Controlling turbulence in a surface chemical reaction by time-delay autosynchronization

Provenance of life: Chemical autonomous agents surviving through associative learning

We present a benchmark study of autonomous, chemical agents exhibiting associative learning of an environmental feature. Associative learning systems have been widely studied in cognitive science and artificial intelligence but are most commonly implemented in highly complex or carefully engineered systems, such as animal brains, artificial neural networks, DNA computing systems, and gene regulatory networks, among others. The ability to encode environmental information and use it to make simple predictions is a benchmark of biological resilience and underpins a plethora of adaptive responses in the living hierarchy, spanning prey animal species anticipating the arrival of predators to epigenetic systems in microorganisms learning environmental correlations. Given the ubiquitous and essential presence of learning behaviors in the biosphere, we aimed to explore whether simple, nonliving dissipative structures could also exhibit associative learning. Inspired by previous modeling of associative learning in chemical networks, we simulated simple systems composed of long- and short-term memory chemical species that could encode the presence or absence of temporal correlations between two external species. The ability to learn this association was implemented in Gray-Scott reaction-diffusion spots, emergent chemical patterns that exhibit self-replication and homeostasis. With the novel ability of associative learning, we demonstrate that simple chemical patterns can exhibit a broad repertoire of lifelike behavior, paving the way for in vitro studies of autonomous chemical learning systems, with potential relevance to artificial life, origins of life, and systems chemistry. The experimental realization of these learning behaviors in protocell or coacervate systems could advance a new research direction in astrobiology, since our system significantly reduces the lower bound on the required complexity for autonomous chemical learning.

Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction

Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh-Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.

Noise-induced standing waves in oscillatory systems with time-delayed feedback

In oscillatory reaction-diffusion systems, time-delay feedback can lead to the instability of uniform oscillations with respect to formation of standing waves. Here, we investigate how the presence of additive, Gaussian white noise can induce the appearance of standing waves. Combining analytical solutions of the model with spatiotemporal simulations, we find that noise can promote standing waves in regimes where the deterministic uniform oscillatory modes are stabilized. As the deterministic phase boundary is approached, the spatiotemporal correlations become stronger, such that even small noise can induce standing waves in this parameter regime. With larger noise strengths, standing waves could be induced at finite distances from the (deterministic) phase boundary. The overall dynamics is defined through the interplay of noisy forcing with the inherent reaction-diffusion dynamics.

Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model

We examine the prototypical Gray-Scott model, which mimics cubic autocatalytic reaction with linear decay of the autocatalyst, to model the kinetics of a reaction-diffusion system subjected to advective streamline flow. For a proper choice of boundary conditions and parameter space, the system admits wave-induced spatiotemporal chaos in the absence of flow. We show that flow above a critical value leads to an arrest of the spatiotemporal chaos due to a change in the instability from absolute to convective type. Furthermore, stationary spatial structures are borne out of a second successive bifurcation for yet another critical flow value. The theoretical formulations are corroborated by extensive numerical simulation of the full reaction-diffusion-advection system in one dimension.

Delayed feedback induced multirhythmicity in the oscillatory electrodissolution of copper

Occurrence of bi- and trirhythmicities (coexistence of two or three stable limit cycles, respectively, with distinctly different periods) has been studied experimentally by applying delayed feedback control to the copper-phosphoric acid electrochemical system oscillating close to a Hopf bifurcation point under potentiostatic condition. The oscillating electrode potential is delayed by τ and the difference between the present and delayed values is fed back to the circuit potential with a feedback gain K. The experiments were performed by determining the period of current oscillations T as a function of (both increasing and decreasing) τ at several fixed values of K. With small delay times, the period exhibits a sinusoidal type dependence on τ. However, with relatively large delays (typically τ ≫ T) for each feedback gain K, there exists a critical delay τcrit above which birhythmicity emerges. The experiments show that for weak feedback, Kτcrit is approximately constant. At very large delays, the dynamics becomes even more complex, and trirhythmicity could be observed. Results of numerical simulations based on a general kinetic model for metal electrodissolution were consistent with the experimental observations. The experimental and numerical results are also interpreted by using a phase model; the model parameters can be obtained from experimental data measured at small delay times. Analytical solutions to the phase model quantitatively predict the parameter regions for the appearance of birhythmicity in the experiments, and explain the almost constant value of Kτcrit for weak feedback.

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection

A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.

Stabilization of standing waves through time-delay feedback

Standing waves are studied as solutions of a complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms. The onset is described as an instability of the uniform oscillations with respect to spatially periodic perturbations. The solution of the standing wave pattern is given analytically and studied through simulations.

Time-delay autosynchronization control of defect turbulence in the cubic-quintic complex Ginzburg-Landau equation

We investigate the effectiveness of a Global time-delay autosynchronization control scheme aimed at stabilizing traveling wave solutions of the cubic-quintic Ginzburg-Landau equation in the Benjamin-Feir-Newell unstable regime. Numerical simulations show that a global control can be efficient and also can create other patterns such as spatiotemporal intermittency regimes, standing waves, or uniform oscillations.

Delay-driven irregular spatiotemporal patterns in a plankton system

An inhomogeneous distribution of species density over physical space is a widely observed scenario in plankton systems. Understanding the mechanisms resulting in these spatial patterns is a central topic in plankton ecology. In this paper we explore the impact of time delay on spatiotemporal patterns in a prey-predator plankton system. We find that time delay can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, a phase transition from a regular spiral pattern to an irregular one was observed and the latter gradually replaced the former and persisted indefinitely. The characteristic length of the emergent spatial pattern is consistent with the scale of plankton patterns observed in field studies.

Spontaneous spiking in an autaptic Hodgkin-Huxley setup

The effect of intrinsic channel noise is investigated for the dynamic response of a neuronal cell with a delayed feedback loop. The loop is based on the so-called autapse phenomenon in which dendrites establish connections not only to neighboring cells but also to its own axon. The biophysical modeling is achieved in terms of a stochastic Hodgkin-Huxley model containing such a built in delayed feedback. The fluctuations stem from intrinsic channel noise, being caused by the stochastic nature of the gating dynamics of ion channels. The influence of the delayed stimulus is systematically analyzed with respect to the coupling parameter and the delay time in terms of the interspike interval histograms and the average interspike interval. The delayed feedback manifests itself in the occurrence of bursting and a rich multimodal interspike interval distribution, exhibiting a delay-induced reduction in the spontaneous spiking activity at characteristic frequencies. Moreover, a specific frequency-locking mechanism is detected for the mean interspike interval.

Theoretical analysis of defect-mediated turbulence in a catalytic surface reaction

We present a statistical analysis of defect-mediated turbulence in a kinetic model of catalytic CO oxidation on Pt(110). A probabilistic description based on the gain and loss rates of defects is derived. For low values of the CO partial pressure the statistics of topological defects agree with earlier results for the complex Ginzburg-Landau equation. For high values of the CO partial pressure, we observe an additional autocatalytic reproduction of defects that results in a linear dependence of the defect creation rate on the number of defects in the system. The role of correlations between defects of opposite topological charge was found to be weaker than in the experimental system.

Source-tract coupling in birdsong production

Birdsong is a complex phenomenon, generated by a nonlinear vocal device capable of displaying complex solutions even under simple physiological motor commands. Among the peripheral physical mechanisms responsible for the generation of complex sounds in songbirds, the understanding of the dynamics emerging from the interaction between the sound source and the upper vocal tract remains most elusive. In this work we study a highly dissipative limit of a simple sound source model interacting with a tract, mathematically described in terms of a delay differential equation. We explore the system numerically and, by means of reducing the problem to a phase equation, we are capable of studying its periodic solutions. Close in parameter space to the point where the resonances of the tract match the frequencies of the uncoupled source solutions, we find coexistence of periodic limit cycles. This hysteresis phenomenon allows us to interpret recently reported features found in the vocalization of some songbirds, in particular, "frequency jumps."

Feedback control of subcritical Turing instability with zero mode

A global feedback control of a system that exhibits a subcritical monotonic instability at a nonzero wave number (short-wave or Turing instability) in the presence of a zero mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero mode. This system is studied analytically and numerically. It is shown that feedback control, based on measuring the maximum of the pattern amplitude over the domain, can stabilize the system and lead to the formation of localized unipulse stationary states or traveling solitary waves. It is found that the unipulse traveling structures result from an instability of the stationary unipulse structures when one of the parameters characterizing the coupling between the periodic pattern and the zero mode exceeds a critical value that is determined by the zero mode damping coefficient.

Robust entrainment phenomena of oscillations by delay time in the photosensitive Oregonator model

The influences of delayed feedback on the oscillating behaviors are numerically investigated by using the photosensitive Oregonator model with a Hopf point. We find that the time delay in the robust entrainment phenomena determines the time scale of the system, that is, T_{m}=(tau+delta)N (N=1,2,cdots, three dots, centered) , where T_{m} is the mean period of the oscillation and delta is a small constant compared with the delay time tau . Further, our numerical simulation shows that, when the system has a characteristic period T0 under the feedback with time delay, there exists an asymptotical line delta=delta_{0}T_{0} ( delta_{0} independent of any parameters) in the entrainment region with increasing strength of the feedback control c ; when the system has no characteristic period, the above linear relation is also kept, and delta decreases with increasing c .

Limit-cycle oscillators subject to a delayed feedback

The coexistence of two stable limit cycles exhibiting different periods is examined for a nonlinear oscillator subject to a delayed feedback. For the case of a weakly nonlinear oscillator, we discuss the validity of a previously determined phase equation. For the case of a strongly nonlinear oscillator, we derive a phase equation and analyze its bifurcation diagram. Our analysis is motivated by previous experimental studies of chemical oscillators controlled by a delayed feedback.

Pattern formation in 4:1 resonance of the periodically forced CO oxidation on Pt(110)

Periodically forced oscillatory reaction-diffusion systems may show complex spatiotemporal patterns. At high-frequency resonant forcing, multiple-phase patterns can be found. In the present work, the dynamics of turbulent CO oxidation on Pt(110), forced with the fourth harmonic of the system's natural frequency, is investigated. Experiments result in subharmonic entrainment, where the system locks to a quarter of the forcing frequency. Cluster patterns are observed, where different parts of the pattern show a defined phase difference. The experimental results are compared with numerical simulations using the realistic Krischer-Eiswirth-Ertl model for catalytic CO oxidation. Using the fourth harmonic of an uncoupled surface element's natural frequency, we find 3:1 entrainment with three-phase cluster patterns in a wide parameter range of forcing amplitudes and frequency detuning. Numerical analysis of the spatially extended, turbulent system reveals a remarkable upshift of the mean oscillation frequency compared to homogeneous oscillations. Using the fourth harmonic of the most prominent frequency found in the turbulent system results in four-phase patterns with partial or full 4:1 entrainment, depending on the forcing parameters chosen.

Suppression of spatiotemporal chaos in the oscillatory CO oxidation on Pt(110) by focused laser light

Chemical turbulence in the oscillatory catalytic CO oxidation on Pt(110) is suppressed by means of focused laser light. The laser locally heats the platinum surface which leads to a local increase of the oscillation frequency, and to the formation of a pacemaker which emits target waves. These waves slowly entrain the medium and suppress the spatiotemporal chaos present in the absence of laser light. Our experimental results are confirmed by a detailed numerical analysis of one- and two-dimensional media using the Krischer-Eiswirth-Ertl model for CO oxidation on Pt(110). Different control regimes are identified and the dispersion relation of the system is determined using the pacemaker as an externally tunable wave source.

Control of spatiotemporal chaos in catalytic CO oxidation by laser-induced pacemakers

Control of spatiotemporal chaos is achieved in the catalytic oxidation of CO on Pt(110) by localized modification of the kinetic properties of the surface chemical reaction. In the experiment, a small temperature heterogeneity is created on the surface by a focused laser beam. This heterogeneity constitutes a pacemaker and starts to emit target waves. These waves slowly entrain the medium and suppress the spatiotemporal chaos that is present in the absence of control. We compare this experimental result with a numerical study of the Krischer-Eiswirth-Ertl model for CO oxidation on Pt(110). We confirm the experimental findings and identify regimes where complete and partial controls are possible.

Pattern transitions induced by delay feedback

Modulated by delay feedback (DF), a reaction-diffusion system is destabilized and undergoes pattern transitions in the parametric region where the undelayed system spontaneously exhibits a bulk oscillation. By varying the feedback parameters, oscillatory hexagon superlattices and stripes, as well as stationary hexagons are observed. Meanwhile, the hexagon superlattices with different wavelengths are found under appropriate feedback parameters. It is demonstrated that, since the DF induces an instability of homogeneous limit cycle with respect to spatial perturbations, the patterns possessing the corresponding spatial modes are formed. Instead of stabilizing the system, here the DF may play a role of destabilization.

Stability and nonlinear dynamics of solitary waves generated by subcritical oscillatory instability under the action of feedback control

We consider the influence of global feedback control which acts on an oscillatory system governed by a subcritical Ginzburg-Landau equation. Exact solutions corresponding to solitary-wave solutions are obtained. A generalized variational approach is applied for the simplification of the whole problem and its reduction to a finite-dimensional dynamical model. The finite-dimensional evolution model is used for studying the indirect interaction between solitary waves caused by the global control. The stability analysis is held in the framework of the finite-dimensional model. The boundaries of monotonic and oscillatory instabilities are obtained. The basic types of dynamics provided by the finite-dimensional model are described and compared with the results of a direct numerical simulation of the original equation.

Control of turbulence in oscillatory reaction-diffusion systems through a combination of global and local feedback

Global time-delay autosynchronization is known to control spatiotemporal turbulence in oscillatory reaction-diffusion systems. Here, we investigate the complex Ginzburg-Landau equation in the regime of spatiotemporal turbulence and study numerically how local or a combination of global and local time-delay autosynchronization can be used to suppress turbulence by inducing uniform oscillations. Numerical simulations show that while a purely local control is unsuitable to produce uniform oscillations, a mixed local and global control can be efficient and also able to create other patterns such as standing waves, amplitude death, or traveling waves.

Global feedback control for pattern-forming systems

Global feedback control of pattern formation in a wide class of systems described by the Swift-Hohenberg (SH) equation is investigated theoretically, by means of stability analysis and numerical simulations. Two cases are considered: (i) feedback control of the competition between hexagon and roll patterns described by a supercritical SH equation, and (ii) the use of feedback control to suppress the blowup in a system described by a subcritical SH equation. In case (i), it is shown that feedback control can change the hexagon and roll stability regions in the parameter space as well as cause a transition from up to down hexagons and stabilize a skewed (mixed-mode) hexagonal pattern. In case (ii), it is demonstrated that feedback control can suppress blowup and lead to the formation of spatially localized patterns in the weakly nonlinear regime. The effects of a delayed feedback are also investigated for both cases, and it is shown that delay can induce temporal oscillations as well as blowup.

Pattern formation and self-organization in a simple precipitation system

Various types of pattern formation and self-organization phenomena can be observed in biological, chemical, and geochemical systems due to the interaction of reaction with diffusion. The appearance of static precipitation patterns was reported first by Liesegang in 1896. Traveling waves and dynamically changing patterns can also exist in reaction-diffusion systems: the Belousov-Zhabotinsky reaction provides a classical example for these phenomena. Until now, no experimental evidence had been found for the presence of such dynamical patterns in precipitation systems. Pattern formation phenomena, as a result of precipitation front coupling with traveling waves, are investigated in a new simple reaction-diffusion system that is based on the precipitation and complex formation of aluminum hydroxide. A unique kind of self-organization, the spontaneous appearance of traveling waves, and spiral formation inside a precipitation front is reported. The newly designed system is a simple one (we need just two inorganic reactants, and the experimental setup is simple), in which dynamically changing pattern formation can be observed. This work could show a new perspective in precipitation pattern formation and geochemical self-organization.

Controlling drift-wave turbulence using time-delay and space-shift autosynchronization feedback

Drift-wave turbulence control in a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave is considered. We apply time-delay and space-shift feedback signals, to suppress turbulence. By using global and local pinning strategies, we show numerically that the turbulent state can be controlled to periodic states effectively if appropriate time-delay length and space-shift distance are chosen. The physical mechanism of the control scheme is understood based on the energy-minimum principle.

Feedback control of subcritical oscillatory instabilities

Feedback control of a subcritical oscillatory instability is investigated in the framework of a globally-controlled complex Ginzburg-Landau equation that describes the nonlinear dynamics near the instability threshold. The control is based on a feedback loop between the system linear growth rate and the maximum of the amplitude of the emerging pattern. It is shown that such control can suppress the blow up and result in the formation of spatially localized pulses similar to oscillons. In the one-dimensional case, depending on the values of the linear and nonlinear dispersion coefficients, several types of the pulse dynamics are possible in which the computational domain contains: (i) a single stationary pulse; (ii) several coexisting stationary pulses; (iii) competing pulses that appear one after another at random locations so that at each moment of time there is only one pulse in the domain; (iv) spatiotemporally chaotic system of short pulses; (v) spatially-synchronized pulses. Similar dynamic behavior is found also in the two-dimensional case. The effect of the feedback delay is also studied. It is shown that the increase of the delay leads to an oscillatory instability of the pulses and the formation of pulses with oscillating amplitude.

Bifurcation to large period oscillations in physical systems controlled by delay

An unusual bifurcation to time-periodic oscillations of a class of delay differential equations is investigated. As we approach the bifurcation point, both the amplitude and the frequency of the oscillations go to zero. The class of delay differential equations is a nonlinear extension of a nonevasive control method and is motivated by a recent study of the foreign exchange rate oscillations. By using asymptotic methods, we determine the bifurcation scaling laws for the amplitude and the period of the oscillations.

Front explosion in a periodically forced surface reaction

Resonantly forced oscillatory reaction-diffusion systems can exhibit fronts with complicated interfacial structure separating phase-locked homogeneous states. For values of the forcing amplitude below a critical value the front "explodes" and the width of the interfacial zone grows without bound. Such front explosion phenomena are investigated for a realistic model of catalytic CO oxidation on a Pt(110) surface in the 2:1 and 3:1 resonantly forced regimes. In the 2:1 regime, the fronts are stationary and the front explosion leads to a defect-mediated turbulent state. In the 3:1 resonantly forced system, the fronts propagate. The front velocity tends to zero as the front explosion point is reached and the final asymptotic state is a 2:1 resonantly locked labyrinthine pattern. The front dynamics described here should be observable in experiment since the model has been shown to capture essential features of the CO oxidation reaction.

Delayed feedback control of chaos: bifurcation analysis

We study the effect of time delayed feedback control in the form proposed by Pyragas on deterministic chaos in the Rössler system. We reveal the general bifurcation diagram in the parameter plane of time delay tau and feedback strength K which allows one to explain the phenomena that have been discovered in some previous works. We show that the bifurcation diagram has essentially a multileaf structure that constitutes multistability: the larger the tau, the larger the number of attractors that can coexist in the phase space. Feedback induces a large variety of regimes nonexistent in the original system, among them tori and chaotic attractors born from them. Finally, we estimate how the parameters of delayed feedback influence the periods of limit cycles in the system.

Self-stabilization of high-frequency oscillations in semiconductor superlattices by time-delay autosynchronization

We present a scheme to stabilize high-frequency domain oscillations in semiconductor superlattices by a time-delayed feedback loop. Applying concepts from chaos control theory we propose to control the spatiotemporal dynamics of fronts of accumulation and depletion layers which are generated at the emitter and may collide and annihilate during their transit, and thereby suppress chaos. The proposed method only requires the feedback of internal global electrical variables, viz., current and voltage, which makes the practical implementation very easy.

Time-delay autosynchronization of the spatiotemporal dynamics in resonant tunneling diodes

The double barrier resonant tunneling diode exhibits complex spatiotemporal patterns including low-dimensional chaos when operated in an active external circuit. We demonstrate how autosynchronization by time-delayed feedback control can be used to select and stabilize specific current density patterns in a noninvasive way. We compare the efficiency of different control schemes involving feedback in either local spatial or global degrees of freedom. The numerically obtained Floquet exponents are explained by analytical results from linear stability analysis.