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Bernold Fiedler

Publications and source records attributed to Bernold Fiedler.

28 records · Page 2Linked to original sources
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.

math.DS↗
arXiv2018

Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay

We study scalar delay equations $$\dot{x} (t) = λf(x(t-1)) + b^{-1} (x(t) + x(t -p/2))$$ with odd nonlinearity $f$, real nonzero parameters $λ, \, b$, and two positive time delays $1,\ p/2$. We assume supercritical Hopf~bifurcation from $x \equiv 0$ in the well-understood single-delay case $b = \infty$. Normalizing $f' (0)=1$, branches of constant minimal period $p_k = 2π/ω_k$ are known to bifurcate from eigenvalues $iω_k = i(k+\tfrac{1}{2})π$ at $λ_k = (-1)^{k+1}ω_k$, for any nonnegative integer $k$. The unstable dimension of these rapidly oscillating periodic solutions is $k$, at the local branch $k$. We obtain stabilization of such branches, for arbitrarily large unstable dimension $k$, and for, necessarily, delicately narrow regions of control amplitudes $b < 0$. For $p$:= $p_k$ the branch $k$ of constant period $p_k$ persists as a solution, for any $b\neq 0$. Indeed the delayed feedback term controlled by $b$ vanishes on branch $k$: the feedback control is noninvasive there. Following an idea of Pyragas (1992), we seek parameter regions $\mathcal{P} = (\underline{b}_k,\overline{b}_k)$ of controls $b \neq 0$ such that the branch $k$ becomes stable, locally at Hopf~bifurcation. We determine rigorous expansions for $\mathcal{P}$ in the limit of large $k$. Our analysis is based on a 2-scale covering lift for the slow and rapid frequencies involved. These results complement earlier results by Fiedler and Oliva (2016) which required control terms $$b^{-1} (x(t-\vartheta) + x(t-\vartheta -p/2))$$ with a third delay $\vartheta$ near 1.

math.DS↗
arXiv2017

Sensitivity of chemical reaction networks: a structural approach 3. Regular multimolecular systems

We present a systematic mathematical analysis of the qualitative steady-state response to rate perturbations in large classes of reaction networks. This includes multimolecular reactions and allows for catalysis, enzymatic reactions, multiple reaction products, nonmonotone rate functions, and non-closed autonomous systems. Our structural sensitivity analysis is based on the stoichiometry of the reaction network, only. It does not require numerical data on reaction rates. Instead, we impose mild and generic nondegeneracy conditions of algebraic type. From the structural data, only, we derive which steady-state concentrations are sensitive to, and hence influenced by, changes of any particular reaction rate - and which are not. We also establish transitivity properties for influences involving rate perturbations. This allows us to derive an influence graph which globally summarizes the influence pattern of the given network. The influence graph allows the computational, but meaningful, automatic identification of functional subunits in general networks, which hierarchically influence each other. We illustrate our results for several variants of the glycolytic citric acid cycle. Biological applications include enzyme knockout experiments, and metabolic control.

math.DS↗
arXiv2017

Sturm 3-ball global attractors 1: Thom-Smale complexes and meanders

This is the first of three papers on the geometric and combinatorial characterization of global Sturm attractors 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 are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. Our geometric description involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres. The combinatorial description is 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 show the implications $$ \text{Sturm attractor}\quad \Longrightarrow \quad \text{Thom-Smale complex} \quad \Longrightarrow \quad \text{meander}\,.$$ The sequel, part 2, closes the cycle of equivalences by the implication $$ \text{meander} \quad \Longrightarrow \quad \text{Sturm attractor}\,.$$ Many explicit examples will be discussed in part 3. The present 3-ball trilogy extends our previous trilogy on planar Sturm global attractors towards the still elusive goal of geometric and combinational characterizations of all Sturm global attractors of arbitrary dimension.

math.DS↗
arXiv2017

Sturm 3-ball global attractors 2: Design of Thom-Smale complexes

This is the second of three papers on the geometric and combinatorial characterization of global Sturm attractors 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 are assumed to be hyperbolic.\\ \newline 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. Our geometric description involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres.\\ \newline The combinatorial description is in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem.\\ \newline In the first paper we showed the implications $$ \text{Sturm attractor}\quad \Longrightarrow \quad \text{Thom-Smale complex} \quad \Longrightarrow \quad \text{meander}\,.$$ The present part 2, closes the cycle of equivalences by the implication $$ \text{meander} \quad \Longrightarrow \quad \text{Sturm attractor}\,.$$ In particular this cycle allows us to construct a unique Sturm 3-ball attractor for any prescribed Thom-Smale complex which satisfies the geometric properties of the bipolar orientation and the hemisphere decomposition. Many explicit examples and illustrations will be discussed in part 3. The present 3-ball trilogy, however, is just another step towards the still elusive geometric and combinational characterization of all Sturm global attractors in arbitrary dimensions.

math.DS↗
arXiv2016

Global Dynamics, Blow-Up, and Bianchi Cosmology

Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type. Specific questions concern the convergence to equilibria, the existence of periodic, homoclinic, and heteroclinic solutions, and the existence and geometric structure of global attractors. On the other hand, many solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities, or to understand their geometry in conjunction with globally bounded solutions. In this context we have also aimed at global qualitative descriptions of blow-up and grow-up phenomena.

math.DS↗
arXiv2015

Stable and transient multi-cluster oscillation death in nonlocally coupled networks

In a network of nonlocally coupled Stuart-Landau oscillators with symmetry-breaking coupling, we study numerically, and explain analytically, a family of inhomogeneous steady states (oscillation death). They exhibit multi-cluster patterns, depending on the cluster distribution prescribed by the initial conditions. Besides stable oscillation death, we also find a regime of long transients asymptotically approaching synchronized oscillations. To explain these phenomena analytically in dependence on the coupling range and the coupling strength, we first use a mean-field approximation which works well for large coupling ranges but fails for coupling ranges which are small compared to the cluster size. Going beyond standard mean-field theory, we predict the boundaries of the different stability regimes as well as the transient times analytically in excellent agreement with numerical results.

nlin.AO↗
arXiv2012

Anisotropic Einstein data with isotropic nonnegative scalar curvature

We construct a black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of initial data contained inside the black hole. The constraints translate into a parabolic equation, with radius as "time" variable, on a metric component u that undergoes blow up. The metric itself is regular up to and including the surface at blow up radius, which is a minimal surface. By applying equivariant bifurcation theory on a self similarly rescaled equation, we show the existence of blow up profiles that are not O(3) symmetric - or anisotropic - although the curvature was isotropically prescribed.

math.DG↗
arXiv2003

Bifurcations without parameters: some ODE and PDE examples

Standard bifurcation theory is concerned with families of vector fields $dx/dt = f(x,λ)$, $x \in \R^n$, involving one or several constant real parameters $λ$. Viewed as a differential equation for the pair $(x,λ)$, we observe a foliation of the total phase space by constant $λ$. Frequently, the presence of a trivial stationary solution $x=0$ is also imposed: $0 = f(0,λ)$. Bifurcation without parameters, in contrast, discards the foliation by a constant parameter $λ$. Instead, we consider systems $dx/dt = f(x,y), dy/dt = g(x,y)$. Standard bifurcation theory then corresponds to the special case $y=λ, g=0$. To preserve only the trivial solution $x=0$, instead, we only require $0 = f(0,y) = g(0,y)$ for all $y$. A rich dynamic phenomenology arises, when normal hyperbolicity of the trivial stationary manifold $x=0$ fails, due to zero or purely imaginary eigenvalues of the Jacobian $f_x(0,y)$. Specifically, we address the cases of failure of normal hyperbolicity due to a simple eigenvalue zero, a simple purely imaginary pair (Hopf bifurcation without parameters), a double eigenvalue zero (Takens-Bogdanov bifurcation without parameters), and due to a double eigenvalue zero with additional time reversal symmetries. The results are joint work with Andrei Afendikov, James C. Alexander, and Stefan Liebscher.

math.DS↗