The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions

Turing instability and spatially homogeneous Hopf bifurcation in a diffusive Brusselator system

The present paper deals with a reaction–diffusion Brusselator system subject to the homogeneous Neumann boundary condition. When the effect of spatial diffusion is neglected, the local asymptotic stability and the detailed Hopf bifurcation of the unique positive equilibrium of the associated ODE system are analyzed. In the stable domain of the ODE system, the effect of spatial diffusion is explored, and local asymptotic stability, Turing instability and existence of Hopf bifurcation of the constant positive equilibrium are demonstrated. In addition, the direction of spatially homogeneous Hopf bifurcation and the stability of the spatially homogeneous bifurcating periodic solutions are also investigated. Finally, numerical simulations are also provided to check the obtained theoretical results

Amplitude Death: The emergence of stationarity in coupled nonlinear systems

When nonlinear dynamical systems are coupled, depending on the intrinsic dynamics and the manner in which the coupling is organized, a host of novel phenomena can arise. In this context, an important emergent phenomenon is the complete suppression of oscillations, formally termed amplitude death (AD). Oscillations of the entire system cease as a consequence of the interaction, leading to stationary behavior. The fixed points that the coupling stabilizes can be the otherwise unstable fixed points of the uncoupled system or can correspond to novel stationary points. Such behaviour is of relevance in areas ranging from laser physics to the dynamics of biological systems. In this review we discuss the characteristics of the different coupling strategies and scenarios that lead to AD in a variety of different situations, and draw attention to several open issues and challenging problems for further study.Comment: Physics Reports (2012

Dynamics in a delayed diffusive cell cycle model

In this paper, we construct a delayed diffusive model to explore the spatial dynamics of cell cycle in G2/M transition. We first obtain the local stability of the unique positive equilibrium for this model, which is irrelevant to the diffusion. Then, through investigating the delay-induced Hopf bifurcation in this model, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. Applying the normal form and center manifold theorem of functional partial differential equations, we also determine the stability and direction of these bifurcating periodic solutions. Finally, numerical simulations are presented to validate our theoretical results

Anomalous Wave Dispersion in the Cyclohexanedione-Bromate Chemical Oscillator

A modified six-variable Oregonator model presented here successfully reproduces a significant portion of the behavior observed in the Ferroin-catalyzed cyclohexanedione variant of the Belousov-Zhabotinsky (CHD-BZ) reaction. The phenomena of anomalous velocity dispersion (in which following waves may catch up to, rather than fall behind an initial excitation wave), wave-stacking, and backfiring have been successfully reproduced numerically as resulting from non-monotonic [Br-] decay to the steady state in the wake of an excitation pulse. The non-monotonic decay is seen as a dip in [Br-] following the passage of a chemical wave. This dip in [Br-] decay curve allows a following wave to accelerate and catch up to the initial wave. The origin of anomalous dispersion as the result of such a non-monotonic decay curve in [Br-] has been suggested previously by Steinbock et al. and Szalai et al. However, the work presented here is the first successful representation of anomalous wave-velocity dispersion using a chemical model. This model is based on the well-understood chemistry of the Oregonator model of the Belousov-Zhabotinsky reaction, coupled to a second pathway (based on chemistry related to uncatalyzed bromate oscillators) for the oxidation of organic substrate to provide the new dynamics

Stochastic switching of delayed feedback suppresses oscillations in genetic regulatory systems

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical, and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.Comment: updated: 13 pages, 5 figure

Controlling turbulence and pattern formation in chemical reactions

Räumlich ausgedehnte Systeme fern des thermodynamischen Gleichgewichts zeichnen sich durch die Fähigkeit aus, spontan raumzeitliche Strukturen und Turbulenz auszubilden. Die vorliegende Arbeit beschäftigt sich theoretisch und experimentell mit der Steuerung und Kontrolle derartiger Phänomene. Als Beispiel wird die katalytische Oxidationsreaktion von Kohlenmonoxid auf einer Platin-Einkristalloberfläche untersucht. Um Turbulenz zu unterdrücken sowie um neuartige Muster in dieses System zu induzieren werden zwei verschiedene Steuerungsverfahren, globale verzögerte Rückkopplung und periodische Forcierung, eingesetzt. Die Effekte einer künstlich implementierten globalen Rückkopplungsschleife werden zunächst in einem mathematischen Reaktions-Diffusions-Modell der CO-Oxidation auf Pt(110) mit Hilfe numerischer Simulationen untersucht. Durch Variation eines globalen Kontrollparameters in Abhängigkeit einer räumlich gemittelten Systemgröße lässt sich chemische Turbulenz in dem Modell unterdrücken und ein homogen oszillierender Zustand stabilisieren. Weiterhin kann eine Vielzahl komplexer raumzeitlicher Strukturen, beispielsweise "phase flips", asynchrone Oszillationen, intermittente Turbulenz in Form chaotischer Kaskaden von Blasen und Ringstrukturen, zelluläre Strukturen und verschiedene Arten von Domänenmustern induziert werden. Die simulierten raumzeitlichen Muster werden mit Hilfe einer zuvor entwickelten Transformation zu Phasen- und Amplitudenvariablen charakterisiert und analysiert. Es zeigt sich, daß die erhaltenen Strukturen große Ähnlichkeit mit dem Verhalten eines generischen Modells, der komplexen Ginzburg-Landau-Gleichung mit globaler Kopplung, aufweisen. Eine globale verzögerte Rückkopplung kann in Experimenten mit der CO-Oxidation auf Pt(110) durch eine externe, zustandsabhängige Variation des CO-Partialdrucks in der Reaktionskammer realisiert werden. Die sich auf der Platinoberfläche ausbildenden Bedeckungsmuster werden dabei mit Hilfe von Photoemissions-Elektronenmikroskopie sichtbar gemacht. In solchen Experimenten kann chemische Spiralwellenturbulenz erstmals unterdrückt und ein Großteil der vorhergesagten Muster - unter anderem intermittente Turbulenz, Domänenmuster und zelluläre Strukturen - tatsächlich nachgewiesen werden. Die experimentell beobachteten Muster werden ebenfalls durch eine Phasen- und Amplitudendarstellung charakterisiert. In weiteren Experimenten wird die Wirkung periodischer Partialdruckmodulationen auf chemische Turbulenz untersucht. Auch mittels dieser Methode läßt sich Spiralwellenturbulenz unterdrücken und eine Vielfalt komplexer Muster induzieren. Als resonante Strukturen sind irreguläre Streifenmuster in subharmonischer Resonanz sowie Domänenmuster mit koexistenten Resonanzen zu nennen. Zudem treten auch nichtresonante Muster in Form intermittenter Turbulenz und ungeordneter zellulärer Strukturen auf. Die Resultate dieser Arbeit zeigen somit, daß sich mit Hilfe globaler Rückkopplung und periodischer Forcierung Turbulenz und Strukturbildung in der betrachteten Oberflächenreaktion wirkungsvoll kontrollieren und manipulieren lassen. Ähnliche Phänomene können auch in anderen Reaktions-Diffusions-Systemen erwartet werden.Spontaneous pattern formation and spatiotemporal chaos (turbulence) are common features of spatially extended nonlinear systems maintained far from equilibrium. The aim of this work is to control and engineer such phenomena. As an example, the catalytic oxidation of carbon monoxide on a platinum (110) single crystal surface is considered. In order to control turbulence and to manipulate pattern formation in this reaction, two different control methods, global delayed feedback and periodic forcing, are employed. The effects of a global delayed feedback on the self-organized behavior of the system are first studied numerically in a reaction-diffusion model of CO oxidation on Pt(110). By applying a global control force generated by the spatially averaged state of one of the system variables, turbulence can be suppressed and uniform oscillations can be stabilized. Moreover, global delayed feedback can be used as a tool to produce a variety of complex spatiotemporal patterns, including phase flips, asynchronous oscillations, intermittent turbulence represented by irregular cascades of ring-shaped objects on a uniformly oscillating background, cellular structures, and different types of cluster patterns. The simulated structures are analyzed using a newly developed transformation to phase and amplitude variables designed for non-harmonic oscillations. The obtained patterns resemble the structures exhibited by a general model, the complex Ginzburg-Landau equation with global feedback. The simulated phenomena of pattern formation are then tested in laboratory experiments with CO oxidation on Pt(110). Global delayed feedback is introduced into the system via a controlled state-dependent variation of the CO partial pressure in the reaction chamber. The spatiotemporal patterns developing on the catalytic surface are imaged by means of photoemission electron microscopy. In such experiments, it is shown that chemical turbulence can be suppressed and a large part of the predicted patterns, including intermittent turbulence, clusters, and cellular structures, can be indeed observed. The experimentally obtained patterns are also transformed into the corresponding spatial distributions of oscillation phase and amplitude. In a further set of experimental investigations, the effects of periodic external forcing on chemical turbulence in CO oxidation on Pt(110) are studied. Using this method, turbulence can be also suppressed and several complex patterns can be induced. The observed frequency locked structures are represented by irregular stripes in subharmonic resonance with the forcing and cluster patterns with coexistent resonances. In addition, non-resonant patterns such as intermittent turbulence and disordered cellular structures are found. Thus, the results of this work demonstrate that by means of global delayed feedback and periodic forcing, turbulence and pattern formation can be effectively controlled and manipulated in the considered surface reaction. Similar phenomena are expected to arise also in other reaction-diffusion systems of various origins