits by representing trajectories as closed curves and introducing a metric structure on a quotient space that accounts for circular phase shifts. Within this framework, we establish the existence of empirical Fréchet means in the resulting infinite-dimensional quotient space. A central finding is that the resulting Fréchet mean depends strongly on the chosen parametrization: time parametrization preserves dynamical information, whereas arc length parametrization emphasizes geometric structure. To reconcile these viewpoints, we propose a decoupled approach that computes a geometric Fréchet mean using arc length parametrization and subsequently reconstructs representative dynamics through harmonic averaging of aligned speed profiles. We further introduce curvature- and medoid-based diagnostic measures that quantify the representativeness of the resulting mean and identify situations in which averaging produces geometric artifacts or fails to capture heterogeneous ensembles. Numerical experiments for the Van der Pol oscillator, the Rosenzweig-MacArthur predator-prey model, and the Morris-Lecar neuronal model demonstrate that the proposed methodology yields robust geometric summaries together with consistent averaged dynamics. The framework provides a principled approach for constructing representative periodic trajectories and assessing their validity in uncertainty quantification and dynamical systems.

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arXiv2026

Thin Domains, Reduction, and Slow Manifolds

We propose a unified framework for dimension reduction of partial differential equations posed on thin domains. Our approach combines three complementary ingredients: a careful boundary-condition analysis, an averaging-based splitting for general thin geometries, and a slow-manifold viewpoint for the resulting fast-slow system. Homogeneous Neumann conditions on the thin faces emerge as the most relevant and physical regime because they preserve the transverse zero mode and therefore lead to a genuine lower-dimensional reduced equation. For general thin domains we derive the averaged fast-slow system and isolate the geometry-induced correction term. We then formulate a splitting-based Lyapunov-Perron construction for an exact slow manifold when a suitable spectral decomposition of the slow variable is available, and we construct approximate slow manifolds and corrected reduced dynamics directly from the invariance equation by asymptotic expansion. Moreover, we propose the Schnakenberg reaction-diffusion system as a canonical test problem for comparing the full thin-domain dynamics, the averaged model, and the manifold-corrected reduced dynamics. Finally, we also extend the framework to thin tubular domains and derive the corresponding rescaled fast-slow system in curved geometry.

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arXiv2026

Graphons, Geometry, and Dynamics: Forward and Inverse Perspectives

In this work, we explore the interplay between graph limit theory, the geometry of underlying probability spaces, spectral theory, and network dynamical systems. We investigate two primary questions concerning forward and inverse perspectives: first, whether a graphon retains information about the geometry of the space on which it is defined, and second, whether spectral properties can distinguish graphons that originate from different geometric spaces. To address these questions, we differentiate between combinatorial equivalence and geometric structure, highlighting how these concepts are captured simultaneously by the class of pure graphons. Furthermore, we construct explicit examples of isospectral graphons -- graphons whose integral operators share the same spectrum -- that differ in their underlying geometry. By utilizing the heat kernels of Neumann- and Dirichlet-isospectral drums, we demonstrate that these graphons are not combinatorially equivalent. Finally, we establish new connections between the geometric aspects of graph limit theory and dynamical systems by analyzing a continuum Kuramoto model with graphon-defined interactions. We demonstrate that while isospectrality implies identical stability properties in certain cases, this correspondence breaks down when the differing boundary conditions of our specific Neumann and Dirichlet constructions are considered.

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arXiv2026

Spectral Selection in Symmetric Self-Attention Dynamics

We study self-attention dynamics on the unit sphere as an interacting particle system arising from an idealized Transformer-type update. Under a symmetry assumption on weight matrices given by $Q^\top K=V=V^\top$, the flow admits a gradient-flow structure and an exact reformulation in the eigenbasis of $V$, revealing a spectral mode-selection mechanism. We show that the dynamics exhibits two distinct asymptotic scenarios: homogeneous alignment toward the dominant eigendirection when one positive eigenvalue strictly dominates all others in modulus, and sign-split polarization toward the most negative eigendirection when $V$ is negative definite. In particular, we obtain local stability criteria for pure-mode equilibria and global selection results in both regimes. These results provide a rigorous finite-particle description of how the spectrum of the weight matrices organizes asymptotic patterns in a symmetric self-attention flow, and highlight how the symmetric setting renders the dynamics amenable to mathematical analysis.

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arXiv2026

Symmetry-Breaking and Hysteresis in a Duplex Voter Model

We introduce and analyze a voter-type model on a two-layer multiplex network, where the presence of a state on one layer acts as a catalyst or inhibitor to the propagation of that state on the other layer. Despite the model's simplicity, our mathematical analysis reveals a rich phase diagram that includes spontaneous symmetry-breaking and a cusp bifurcation, which arises when noise is introduced into the model. In particular, this bifurcation mechanism can be viewed as a prototypical unfolding of the change between explosive and non-explosive transitions observed in various other network models. We cross-validate our analytic results by numerical simulations.

nlin.AO↗

arXiv2026

Universal Approximation Constraints of Narrow ResNets: The Tunnel Effect

We analyze the universal approximation constraints of narrow Residual Neural Networks (ResNets) both theoretically and numerically. For deep neural networks without input space augmentation, a central constraint is the inability to represent critical points of the input-output map. We prove that this has global consequences for target function approximations and show that the manifestation of this defect is typically a shift of the critical point to infinity, which we call the ``tunnel effect'' in the context of classification tasks. While ResNets offer greater expressivity than standard multilayer perceptrons (MLPs), their capability strongly depends on the signal ratio between the skip and residual channels. We establish quantitative approximation bounds for both the residual-dominant (close to MLP) and skip-dominant (close to neural ODE) regimes. These estimates depend explicitly on the channel ratio and uniform network weight bounds. Low-dimensional examples further provide a detailed analysis of the different ResNet regimes and how architecture-target incompatibility influences the approximation error.

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arXiv2026

Statistical warning indicators for abrupt transitions in dynamical systems with slow periodic forcing

There is growing interest in anticipating critical transitions in natural systems, often pursued through statistical detection of early warning signals associated with dynamical bifurcations. In stochastic dynamical systems, such signals commonly rely on manifestations of critical slowing down. However, we still need additional development for the underlying theory for critical transitions in non-autonomous systems. This extension is relevant for natural systems, whose behaviour often emerges from seasonal periodic forcing. In this study, we systematically investigate the feasibility of anticipating the termination of oscillatory behavior in a bistable system with slow periodic forcing. In this setting, existing approaches of estimating linear characteristics of the return map fail in practical scenarios due to the unfavourable time-scale separation. Instead, we propose two statistical indicators for the anticipation of critical transitions in the periodic behaviour: (i) conventional early warning indicators, such as increasing variance and autocorrelation, evaluated across system cycles, and (ii) indicators derived from the phase of the seasonal forcing. By statistically comparing their predictive performance, we find that phase-based indicators provide the strongest early warning capability. Our results offer guidance for the detection of critical transitions in periodically forced systems and, more broadly, systematically extend early-warning signs towards non-autonomous dynamical systems.

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arXiv2026

Emergent Higher-Order Structure from Fast Adaptive Networks

We study adaptive network models in which coupling weights evolve on a fast time scale relative to the phase dynamics of the nodes. Using Geometric Singular Perturbation Theory (GSPT), we prove that, although the microscopic system is strictly pairwise, the effective slow dynamics on the invariant slow manifold can exhibit genuinely higher-order structure. More precisely, Fenichel reduction produces explicit $O(\varepsilon)$ triplet terms in the reduced phase dynamics. In addition, we give a rigorous criterion ensuring that these terms are irreducible, in the sense that the reduced vector field does not admit a pairwise decomposition in node coordinates. We derive the first-order slow-manifold correction explicitly, formulate the irreducibility criterion via mixed second derivatives, and verify it for the adaptive Kuramoto phase oscillator model. The results show that the class of pairwise-coupled fast--slow adaptive network systems is not closed under slow-manifold reduction.

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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.

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