arXiv subjects

Harald Engel

Publications and source records attributed to Harald Engel.

16 recordsLinked to original sources

arXiv2018

First-order synchronization transition in a large population of strongly coupled relaxation oscillators

Onset and loss of synchronization in coupled oscillators are of fundamental importance in understanding emergent behavior in natural and man-made systems, which range from neural networks to power grids. We report on experiments with hundreds of strongly coupled photochemical relaxation oscillators that exhibit a discontinuous synchronization transition with hysteresis, as opposed to the paradigmatic continuous transition expected from the widely used weak coupling theory. The resulting first-order transition is robust with respect to changes in network connectivity and natural frequency distribution. This allows us to identify the relaxation character of the oscillators as the essential parameter that determines the nature of the synchronization transition. We further support this hypothesis by revealing the mechanism of the transition, which cannot be accounted for by standard phase reduction techniques.

nlin.AO↗

arXiv2018

Control of traveling localized spots

Traveling localized spots represent an important class of self-organized two-dimensional patterns in reaction-diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot's concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise time-consuming calculations.

nlin.PS↗

arXiv2018

Oscillatory motion of a droplet in an active poroelastic two-phase model

We investigate flow-driven amoeboid motility as exhibited by microplasmodia of Physarum polycephalum. A poroelastic two-phase model with rigid boundaries is extended to the case of free boundaries and substrate friction. The cytoskeleton is modeled as an active viscoelastic solid permeated by a fluid phase describing the cytosol. A feedback loop between a chemical regulator, active mechanical deformations, and induced flows gives rise to oscillatory and irregular motion accompanied by spatio-temporal contraction patterns. We cover extended parameter regimes of active tension and substrate friction by numerical simulations in one spatial dimension and reproduce experimentally observed oscillation periods and amplitudes. In line with experiments, the model predicts alternating forward and backward ectoplasmatic flow at the boundaries with reversed flow in the center. However, for all cases of periodic and irregular motion, we observe practically no net motion. A simple theoretical argument shows that directed motion is not possible with a spatially independent substrate friction.

cond-mat.soft↗

arXiv2018

Control of Nonlinear Wave Solutions to Neural Field Equations

Neural field equations offer a continuous description of the dynamics of large populations of synaptically coupled neurons. This makes them a convenient tool to describe various neural processes, such as working memory, motion perception, and visual hallucinations, to name a few. Due to the important applications, the question arises how to effectively control solutions in such systems. In this work, we investigate the problem of position control of traveling wave solutions to scalar neural field equations on the basis of singular perturbation analysis. Thereby, we consider different means of control such as spatio-temporal modulations of the neural firing threshold, asymmetric synaptic coupling kernels, and additive inputs. Treating these controls as perturbations to the neural field system, one obtains an equation of motion for traveling wave solutions in response to the applied controls. Subsequently, we pose the inverse question of how to design controls that lead to a propagation of the solution following a desired velocity protocol. In particular, we make use of a specific excitation of the solution's translational modes which enables an explicit calculation of a necessary control signal for a given velocity protocol without evoking shape deformations. Moreover, we derive an equivalent control method relying on a modulation of the neurons' synaptic footprint.

nlin.PS↗

arXiv2016

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

This work deals with the position control of selected patterns in reaction-diffusion systems. Exemplarily, the Schlögl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions.

nlin.PS↗

arXiv2015

Spatial Confinement Causes Lifetime Enhancement and Expansion of Vortex Rings with Positive Filament Tension

We study the impact of spatial confinement on the dynamics of three-dimensional excitation vortices with circular filaments. In a chemically active medium we observe a decreased contraction of such scroll rings and even expanding ones, despite of their positive filament tension. We propose a kinematical model which takes into account the interaction of the scroll ring with a confining Neumann boundary. The model reproduces all experimentally observed regimes of ring evolution, and correctly predicts the results obtained by numerical simulations of the underlying reaction-diffusion equations.

nlin.PS↗

arXiv2015

Control of transversal instabilities in reaction-diffusion systems

In two-dimensional reaction-diffusion systems, local curvature perturbations in the shape of traveling waves are typically damped out and disappear in the course of time. If, however, the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated wave shapes and to spreading spiral turbulence. For experimentally relevant parameter values, the photosensitive Belousov-Zhabotinsky reaction (PBZR) does not exhibit transversal wave instabilities. Here, we propose a mechanism to artificially induce these instabilities via a wave shape dependent spatio-temporal feedback loop, and study the emerging wave patterns. In numerical simulations with the modified Oregonator model for the PBZR using experimentally realistic parameter values we demonstrate the feasibility of this control scheme. Conversely, in a piecewise-linear version of the FitzHugh-Nagumo model transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilising flat wave propagation.

nlin.PS↗

arXiv2014

Shaping wave patterns in reaction-diffusion systems

We present a method to control the two-dimensional shape of traveling wave solutions to reaction-diffusion systems, as e.g. interfaces and excitation pulses. Control signals that realize a pre-given wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation. While the control enforces a desired wave shape perpendicular to the local propagation direction, the wave profile along the propagation direction itself remains almost unaffected. Provided that the one-dimensional wave profile and its propagation velocity can be measured experimentally, and the diffusion coefficients of the reacting species are given, the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown.

nlin.PS↗

arXiv2014

Three-Dimensional Autonomous Pacemaker in the Photosensitive Belousov-Zhabotinsky medium

In experiments with the photosensitive Belousov-Zhabotinsky reaction (PBZR) we found a stable three-dimensional organizing center that periodically emits trigger waves of chemical concentration. The experiments are performed in a parameter regime with negative line tension using an open gel reactor to maintain stationary non-equilibrium conditions. The observed periodic wave source is formed by a scroll ring stabilized due to its interaction with a no-flux boundary. Sufficiently far from the boundary, the scroll ring expands and undergoes the negative line tension instability before it finally develops into scroll wave turbulence. Our experimental results are reproduced by numerical integration of the modified Oregonator model for the PBZR. Stationary and breathing self-organized pacemakers have been found in these numerical simulations. In the latter case, both the radius of the scroll ring and the distance of its filament plane to the no-flux boundary after some transient undergo undamped stable limit cycle oscillations. So far, in contrary to their stationary counterpart, the numerically predicted breathing autonomous pacemaker has not been observed in the chemical experiment.

nlin.PS↗

arXiv2014

Control of chemical wave propagation

Using the Schlögl model as a paradigmatic example of a bistable reaction-diffusion system, we discuss some physically feasible options of open and closed loop spatio-temporal control of chemical wave propagation.

nlin.PS↗

arXiv2014

No-flux boundaries stabilize scroll rings in excitable media with negative filament tension

Scroll rings in an unbounded excitable medium with negative line tension undergo an instability ending eventually in a "turbulent" state, known as scroll wave (Winfree) turbulence. In this paper we demonstrate by numerical simulations based upon the Oregonator model for the photosensitive Belousov-Zhabotinskii reaction (PBZR) that the Winfree turbulence is suppressed by the interaction of the scroll ring with a confining Neumann boundary. Instead of the Winfree turbulence a stable scroll ring forms due to the boundary interaction. Furthermore, we will discuss the conditions under which boundary-stabilized scroll rings could be observed in the PBZR, taking into account a light-induced excitability gradient in parallel to the scroll ring's symmetry axis.

nlin.PS↗

arXiv2014

Controlling the position of traveling waves in reaction-diffusion systems

We present a method to control the position as a function of time of one-dimensional traveling wave solutions to reaction-diffusion systems according to a pre-specified protocol of motion. Given this protocol, the control function is found as the solution of a perturbatively derived integral equation. Two cases are considered. First, we derive an analytical expression for the space ($x$) and time ($t$) dependent control function $f\left(x,t\right)$ that is valid for arbitrary protocols and many reaction-diffusion systems. These results are close to numerically computed optimal controls. Second, for stationary control of traveling waves in one-component systems, the integral equation reduces to a Fredholm integral equation of the first kind. In both cases, the control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.

nlin.PS↗

arXiv2013

Analytical approximations for spiral waves

We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency $Ω$ and core radius $R_{0}$. For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent $Ω\left(R_{+}\right)$ dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect with radius $R_{+}$ with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the result for the dependence of the rotation frequency on the excitability of the medium.

nlin.PS↗

arXiv2013

An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum

Motivated by recent experimental studies, we derive and analyze a twodimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a model of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The model equations for the poroelastic medium are obtained from continuum force-balance equations that include the relevant mechanical fields and an incompressibility relation for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stresses. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a chemomechanical feedback. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.

nlin.PS↗

arXiv2013

Oscillatory motion of sheared nanorods beyond the nematic phase

We study the role of the control parameter triggering nematic order (temperature or concentration) on the dynamical behavior of a system of nanorods under shear. Our study is based on a set of mesoscopic equations of motion for the components of the tensorial orientational order parameter. We investigating these equations via a systematic bifurcation analysis based on a numerical continuation technique, focusing on spatially homogeneous states. Exploring a wide range of parameters we find, unexpectedly, that states with oscillatory motion can exist even under conditions where the equilibrium system is isotropic. These oscillatory states are characterized by wagging motion of the paranematic director, and they occur if the tumbling parameter is sufficiently small. We also present full non-equilibrium phase diagrams, in the plane spanned by the concentration and the shear rate.

cond-mat.soft↗

arXiv2012

Propagation Of Waves In Periodic-Heterogeneous Bistable Systems

Wave propagation in one-dimensional heterogeneous bistable media is studied using the Schlögl model as a representative example. Starting from the analytically known traveling wave solution for the homogeneous medium, infinitely extended, spatially periodic variations in kinetic parameters as the excitation threshold, for example, are taken into account perturbatively. Two different multiple scale perturbation methods are applied to derive a differential equation for the position of the front under perturbations. This equation allows the computation of a time independent average velocity, depending on the spatial period length and the amplitude of the heterogeneities. The projection method reveals to be applicable in the range of intermediate and large period lengths but fails when the spatial period becomes smaller than the front width. Then, a second order averaging method must be applied. These analytical results are capable to predict propagation failure, velocity overshoot, and the asymptotic value for the front velocity in the limit of large period lengths in qualitative, often quantitative agreement with the results of numerical simulations of the underlying reaction-diffusion equation. Very good agreement between numerical and analytical results has been obtained for waves propagating through a medium with periodically varied excitation threshold.

nlin.PS↗