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Bernold Fiedler

Publications and source records attributed to Bernold Fiedler.

At least 19 recordsLinked to original sources
arXiv2025

Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type

The celebrated 1975 Kuramoto model of $N$ identical oscillators with phase angle vector $\boldsymbol{\vartheta}=(\vartheta_1,\ldots,\vartheta_N)$ and all-to-all coupling reads \begin{equation} \label{*} \dot\vartheta_j\,=\tfrac{1}{N}\sum_{k=1}^N \sin(\vartheta_k-\vartheta_j). \tag{*} \end{equation} Here we have passed to co-rotating coordinates in normalized time scale. The model is highly accessible to rigorous mathematical analysis, and has been studied as a paradigm for effects like total and partial synchronization. Most initial conditions $\boldsymbol{\vartheta}$ lead to total synchronization. The plethora of $2^N-1$ (circles of) partially synchronized states, however, is unstable. The precise behavior of transitions to synchrony seems to have eluded description. In the present paper, we address this gap. By the gradient structure of (*), the global dynamics decompose into equilibria and heteroclinic orbits between them. Except for the extremes of total instability and total synchrony, all equilibria are 2-cluster solutions: their phase angles $\vartheta_j$ attain only two values, with a phase difference of $\pi$ between them. Any heteroclinic orbit between 2-cluster equilibria, or towards synchrony, can be realized as a 3-cluster rebellion. Cluster rebellions split the smaller, "slim", minority cluster of the source equilibrium and send the rebellious part to join the bigger, "fat", majority cluster. Heteroclinic transversality identifies the Kuramoto model as a structurally stable Morse-Smale system. In particular, heteroclinic orbits can be concatenated in finite time. The options involved in successive cluster rebellions amount to finite symbol sequences of 2-shift type. The paper is dedicated to Professor Yoshiki Kuramoto, with admiration.

math.DS↗
arXiv2025

Beyond real blow-up: Masuda detours and complex holonomy

For real $\mathbf{b}$, consider quadratic heat equations like \begin{equation*} \mathbf{w}_t=\mathbf{w}_{\boldsymbol{\xi}\boldsymbol{\xi}} + \mathbf{b}(\boldsymbol{\xi})\,\mathbf{w}^2 \end{equation*} on $\boldsymbol{\xi}\in(0,\pi)$ with Neumann boundary conditions. For $\mathbf{b}$=1, pioneering work by Ky\^uya Masuda in the 1980s aimed to circumvent PDE blow-up, which occurs in finite real time, by a detour which ventures through complex time. Naive projection onto the first two Galerkin modes $\mathbf{w}=x+y \cos\boldsymbol{\xi}$ leads us to an ODE caricature. As in the PDE, spatially homogeneous solutions $y=0\neq x\in\mathbb{R}$ starting at $x_0$ blow up at finite real time $t=T=1/x_0$. We aim for ODE "linearization at infinity". Since iterated complex time loops are not feasible, for parabolic PDEs, our PDE-motivated approach is currently limited to ODEs. On the other hand, all ODE results of the present paper exactly embed into certain PDEs of parabolic type, which possess a PDE-invariant Galerkin subspace. In the spirit of Masuda, we extend real analytic ODE solutions to complex time, and to real 4-dimensional $(x,y)\in\mathbb{C}^2$, to circumvent the real blow-up singularity at $t=T$. We therefore study complex foliations of general polynomial ODEs for $(x,y)\in\mathbb{C}^2$, in projective compactifications like $u=1/x,\ z=y/x$, including their holonomy at blow-up $u=0$. We obtain linearizations, at blow-up equilibria of Poincar\'e and Siegel type, based on spectral nonresonance. We discuss the consequences of rational periodic nonresonance, and of irrational quasiperiodic nonresonance of Diophantine type, for iterated Masuda detours in the ODE caricature. We conclude with some comments on global aspects, PDEs, discretizations, and other applications.

math.DS↗
arXiv2025

Classification of global attractors for $\mathbb{S}^1$-equivariant parabolic equations: a survey

We survey the global dynamics of semiflows generated by scalar semilinear parabolic equations which are $\mathbb{SO}(2)$ equivariant under spatial shifts of $x\in \mathbb{S}^1=\mathbb{R}/2\pi\mathbb{Z}$, i.e. $$ u_t = u_{xx} + f(u,u_x),\qquad x\in \mathbb{S}^1.$$ For dissipative $C^2$ nonlinearities $f$, the semiflow possesses a compact global attractor $\mathcal{A}=\mathcal{A}^\mathcal{P}$ which we call Sturm attractor. The Sturm attractor $\mathcal{A}^\mathcal{P}$ decomposes as $$ \mathcal{A}^\mathcal{P}=\mathcal{E}\cup\mathcal{F}^\mathcal{P}\cup\mathcal{R}^\mathcal{P}\cup\mathcal{H}^\mathcal{P},$$ where $\mathcal{H}^\mathcal{P}$ denotes heteroclinic orbits between distinct elements of spatially homogeneous equilibria $\mathcal{E}$, rigidly rotating waves $\mathcal{R}^\mathcal{P}$ and, as their non-rotating counterparts, frozen waves $\mathcal{F}^\mathcal{P}$. We therefore represent $\mathcal{A}^\mathcal{P}$ by its connection graph $\mathcal{C}^\mathcal{P}$, with vertices in $\mathcal{E},\mathcal{F}^\mathcal{P},\mathcal{R}^\mathcal{P}$ and edges $\mathcal{H}^\mathcal{P}$. Under mild hyperbolicity assumptions, the directed graphs $\mathcal{C}^\mathcal{P}$ are finite and transitive. For illustration, we enumerate all 21 connection graphs $\mathcal{C}^\mathcal{P}$ with up to seven vertices. The result uses a lap signature of period maps associated to integrable versions of the steady state ODE of our PDE. As an example, we freeze and reconstruct the connection graph of the Vas tulip attractor, known from delay differential equations, in our PDE setting.

math.DS↗
arXiv2025

Real-time blow-up and connection graphs of rational vector fields on the Riemann sphere

Inspired by pioneering work of Ky\^uya Masuda in the 1980s, only much more recent PDE studies address global boundedness versus finite-time blow-up in complex time. The two phenomena are related by passage from real to purely imaginary time. As a most simplistic ODE example, we study scalar rational vector fields \begin{equation*} \label{*} \dot{w}=P(w)/Q(w), \tag{*} \end{equation*} for complex polynomials $P,Q$. We impose mild generic nondegeneracy conditions, including simplicity of poles and hyperbolicity of zeros. Generically, the real-time dynamics become gradient-like Morse. Poles play the role of hyperbolic saddle points. At poles, however, solutions may blow up in finite time. On the Riemann sphere $w\in\widehat{\mathbb{C}}$, we classify the resulting global dynamics up to $C^0$ orbit equivalence, in real time. This relies on a global description of the connection graph of blow-up orbits, from sources towards saddles/poles, in forward time. Time reversal identifies the dual graph of blow-down orbits. We show that the blow-up and blow-down graphs of (*) realize all finite multi-graphs on $\mathbb{S}^2$, equivalently. The purely polynomial case $Q=1$ realizes all planar trees, alias diagrams of non-intersecting circle chords. The anti-holomorphic cousin $P=1$ realizes all noncrossing trees with vertices restricted to circles. This classification provide combinatorial counts for the number of global phase portraits, which only depend on the degrees of $P$ and $Q$, respectively.

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arXiv2024

Scalar polynomial vector fields in real and complex time

In the present paper, the simplest scalar ODE case is studied for polynomials $$ \dot{w}=f(w)=(w-e_0)\cdot\ldots\cdot(w-e_{d-1}) $$ of degree $d$ with $d$ simple complex zeros. The explicit solution by separation of variables and explicit integration is an almost trivial matter. In a classical spirit, indeed, we describe the complex Riemann surface $\mathcal{R}$ of the global nontrivial solution $(t,w(t))$ in complex time, as an unbranched cover of the punctured Riemann sphere. The flow property, however, fails at $w=\infty$. The global consequences depend on the period map of the residues $2\pi\mathrm{i}/f'(e_j)$ of $1/f$ at the punctures, in detail. We therefore show that polynomials $f$ exist for arbitrarily prescribed residues with zero sum. This result is not covered by standard interpolation theory. Motivated by the PDE case, we also classify the planar real-time phase portraits of the above ODE. Poincar\'e compactification of $w$ regularizes $w=\infty$ by $2(d-1)$ equilibria, alternatingly stable and unstable within the invariant circle boundary at infinity. In structurally stable cases, we classify all compactified phase portraits, up to orientation preserving orbit equivalence and time reversal. Combinatorially, their connection graphs are equivalent to certain unrooted, unlabeled, undirected planar trees of $d$ vertices or, dually, to certain chord diagrams with $d-1$ nonintersecting chords. We sketch a proof that all planar trees are actually realized by Poincar\'e compactifications. Not least, we offer a 1,000 Euro reward for the discovery, or refutation, of complex entire homoclinic orbits.

math.DS↗
arXiv2024

Real chaos and complex time

Real vector fields $\dot{z} = f(z)$ in $\mathbb{R}^N$ extend to $\mathbb{C}^N$, for complex entire $f$. One known consequence are exponentially small upper bounds \begin{equation*} \label{*} C_η\exp(-η/\varepsilon) \tag{*} \end{equation*} on homoclinic splittings under discretizations of step size $\varepsilon>0$, or under rapid forcings of that period. Here the complex time extension of $Γ(t)$ is assumed to be analytic in the complex horizontal strip $|\mathrm{Im}\, t|\leq η$. The phenomenon relates to adiabatic elimination, infinite order averaging, invisible chaos, and backward error analysis. However, what if $Γ(t)$ itself were complex entire? Then $η$ could be chosen arbitrarily large. We consider connecting orbits $Γ(t)$ between limiting hyperbolic equilibria $f(v_\pm)=0$, for real $t\rightarrow\pm\infty$. For the linearizations $f'(v_\pm)$, we assume real eigenvalues which are nonresonant, separately at $v_\pm$. We then show the existence of singularities of $Γ(t)$ in complex time $t$. In that sense, real connecting orbits are accompanied by finite time blow-up, in imaginary time. Moreover, the singularities bound admissible $η$ in exponential estimates \eqref{*}. The cases of complex or resonant eigenvalues are completely open. We therefore offer a 1,000 Euro reward to any mathematician, up to and including non-permanent PostDoc level, who first comes up with a complex entire homoclinic orbit $Γ(t)$, in the above setting. Such an example would exhibit ultra-exponentially small separatrix splittings, and ultra-invisible chaos, under discretization. We also provide a time-reversible example of an entire periodic orbit with ultra-sharp Arnold tongues, alias ultra-invisible phaselocking, under discretization.

math.DS↗
arXiv2024

Real eternal PDE solutions are not complex entire: a quadratic parabolic example

In parabolic or hyperbolic PDEs, solutions which remain uniformly bounded for all real times $t=r\in\mathbb{R}$ are often called PDE entire or eternal. For example, consider the quadratic parabolic PDE \begin{equation*} \label{*} w_t=w_{xx}+6w^2-\lambda, \tag{*} \end{equation*} for $0 0$, and are currently limited to unstable dimensions $n\leq22$, or to fast unstable manifolds of dimensions $d<1+\tfrac{1}{\sqrt{2}}n$.

math.AP↗
arXiv2023

Design of Sturm global attractors 2: Time-reversible Chafee-Infante lattices of 3-nose meanders

This sequel continues our exploration arxiv:2302.12531 of a deceptively ``simple'' class of global attractors, called Sturm due to nodal properties. They arise for the semilinear scalar parabolic PDE \begin{equation}\label{eq:*} u_t = u_{xx} + f(x,u,u_x) \tag{$*$} \end{equation} on the unit interval $0 < x<1$, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ODE boundary value problem of equilibrium solutions $u=v(x)$. Specifically, we address meanders with only three ``noses'', each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm of 1974, with cubic nonlinearity $f=f(u)$, features just two noses. We present, and fully prove, a precise description of global PDE connection graphs, graded by Morse index, for such gradient-like Morse-Smale systems \eqref{eq:*}. The directed edges denote PDE heteroclinic orbits $v_1 \leadsto v_2$ between equilibrium vertices $v_1, v_2$ of adjacent Morse index. The connection graphs can be described as a lattice-like structure of Chafee-Infante subgraphs. However, this simple description requires us to adjoin a single ``equilibrium'' vertex, formally, at Morse level -1. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graphs then also exhibit global time reversibility.

math.AP↗
arXiv2023

Design of Sturm global attractors 1: Meanders with three noses, and reversibility

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic PDE \begin{equation*}\label{eq:*} u_t = u_{xx} + f(x,u,u_x) %\tag{$*$} \end{equation*} on the unit interval $0 < x<1$, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ODE boundary value problem of equilibrium solutions $u_t=0$. Specifically, we address meanders with only three "noses", each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity $f=f(u)$, features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits $v_1 \leadsto v_2$ between equilibrium vertices $v_1, v_2$ of adjacent Morse index. The global attractor turns out to be a ball of dimension $d$, given as the closure of the unstable manifold $W^u(\mathcal{O})$ of the unique equilibrium with maximal Morse index $d$. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the ($d$-1)-sphere boundary of the global attractor.

math.AP↗
arXiv2021

Meanders, zero numbers and the cell structure of Sturm global attractors

We study global attractors $\mathcal{A}=\mathcal{A}_f$ of semiflows generated by semilinear partial parabolic differential equations of the form $u_t = u_{xx} + f(x,u,u_x), 0<x<1$, satisfying Neumann boundary conditions. The equilibria $v\in\mathcal{E}\subset\mathcal{A}$ of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of $\mathcal{A}$ into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors $\mathcal{A}$ as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation $σ=σ_f$ derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries $x=0,1$, respectively, completely determines the Sturm global attractor $\mathcal{A}$. Equivalently, we use a planar curve, the meander $\mathcal{M}=\mathcal{M}_f$, associated to the the ODE boundary value problem by shooting. The main objective of this paper is to derive a minimax property which identifies the equilibria on the cell boundary of $\mathcal{O}$ which are closest or most distant from $\mathcal{O}$ at the boundaries $x=0,1$, directly from the permutation $σ$, the Sturm permutation, or equivalently from the meander $\mathcal{M}$, the Sturm meander, based on the Sturm nodal properties of the solutions of the ODE boundary value problem. We emphasize the local aspect of this result by applying it to an example for which the identification of the equilibria in the cell boundary of $\mathcal{O}$ is obtained from the knowledge of only a section of the Sturm meander $\mathcal{M}$.

math.DS↗
arXiv2021

2-Cluster Fixed-Point Analysis of Mean-Coupled Stuart-Landau Oscillators in the Center Manifold

We reduce the dynamics of an ensemble of mean-coupled Stuart-Landau oscillators close to the synchronized solution. In particular, we map the system onto the center manifold of the Benjamin-Feir instability, the bifurcation destabilizing the synchronized oscillation. Using symmetry arguments, we describe the structure of the dynamics on this center manifold up to cubic order, and derive expressions for its parameters. This allows us to investigate phenomena described by the Stuart-Landau ensemble, such as clustering and cluster singularities, in the lower-dimensional center manifold, providing further insights into the symmetry-broken dynamics of coupled oscillators. We show that cluster singularities in the Stuart-Landau ensemble correspond to vanishing quadratic terms in the center manifold dynamics. In addition, they act as organizing centers for the saddle-node bifurcations creating unbalanced cluster states as well for the transverse bifurcations altering the cluster stability. Furthermore, we show that bistability of different solutions with the same cluster-size distribution can only occur when either cluster contains at least $1/3$ of the oscillators, independent of the system parameters.

nlin.CD↗
arXiv2021

Global heteroclinic rebel dynamics among large 2-clusters in permutation equivariant systems

We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study one-parameter vector fields, up to cubic order, which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the linearization at the trivial 1-cluster equilibrium of total synchrony. All equilibria are cluster solutions involving up to three clusters. The resulting global dynamics is of gradient type: all bounded solutions are cluster equilibria and heteroclinic orbits between them. In the limit of large $N$, we present a detailed analysis of the web of heteroclinic orbits among the plethora of 2-cluster equilibria. Our focus is on the global dynamics of 3-cluster solutions with one rebel cluster of small size. These solutions describe slow relative growth and decay of 2-cluster states. For $N\rightarrow\infty$, the limiting heteroclinic web defines an integrable \emph{rebel flow} in the space of 2-cluster equilibrium configurations. We identify and study the seven qualitatively distinct global rebel flows which arise in this setting. Applications include oscillators with all-to-all coupling, and electrochemistry. For illustration we consider synchronization clusters among $N$ complex Stuart-Landau oscillators with complex linear global coupling.

nlin.CD↗
arXiv2020

Boundary orders and geometry of the signed Thom-Smale complex for Sturm global attractors

We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE \begin{equation}\label{eq:*} u_t = u_{xx} + f(x,u,u_x) \tag{$*$} \end{equation} on the unit interval $0 < x<1$ with Neumann boundary conditions. We assume $f$ to be dissipative with $N$ hyperbolic equilibria $v\in\mathcal{E}$. The global attractor $\mathcal{A}$ of \eqref{eq:*}, also called \emph{Sturm global attractor}, consists of the unstable manifolds of all equilibria $v$. As cells, these form the \emph{Thom-Smale complex} $\mathcal{C}$. Based on the fast unstable manifolds of $v$, we introduce a refinement $\mathcal{C}^s$ of the regular cell complex $\mathcal{C}$, which we call the \emph{signed Thom-Smale complex}. Given the signed cell complex $\mathcal{C}^s$ and its underlying partial order, only, we derive the two total boundary orders $h_ι:\{1,\ldots , N\}\rightarrow\mathcal{E}$ of the equilibrium values $v(x)$ at the two Neumann boundaries $ι=x=0,1$. In previous work we have already established how the resulting Sturm permutation \[σ:=h_{0}^{-1} \circ h_1,\] conversely, determines the global attractor $\mathcal{A}$ uniquely, up to topological conjugacy.

math.DS↗
arXiv2020

Global Hopf bifurcation in networks with fast feedback cycles

Autonomous sustained oscillations are ubiquitous in living and nonliving systems. As open systems, far from thermodynamic equilibrium, they defy entropic laws which mandate convergence to stationarity. We present structural conditions on network cycles which support global Hopf bifurcation, i.e. global bifurcation of non-stationary time-periodic solutions from stationary solutions. Specifically, we show how monotone feedback cycles of the linearization at stationary solutions give rise to global Hopf bifurcation, for sufficiently dominant coefficients along the cycle. We include four example networks which feature such strong feedback cycles of length three and larger: Oregonator chemical reaction networks, Lotka-Volterra ecological population dynamics, citric acid cycles, and a circadian gene regulatory network in mammals. Reaction kinetics in our approach are not limited to mass action or Michaelis-Menten type.

math.DS↗
arXiv2019

Coexistence of infinitely many large, stable, rapidly oscillating periodic solutions in time-delayed Duffing oscillators

We explore stability and instability of rapidly oscillating solutions $x(t)$ for the hard spring delayed Duffing oscillator $$x''(t)+ ax(t)+bx(t-T)+x^3(t)=0.$$ Fix $T>0$. We target periodic solutions $x_n(t)$ of small minimal periods $p_n=2T/n$, for integer $n\rightarrow \infty$, and with correspondingly large amplitudes. Note how $x_n(t)$ are also marginally stable solutions, respectively, of the two standard, non-delayed, Hamiltonian Duffing oscillators $$x''+ ax+(-1)^nbx+x^3=0.$$ Stability changes for the delayed Duffing oscillator. Simultaneously for all sufficiently large $n\geq n_0$, we obtain local exponential stability for $(-1)^nb<0$, and exponential instability for $(-1)^nb>0$, provided that $$0 \neq (-1)^{n+1}b\,T^2< \tfrac{3}{2}π^2.$$ We interpret our results in terms of noninvasive delayed feedback stabilization and destabilization for large amplitude rapidly periodic solutions of the standard Duffing oscillators. We conclude with numerical illustrations of our results for small and moderate $n$ which also indicate a Neimark-Sacker torus bifurcation at the validity boundary of our theoretical results.

math.DS↗
arXiv2018

A Lyapunov function for fully nonlinear parabolic equations in one spatial variable

Lyapunov functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano's method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.

math.DS↗
arXiv2018

Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$ u_t = u_{xx} + f(x,u,u_x)\,, $$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria $v_t=0$ are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors $\mathcal{A}$ as 3-cell templates $\mathcal{C}$. The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $x=0$ and $x=1$, respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors $\mathcal{A}$ with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.

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