arXiv subjects

Wolf-Jürgen Beyn

Publications and source records attributed to Wolf-Jürgen Beyn.

At least 19 recordsLinked to original sources

arXiv2026

Angular maps for visualizing rotational dynamics

We develop and analyze angular maps as a numerical tool for visualizing and detecting rotational dynamics in nonlinear, nonautonomous dynamical systems in discrete and continuous time. An angular map assigns to each initial point the angular spectrum of the variational equation along the corresponding trajectory. This spectrum describes the long-time average rotation of subspaces transported by the linearized dynamics, measured by maximal principal angles. Building on the theory of angular spectra, we develop efficient numerical algorithms based on forward and backward subspace iteration. We prove that these algorithms asymptotically provide angular spectral values for subspaces that are dominant in either forward or backward time. Applications to Hénon maps, a planar flow, and the Lorenz system illustrate how angular maps reveal rotational features across phase space. For continuous-time systems, we prove that suitably rescaled angular spectra of exact time-step maps converge to the continuous angular spectrum in the Hausdorff metric as the step size tends to zero. For autonomous systems, we also justify a simplified algorithm that uses successive trajectory points to approximate the angular range associated with the flow direction.

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arXiv2024

On generalized metrics of Vandermonde type

In a series of papers in the 1960's, S. Gähler defined and investigated so-called m-metric spaces and their topological properties. An m-metric assigns to any tuple of m+1 elements a real value (more generally an element in a partially odered set) which satisfies the generalized metric axioms of semidefiniteness, symmetry, and simplex inequality. In this contribution we consider a new type of generalized metric which is based on the Vandermonde determinant. We present some remarkable geometric consequences of the corresponding simplex inequality in the complex plane. Then we show that the Vandermonde principle of construction extends to linear spaces of arbitrary dimension by using symmetric multilinear maps of degree m(m + 1)/2. In particular, we analyze when this generalized metrics has the stronger property of definiteness. Finally, an application is provided to the m-metric of point sets when driven by the same linear ordinary differential equation.

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arXiv2024

Angular spectra of linear dynamical systems in discrete time

In this work we introduce the notion of an angular spectrum for a linear discrete time nonautonomous dynamical system. The angular spectrum comprises all accumulation points of longtime averages formed by maximal principal angles between successive subspaces generated by the dynamical system. The angular spectrum is bounded by angular values which have previously been investigated by the authors. In this contribution we derive explicit formulas for the angular spectrum of some autonomous and specific nonautonomous systems. Based on a reduction principle we set up a numerical method for the general case; we investigate its convergence and apply the method to systems with a homoclinic orbit and a strange attractor. Our main theoretical result is a theorem on the invariance of the angular spectrum under summable perturbations of the given matrices (roughness theorem). It applies to systems with a so-called complete exponential dichotomy (CED), a concept which we introduce in this paper and which imposes more stringent conditions than those underlying the exponential dichotomy spectrum.

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arXiv2024

Algebraic rates of stability for front-type modulated waves in Ginzburg Landau equations

We consider the stability of front-type modulated waves in the complex Ginzburg-Landau equation (CGL). The waves occur in the bistable regime (e.g. of the quintic CGL) and connect the zero state to a spatially homogenous state oscillating in time. For initial perturbations that decay at a certain algebraic rate, we prove convergence to the wave with asymptotic phase. The convergence holds in algebraically weighted Sobolev norms and with an algebraic rate in time, where the asymptotic phase is approached by one order less than the profile. On the technical side we use the theory of exponential trichotomies to separate the spatial modes into growing, weakly decaying, and strongly decaying ones. This allows us to derive resolvent and semigroup estimates in weighted Sobolev norms and to close the argument with a Gronwall lemma involving algebraic weights.

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arXiv2023

On a generalized notion of metrics

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n points. We present two principles of constructing pseudo n-metrics. The first one uses the Vandermonde determinant while the second one uses exterior products and is related to the volume of the simplex spanned by the given points. We show that the second class of examples induces pseudo n-metrics on the unit sphere of a Hilbert space and on matrix manifolds such as the Stiefel and the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.

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arXiv2023

Smoothness properties of principal angles between subspaces with applications to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyze their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in [W.-J. Beyn, G. Froyland, and T. Hüls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245--1286], [W.-J. Beyn, and T. Hüls, SIAM J. Appl. Dyn. Syst., 22 (2023), pp. 162--198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.

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arXiv2023

Angular Values of Nonautonomous Linear Dynamical Systems: Part II -Reduction Theory and Algorithm

This work focuses on angular values of nonautonomous dynamical systems which have been introduced for general random and (non)autonomous dynamical systems in a previous publication [W.-J. Beyn, G. Froyland, and T. Hüls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245--1286]. The angular value of dimension $s$ measures the maximal average rotation which an $s$-dimensional subspace of the phase space experiences through the dynamics of a discrete-time linear system. Our main results relate the notion of angular value to the well-known dichotomy (or Sacker--Sell) spectrum and its associated spectral bundles. In particular, we prove a reduction theorem which shows that instead of maximizing over all subspaces, it suffices to maximize over so-called trace spaces which have their basis in the spectral fibers. The reduction leads to an algorithm for computing angular values of dimensions one and two. We apply the algorithm to several systems of dimension up to 4 and demonstrate its efficiency to detect the fastest rotating subspace even if it is not dominant under the forward dynamics.

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arXiv2022

Angular values of nonautonomous and random linear dynamical systems: Part I -- Fundamentals

We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\fracπ{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.

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arXiv2021

Stability of Traveling Oscillating Fronts in Complex Ginzburg Landau Equations

Traveling oscillating fronts (TOFs) are specific waves of the form $U_\star (x,t) = e^{-i ωt} V_\star(x - ct)$ with a profile $V_{\star}$ which decays at $- \infty$ but approaches a nonzero limit at $+\infty$. TOFs usually appear in complex Ginzburg Landau equations of the type $U_t = αU_{xx} + G(|U|^2)U$. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at $+ \infty$. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion and that further isolated eigenvalues are bounded away from the imaginary axis. The basic idea of the proof is to consider the problem in an extended phase space which couples the wave dynamics on the real line to the ODE dynamics at $+ \infty$. Using slowly decaying exponential weights, the framework allows to derive appropriate resolvent estimates, semigroup techniques, and Gronwall estimates.

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arXiv2018

Computation and stability of waves in equivariant evolution equations

Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant evolution equa- tion. In numerical computations the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain degree by the project: linear stability implies non-linear (asymp- totic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of non-linear eigenvalue problems via a contour method.

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arXiv2017

Semilinear Parabolic Differential Inclusions with One-sided Lipschitz Nonlinearities

We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets.

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arXiv2017

Freezing Traveling and Rotating Waves in Second Order Evolution Equations

In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave.

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arXiv2017

Computation and Stability of Traveling Waves in Second Order Evolution Equations

The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type.

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arXiv2016

Fredholm Properties and $L^p$-Spectra of Localized Rotating Waves in Parabolic Systems

In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators $$\mathcal{L}v(x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle+Df(v_{\star}(x))v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2$$ where $v_{\star}:\mathbb{R}^d\rightarrow\mathbb{R}^m$ is a rotating wave profile with $v_{\star}(x)\to v_{\infty}\in\mathbb{R}^m$ as $|x|\to\infty$, $f:\mathbb{R}^m\rightarrow\mathbb{R}^m$ is smooth, $A\in\mathbb{R}^{m,m}$ has eigenvalues with positive real parts and commutes with the limit matrix $Df(v_{\infty})$. The matrix $S\in\mathbb{R}^{d,d}$ is assumed to be skew-symmetric with eigenvalues $(λ_1,\ldots,λ_d)=(\pm iσ_1,\ldots,\pm i σ_k,0,\ldots,0)$. The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction diffusion systems. We prove under suitable conditions that every $λ\in\mathbb{C}$ satisfying the dispersion relation $$\det\Big(λI_m + η^2 A - Df(v_{\infty}) + i\langle n,σ\rangle I_m\Big)=0\quad\text{for some $η\in\mathbb{R}$ and $n\in\mathbb{Z}^k$}$$ belongs to the essential spectrum $σ_{\mathrm{ess}}(\mathcal{L})$ in $L^p$. For values $\mathrm{Re}\,λ$ to the right of the spectral bound for $Df(v_{\infty})$ we show that the operator $λI-\mathcal{L}$ is Fredholm of index $0$, solve the identification problem for the adjoint operator $(λI-\mathcal{L})^*$, and formulate the Fredholm alternative. Moreover, we show that the set $$σ(S)\cup\{λ_i+λ_j:\;λ_i,λ_j\inσ(S),\,1\leqslant i<j\leqslant d\}$$ belongs to the point spectrum $σ_{\mathrm{pt}}(\mathcal{L})$ in $L^p$. We determine their eigenfunctions and show that they decay exponentially in space. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation and compute their numerical spectra as well as associated eigenfunctions.

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arXiv2016

Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in [Beyn, Isaak, Kruse, J. Sci. Comp., 2015]. As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.

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arXiv2016

Spatial Decay of Rotating Waves in Reaction Diffusion Systems

In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators \begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + f(v(x)) = 0,\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*} where the matrix $A\in\mathbb{R}^{N,N}$ is diagonalizable and has eigenvalues with positive real part, the map $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is sufficiently smooth and the matrix $S\in\mathbb{R}^{d,d}$ in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating waves in time-dependent reaction diffusion systems. We prove under appropriate conditions that every bounded classical solution $v_{\star}$ of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that $v_{\star}$ belongs to an exponentially weighted Sobolev space $ W^{1,p}_θ(\mathbb{R}^d,\mathbb{R}^N)$. Several extensions of this basic result are presented: to complex-valued systems, to exponential decay in higher order Sobolev spaces and to pointwise estimates. We also prove that every bounded classical solution $v$ of the eigenvalue problem \begin{align*} A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle + Df(v_{\star}(x))v(x) = λv(x),\,x\in\mathbb{R}^d,\,d\geqslant 2, \end{align*} decays exponentially in space, provided $\mathrm{Re}\,λ$ lies to the right of the essential spectrum. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation. Our results form the basis for investigating nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains.

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arXiv2015

Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg-Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (2002) and a newly proposed explicit variant of the Euler-Maruyama scheme, the so called projected Euler-Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.

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arXiv2013

A numerical method for the solution of relaxed one-sided Lipschitz algebraic inclusions

An existing solvability result for relaxed one-sided Lipschitz algebraic inclusions is substantially improved. This enhanced solvability result allows the design of a very robust numerical method for the approximation of a solution of the algebraic inclusion. Sharp error estimates for this method, illustrative analytic examples and a numerical example are provided.

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