arXiv subjects

Christian Kuehn

Publications and source records attributed to Christian Kuehn.

At least 73 records · Page 4Linked to original sources

arXiv2024

Resilience of the Atlantic Meridional Overturning Circulation

We address the issue of resilience of the Atlantic Meridional Overturning Circulation (AMOC) given the many indications that this dynamical system is in a multi-stable regime. A novel approach to resilience based on rare event techniques is presented which leads to a measure capturing `resistance to change` and `ability to return' aspects in a probabilistic way. The application of this measure to a conceptual model demonstrates its suitability for assessing AMOC resilience but also shows its potential use in many other non-autonomous dynamical systems. This framework is then extended to compute the probability that the AMOC undergoes a transition conditioned on an external forcing. Such conditional probability can be estimated by exploiting the information available when computing the resilience of this system. This allows us to provide a probabilistic view on safe operating spaces by defining a conditional safe operating space as a subset of the parameter space of the (possibly transient) imposed forcing.

physics.ao-ph↗

arXiv2024

Warning Signs for Boundary Noise and their Application to an Ocean Boussinesq Model

In this paper, we construct and discuss early-warning signs of the approach of a parameter to a deterministic bifurcation on a stochastic partial differential equation (SPDE) model with Gaussian white-noise on the boundary. We specifically focus on the qualitative behaviour of the time-asymptotic autocovariance and autocorrelation of the solutions of the linearized system. We also discuss the reliability of the tools from an analytic perspective and through various examples. Among those, the application of the early-warning signs for an ocean Boussinesq model is explored through numerical simulations. The analytic results obtained expand on previous work and present valuable early-warning signs for various applications.

math.PR↗

arXiv2024

Infinite dimensional Slow Manifolds for a Linear Fast-Reaction System

The aim of this expository paper is twofold. We start with a concise overview of the theory of invariant slow manifolds for fast-slow dynamical systems starting with the work by Tikhonov and Fenichel to the most recent works on infinite-dimensional fast-slow systems. The main part focuses on a class of linear fast-reaction PDE, which are particular forms of fast-reaction systems. The first result shows the convergence of solutions of the linear system to the limit system as the time-scale parameter $\varepsilon$ goes to zero. Moreover, from the explicit solutions the slow manifold is constructed and the convergence to the critical manifold is proven. The subsequent result, then, states a generalized version of the Fenichel-Tikhonov theorem for linear fast-reaction systems.

math.AP↗

arXiv2024

Adjoint-Based Projections for Uncertainty Quantification near Stochastically Perturbed Limit Cycles and Tori

This paper presents a new boundary-value problem formulation for quantifying uncertainty induced by the presence of small Brownian noise near transversally stable periodic orbits (limit cycles) and quasiperiodic invariant tori of the deterministic dynamical systems obtained in the absence of noise. The formulation uses adjoints to construct a continuous family of transversal hyperplanes that are invariant under the linearized deterministic flow near the limit cycle or quasiperiodic invariant torus. The intersections with each hyperplane of stochastic trajectories that remain near the deterministic cycle or torus over intermediate times may be approximated by a Gaussian distribution whose covariance matrix can be obtained from the solution to the corresponding boundary-value problem. In the case of limit cycles, the analysis improves upon results in the literature through the explicit use of state-space projections, transversality constraints, and symmetry-breaking parameters that ensure uniqueness of the solution despite the lack of hyperbolicity along the limit cycle. These same innovations are then generalized to the case of a quasiperiodic invariant torus of arbitrary dimension. In each case, a closed-form solution to the covariance boundary-value problem is found in terms of a convergent series. The methodology is validated against the results of numerical integration for two examples of stochastically perturbed limit cycles and one example of a stochastically perturbed two-dimensional quasiperiodic invariant torus. Finally, an implementation of the covariance boundary-value problem in the numerical continuation package coco is applied to analyze the small-noise limit near a two-dimensional quasiperiodic invariant torus in a nonlinear deterministic dynamical system in $\mathbb{R}^4$ that does not support closed-form analysis.

math.DS↗

arXiv2024

Mean-Field Limits for Stochastic Interacting Particles on Digraph Measures

Many natural phenomena are effectively described by interacting particle systems, which can be modeled using either deterministic or stochastic differential equations (SDEs). In this study, we specifically investigate particle systems modeled by SDEs, wherein the mean field limit converges to a Vlasov-Fokker-Planck-type equation. Departing from conventional approaches in stochastic analysis, we explore the network connectivity between particles using diagraph measures (DGMs). DGMs are one possible tool to capture sparse, intermediate and dense network/graph interactions in the mean-field thereby going beyond more classical approaches such as graphons. Since the main goal is to capture large classes of mean-field limits, we set up our approach using measure-theoretic arguments and combine them with suitable moment estimates to ensure approximation results for the mean-field.

math.AP↗

arXiv2024

Preserving Bifurcations through Moment Closures

Moment systems arise in a wide range of contexts and applications, e.g. in network modeling of complex systems. Since moment systems consist of a high or even infinite number of coupled equations, an indispensable step in obtaining a low-dimensional representation that is amenable to further analysis is, in many cases, to select a moment closure. A moment closure consists of a set of approximations that express certain higher-order moments in terms of lower-order ones, so that applying those leads to a closed system of equations for only the lower-order moments. Closures are frequently found drawing on intuition and heuristics to come up with quantitatively good approximations. In contrast to that, we propose an alternative approach where we instead focus on closures giving rise to certain qualitative features, such as bifurcations. Importantly, this fundamental change of perspective provides one with the possibility of classifying moment closures rigorously in regard to these features. This makes the design and selection of closures more algorithmic, precise, and reliable. In this work, we carefully study the moment systems that arise in the mean-field descriptions of two widely known network dynamical systems, the SIS epidemic and the adaptive voter model. We derive conditions that any moment closure has to satisfy so that the corresponding closed systems exhibit the transcritical bifurcation that one expects in these systems coming from the stochastic particle model.

nlin.AO↗

arXiv2024

Fast Reactions and Slow Manifolds

In this paper we generalize the Fenichel theory for attracting critical/slow manifolds to fast-reaction systems in infinite dimensions. In particular, we generalize the theory of invariant manifolds for fast-slow partial differential equations in standard form to the case of fast reaction terms. We show that the solution of the fast-reaction system can be approximated by the corresponding slow flow of the limit system. Introducing an additional parameter that stems from a splitting in the slow variable space, we construct a family of slow manifolds and we prove that the slow manifolds are close to the critical manifold. Moreover, the semi-flow on the slow manifold converges to the semi-flow on the critical manifold. Finally, we apply these results to an example and show that the underlying assumptions can be verified in a straightforward way.

math.DS↗

arXiv2024

The Amplitude Equation for the Space-Fractional Swift-Hohenberg Equation

Non-local reaction-diffusion partial differential equations (PDEs) involving the fractional Laplacian have arisen in a wide variety of applications. One common tool to analyse the dynamics of classical local PDEs near instability is to derive local amplitude/modulation approximations, which provide local normal forms classifying a wide variety of pattern-formation phenomena. In this work, we study amplitude equations for the space-fractional Swift-Hohenberg equation. The Swift-Hohenberg equation is a basic model problem motivated by pattern formation in fluid dynamics and has served as one of the main PDEs to develop general techniques to derive amplitude equations. We prove that there exists near the first bifurcation point an approximation by a (real) Ginzburg-Landau equation. Interestingly, this Ginzburg-Landau equation is a local PDE, which provides a rigorous justification of the physical conjecture that suitably localized unstable modes can out-compete superdiffusion and re-localize a PDE near instability. Our main technical contributions are to provide a suitable function space setting for the approximation problem, and to then bound the residual between the original PDE and its amplitude equation.

math.AP↗

arXiv2024

Higher-Order Network Interactions through Phase Reduction for Oscillators with Phase-Dependent Amplitude

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction -- the reduction of the dynamics onto an invariant torus -- captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart--Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.

math.DS↗

arXiv2023

Fractional Dissipative PDEs

In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also provide an outlook on closely related classes of PDEs. To simplify the exposition, we only discuss the cases of fractional time derivatives and fractional space derivatives in the PDE separately. As our main tools, we describe analytical as well as numerical methods, which are generically necessary to study nonlinear dynamics. We start with the analytical study of steady states and local linear stability for fractional time derivatives. Then we extend this view to a global perspective and consider time-fractional PDEs and gradient flows. Next, we continue to steady states, linear stability analysis and bifurcations for space-fractional PDEs. As a final analytical consideration we discuss existence and stability of traveling waves for space-fractional PDEs. In the last parts, we provide numerical discretization schemes for fractional (dissipative) PDEs and we utilize these techniques within numerical continuation in applied examples of fractional reaction-diffusion PDEs. We conclude with a brief summary and outlook on open questions in the field.

math.AP↗

arXiv2023

Early warning signs of critical transitions -- The $α$-stable case

Statistical early warning signs can be used to identify an approaching bifurcation in stochastic dynamical systems and are now regularly employed in applications concerned with the identification of potential rapid, non-linear change or tipping points. However, the reliability of these early warning signs relies on a number of key mathematical assumptions, most notably the presence of Gaussian noise. We here show that for systems driven by non-Gaussian, $α$-stable noise, the classical early warning signs of rising variance and autocorrelation are not supported by mathematical theory and their use poses the danger of spurious, false-positive results. To address this, we provide a generalized approach by introduce the scaling factor $γ_X$ as an alternative early warning sign. We show that in the case of the Ornstein-Uhlenbeck process, there exists a direct inverse relationship between $γ_{X}$ and the bifurcation parameter, telling us that $γ_{X}$ will increase as we approach the bifurcation. Our numerical simulations confirm theoretical results and show that our findings generalize well to non-linear, non-equilibrium systems. We thus provide a generalized, robust and applicable statistical early warning sign for systems driven by Gaussian and non-Gaussian $α$-stable noise.

math.DS↗

arXiv2023

Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of 'stationary' multiple time-scale systems which possess a critical manifold, i.e. a smooth manifold of steady states for the limiting fast subsystem, particularly when combined with a method of desingularization known as blow-up. The theory for 'oscillatory' multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. We use the blow-up method to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale 'semi-oscillatory' systems with two small parameters. The systems considered behave like oscillatory systems as the smallest perturbation parameter tends to zero, and stationary systems as both perturbation parameters tend to zero. The additional time-scale structure is crucial for the applicability of the blow-up method, which cannot be applied directly to the two time-scale oscillatory counterpart of the problem. Our methods allow us to describe the asymptotics and strong contractivity of all solutions which traverse a neighbourhood of the global singularity. Our main results cover a range of different cases with respect to the relative time-scale of the angular dynamics and the parameter drift. We demonstrate the applicability of our results for systems with periodic forcing in the slow equation, in particular for a class of Liénard equations. Finally, we consider a toy model used to study tipping phenomena in climate systems with periodic forcing in the fast equation, which violates the conditions of our main results, in order to demonstrate the applicability of classical (two time-scale) theory for problems of this kind.

math.DS↗

arXiv2023

On network dynamical systems with a nilpotent singularity

Network dynamics is nowadays of extreme relevance to model and analyze complex systems. From a dynamical systems perspective, understanding the local behavior near equilibria is of utmost importance. In particular, equilibria with at least one zero eigenvalue play a crucial role in bifurcation analysis. In this paper, we want to shed some light on nilpotent equilibria of network dynamical systems. As a main result, we show that the blow-up technique, which has proven to be extremely useful in understanding degenerate singularities in low-dimensional ordinary differential equations, is also suitable in the framework of network dynamical systems. Most importantly, we show that the blow-up technique preserves the network structure. The further usefulness of the blow-up technique, especially with regard to the desingularization of a nilpotent point, is showcased through several examples including linear diffusive systems, systems with nilpotent internal dynamics, and an adaptive network of Kuramoto oscillators.

math.DS↗

arXiv2023

Embedding Capabilities of Neural ODEs

A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results about the exact embedding of maps in different neural ODE architectures in low and high dimension. The embedding capability of a neural ODE architecture can be increased by adding, for example, a linear layer, or augmenting the phase space. Yet, there is currently no systematic theory available and our work contributes towards this goal by developing various embedding results as well as identifying situations, where no embedding is possible. The mathematical techniques used include as main components iterative functional equations, Morse functions and suspension flows, as well as several further ideas from analysis. Although practically, mainly universal approximation theorems are used, our geometric dynamical systems viewpoint on universal embedding provides a fundamental understanding, why certain neural ODE architectures perform better than others.

math.DS↗

arXiv2023

Computer Validation of Neural Network Dynamics: A First Case Study

A large number of current machine learning methods rely upon deep neural networks. Yet, viewing neural networks as nonlinear dynamical systems, it becomes quickly apparent that mathematically rigorously establishing certain patterns generated by the nodes in the network is extremely difficult. Indeed, it is well-understood in the nonlinear dynamics of complex systems that, even in low-dimensional models, analytical techniques rooted in pencil-and-paper approaches frequently reach their limits. In this work, we propose a completely different perspective via the paradigm of validated numerical methods of nonlinear dynamics. The idea is to use computer-assisted proofs to validate mathematically the existence of nonlinear patterns in neural networks. As a case study, we consider a class of recurrent neural networks, where we prove via computer assistance the existence of several hundred Hopf bifurcation points, their non-degeneracy, and hence also the existence of several hundred periodic orbits. Our paradigm has the capability to rigorously verify complex nonlinear behaviour of neural networks, which provides a first step to explain the full abilities, as well as potential sensitivities, of machine learning methods via computer-assisted proofs. We showcase how validated numerical techniques can shed light on the internal working of recurrent neural networks (RNNs). For this, proofs of Hopf bifurcations are a first step towards an integration of dynamical system theory in practical application of RNNs, by proving the existence of periodic orbits in a variety of settings.

math.DS↗

arXiv2023

Cross-diffusion induced instability on networks

The concept of Turing instability, namely that diffusion can destabilize the uniform steady state, is well known either in the context of partial differential equations (PDEs) or in networks of dynamical systems. Recently reaction-diffusion equations with cross-diffusion terms have been investigated, showing an analogous effect called cross-diffusion induced instability. In this paper, we extend this concept to networks of dynamical systems, showing that the spectrum of the graph Laplacian determines the instability appearance, as well as the spectrum of the Laplace operator in reaction-diffusion equations. We extend to network dynamics a particular network model for competing species, coming from the PDEs context. In particular, the influence of different topology structures on the cross-diffusion induced instability is highlighted, considering regular rings and lattices, and also small-world, Erdős-Réyni, and Barabási-Albert networks.

math.DS↗

arXiv2023

Stability and Fluctuations in Complex Ecological Systems

From 08-12 August, 2022, 32 individuals participated in a workshop, Stability and Fluctuations in Complex Ecological Systems, at the Lorentz Center, located in Leiden, The Netherlands. An interdisciplinary dialogue between ecologists, mathematicians, and physicists provided a foundation of important problems to consider over the next 5-10 years. This paper outlines eight areas including (1) improving our understanding of the effect of scale, both temporal and spatial, for both deterministic and stochastic problems; (2) clarifying the different terminologies and definitions used in different scientific fields; (3) developing a comprehensive set of data analysis techniques arising from different fields but which can be used together to improve our understanding of existing data sets; (4) having theoreticians/computational scientists collaborate closely with empirical ecologists to determine what new data should be collected; (5) improving our knowledge of how to protect and/or restore ecosystems; (6) incorporating socio-economic effects into models of ecosystems; (7) improving our understanding of the role of deterministic and stochastic fluctuations; (8) studying the current state of biodiversity at the functional level, taxa level and genome level.

q-bio.PE↗

arXiv2023

Adaptive Dynamical Networks

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks.

nlin.AO↗