arXiv subjects

Arnd Scheel

Publications and source records attributed to Arnd Scheel.

At least 19 recordsLinked to original sources

arXiv2026

Oscillations in a scalar differential equation coupled to a diffusive field

We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.

math.AP↗

arXiv2026

Diffusive synchronization of phase waves in the FitzHugh-Nagumo system

We analyze synchronization of relaxation oscillations in multiple-timescale reaction-diffusion systems. Interpreting synchronization as convergence to frequency-synchronized wave-train solutions, we resolve for the first time the case of phase waves. These waves are nearly phase-synchronized relaxation oscillations, featuring quasistationary plateaus of length $\varepsilon^{-1}$ separated by fast transition layers, where $\varepsilon\ll1$ is the timescale separation parameter. Tracking the decay of modulations via a Bloch-wave eigenfunction analysis, we find a remarkably weak interaction strength of order $\varepsilon^{8/3}$. This weak layer interaction and many of the technical difficulties arise from repeated scattering of eigenfunctions through fold points at the ends of the quasistationary plateaus. We capture this by combining a novel geometric desingularization approach with Lin's method, exponential trichotomies, and the Riccati transform. While our spectral stability analysis yields diffusive synchronization of all phase waves in the FitzHugh-Nagumo system, it also identifies potential finite-wavelength instabilities, which we realize in a system variant.

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arXiv2025

Selection mechanisms in front invasion

We review progress on questions related to front propagation into unstable states and point out open problems in the area. We strive to highlight different theoretical perspectives and challenges while also addressing more practical questions with examples and guides to computational methods. Throughout we take a dynamical systems point of view that focuses on the ability of invasion processes to act as a selection mechanism in complex systems.

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arXiv2025

Pulled fronts are not (just) pulled

Front propagation into unstable states is often determined by the linearization, that is, propagation speeds agree with predictions from the linearized equation at the unstable state. The leading edge behavior is then a Gaussian tail propagating with the linear spreading speed. Fronts following this leading edge are commonly referred to as pulled fronts, alluding to the idea that they are ``pulled'' by this leading-edge Gaussian tail. We describe here a class of examples that exhibits how these leading-order effects do not completely describe the dynamics in the wake of the front. In fact, leading edge behavior predicts at most two possible invasion scenarios, associated with positive and negative amplitudes of the Gaussian tail, but our examples exhibit three or more invasion fronts with different states in the wake. The resulting invasion process therefore leaves behind a state that is not solely determined by the leading edge, and thus not just pulled by the Gaussian tail.

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arXiv2025

Vacuum bubble and fissure formation in collective motion with competing attractive and repulsive forces

We study the continuum limit of the motion of agents in the plane driven by competing short-range repulsion and long-range attractive forces. At a critical parameter value, we find destabilization of a trivial branch of uniformly distributed solutions and analyze bifurcating solutions. Curiously, the bifurcating branch is vertical, leading to a reversible, non-hysteretic phase transition. Near the bifurcation point, we demonstrate scaling laws for the size of vacuum regions, which can form fissures or bubbles. We also study the effect of small noise and the eventual topological transition from vacuum bubbles to isolated particle clusters.

nlin.PS↗

arXiv2025

Reversible switching due to attraction and repulsion: clusters, gaps, sorting, and mixing

We describe a phase transition in continuum limits of interacting particle systems that exhibits a vertical bifurcation diagram. The transition is mediated by a competition short-range repulsion and long-range attraction. As a consequence of the transition, infinitesimal parameter variations allow switching between uniform distribution and clusters in single-species models, and between mixed and sorted states in multi-species contexts, without hysteresis. Our main technical contribution is a universal expansion for the size of vacuum bubbles that arise in this phase transition and a quantitative analysis of the effect of noise.

math.DS↗

arXiv2025

Instability of anchored spirals in geometric flows

We investigate existence, stability, and instability of anchored rotating spiral waves in a model for geometric curve evolution. We find existence in a parameter regime limiting on a purely eikonal curve evolution. We study stability and instability theoretically, in the aforementioned limiting regime, and numerically. We find convective and absolute oscillatory instability, as well as saddle-node bifurcations. Our results in particular shed light on the instability of spiral waves in reaction-diffusion systems caused by an instability of wave trains against transverse modulations.

nlin.PS↗

arXiv2024

Pinning and dipole asymptotics of locally deformed striped phases

We investigate the effect of spatial inhomogeneity on perfectly periodic, self-organized striped patterns in spatially extended systems. We demonstrate that inhomogeneities select a specific translate of the striped patterns and induce algebraically decaying, dipole-type farfield deformations. Phase shifts and leading order terms are determined by effective moments of the spatial inhomogeneity. Farfield decay is proportional to the derivatives of the Green's function of an effective Laplacian. Technically, we use mode filters and conjugacies to an effective Laplacian to establish Fredholm properties of the linearization in Kondratiev spaces. Spatial localization in a contraction argument is gained through the use of an explicit deformation ansatz and a subtle cancellation in Bloch wave space.

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arXiv2024

Pitchfork bifurcation along a slow parameter ramp: coherent structures in the critical scaling

We investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed $c\sim \varepsilon^{1/3}$, where $\varepsilon$ is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of $c$, that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings-McLeod solution of the Painlev\'e-II equation to Painlev\'e-II equations with a drift term.

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arXiv2023

Anchored spirals in the driven curvature flow approximation

We study existence, asymptotics, and stability of spiral waves in a driven curvature approximation, supplemented with an anchoring condition on a circle of finite radius. We analyze the motion of curves written as graphs in polar coordinates, finding spiral waves as rigidly rotating shapes. The existence analysis reduces to a planar ODE and asymptotics are given through center manifold expansions. In the limit of a large core, we find rotation frequencies and corrections starting form a problem without curvature corrections. Finally, we demonstrate orbital stability of spiral waves by exploiting a comparison principle inherent to curvature driven flow. \end{abstract}

nlin.PS↗

arXiv2023

Stability of coherent pattern formation through invasion in the FitzHugh-Nagumo system

We establish sharp nonlinear stability results for fronts that describe the creation of a periodic pattern through the invasion of an unstable state. The fronts we consider are critical, in the sense that they are expected to mediate pattern selection from compactly supported or steep initial data. We focus on pulled fronts, that is, on fronts whose propagation speed is determined by the linearization about the unstable state in the leading edge, only. We present our analysis in the specific setting of the FitzHugh-Nagumo system, where pattern-forming uniformly translating fronts have recently been constructed rigorously, but our methods can be used to establish nonlinear stability of pulled pattern-forming fronts in general reaction-diffusion systems. This is the first stability result of critical pattern-selecting fronts and provides a rigorous foundation for heuristic, universal wave number selection laws in growth processes based on a marginal stability conjecture. The main technical challenge is to describe the interaction between two separate modes of marginal stability, one associated with the spreading process in the leading edge, and one associated with the pattern in the wake. We develop tools based on far-field/core decompositions to characterize, and eventually control, the interaction between these two different types of diffusive modes. Linear decay rates are insufficient to close a nonlinear stability argument and we therefore need a sharper description of the relaxation in the wake of the front using a phase modulation ansatz. We control regularity in the resulting quasilinear equation for the modulated perturbation using nonlinear damping estimates.

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arXiv2023

Slow passage through the Busse balloon -- predicting steps on the Eckhaus staircase

Motivated by the impact of worsening climate conditions on vegetation patches, we study dynamic instabilities in an idealized Ginzburg-Landau model. Our main results predict time instances of sudden drops in wavenumber and the resulting target states. The changes in wavenumber correspond to the annihilation of individual vegetation patches when resources are scarce and cannot support the original number of patches. Drops happen well after the primary pattern has destabilized at the Eckhaus boundary and key to distinguishing between the disappearance of 1,2, or more patches during the drop are complex spatio-temporal resonances in the linearization at the unstable pattern. We support our results with numerical simulations and expect our results to be conceptually applicable universally near the Eckhaus boundary, in particular in more realistic models.

nlin.AO↗

arXiv2023

Pushed and pulled fronts in a logistic Keller-Segel model with chemorepulsion

We analyze spatial spreading in a population model with logistic growth and chemorepulsion. In a parameter range of short-range chemo-diffusion, we use geometric singular perturbation theory and functional-analytic farfield-core decompositions to identify spreading speeds with marginally stable front profiles. In particular, we identify a sharp boundary between between linearly determined, pulled propagation, and nonlinearly determined, pushed propagation, induced by the chemorepulsion. The results are motivated by recent work on singular limits in this regime using PDE methods.

math.AP↗

arXiv2023

Viscous shocks and long-time behavior of scalar conservation laws

We study the long-time behavior of scalar viscous conservation laws via the structure of $\omega$-limit sets. We show that $\omega$-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $\omega$-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.

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arXiv2023

Growing patterns

Pattern forming systems allow for a wealth of states, where wavelengths and orientation of patterns varies and defects disrupt patches of monocrystalline regions. Growth of patterns has long been recognized as a strong selection mechanism. We present here recent and new results on the selection of patterns in situations where the pattern-forming region expands in time. The wealth of phenomena is roughly organized in bifurcation diagrams that depict wavenumbers of selected crystalline states as functions of growth rates. We show how a broad set of mathematical and numerical tools can help shed light into the complexity of this selection process.

nlin.PS↗

arXiv2023

Shifting consensus in a biased compromise model

We investigate the effect of bias on the formation and dynamics of political parties in the bounded confidence model. For weak bias, we quantify the change in average opinion and potential dispersion and decrease in party size. For nonlinear bias modeling self-incitement, we establish coherent drifting motion of parties on a background of uniform opinion distribution for biases below a critical threshold where parties dissolve. Technically, we use geometric singular perturbation theory to derive drift speeds, we rely on a nonlocal center manifold analysis to construct drifting parties near threshold, and we implement numerical continuation in a forward-backward delay equation to connect asymptotic regimes.

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arXiv2022

Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcations

This work studies front formation in the Allen-Cahn equation with a parameter heterogeneity which slowly varies in space. In particular, we consider a heterogeneity which mediates the local stability of the zero state and subsequent pitchfork bifurcation to a non-trivial state. For slowly-varying ramps which are either rigidly propagating in time or stationary, we rigorously establish existence and stability of positive, monotone fronts and give leading order expansions for their interface location. For non-zero ramp speeds, and sufficiently small ramp slopes, the front location is determined by the local transition between convective and absolute instability of the base state and leads to an O(1) delay beyond the instantaneous pitchfork location before the system jumps to a nontrivial state. The slow ramp induces a further delay of the interface controlled by a slow-passage through a fold of strong- and weak-stable eigenspaces of the associated linearization. We introduce projective coordinates to de-singularize the dynamics near the trivial state and track relevant invariant manifolds all the way to the fold point. We then use geometric singular perturbation theory and blow-up techniques to locate the desired intersection of invariant manifolds. For stationary ramps, the front is governed by the slow passage through the instantaneous pitchfork bifurcation with inner expansion given by the unique Hastings-McLeod connecting solution of Painlev\'{e}'s second equation. We once again use geometric singular perturbation theory and blow-up to track invariant manifolds into a neighborhood of the non-hyperbolic point where the ramp passes through zero and to locate intersections.

math.DS↗

arXiv2022

Coherent structures in nonlocal systems -- functional analytic tools

We develop tools for the analysis of fronts, pulses, and wave trains in spatially extended systems with nonlocal coupling. We first determine Fredholm properties of linear operators, thereby identifying pointwise invertibility of the principal part together with invertibility at spatial infinity as necessary and sufficient conditions. We then build on the Fredholm theory to construct center manifolds for nonlocal spatial dynamics under optimal regularity assumptions, with reduced vector fields and phase space identified a posteriori through the shift on bounded solutions. As an application, we establish uniqueness of small periodic wave trains in a Lyapunov center theorem using only $C^1$-regularity of the nonlinearity.

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