Belousov-Zhabotinsky Reaction Research Articles

Spiral waves in fractal dimensions and their elimination in λ − ω systems with less damaging intervention

The theory of spiral waves in excitable media is considered an important topic in several dynamical systems. They have been observed in various physicochemical and biological systems such as the Belousov-Zhabotinsky chemical reactions, lethal arrhythmia in the heart, amoebas Dictyostelium discoideium, disinhibited mammalian neocortex retinal spreading depression, and seizures in the brain. In this study, spiral waves in an excitable system are analytically studied in fractal dimensions based on the theory of integration and differentiation for a non-integer dimensional space. We consider the generic oscillatory model λ−ω system characterized by a rotational symmetry where spiral waves are expected to occur. It was observed that for lower fractal dimensions with respect to unity, spiral waves may be suppressed in an excitable media without destroying the propagation of normal waves. This may be considered as an alternative approach to experimental and numerical approaches found in the literature. Their eliminations are important since they forbid the transition to a chaotic state in excitable media. Our approach could contribute to an improved therapy of clinical conditions such as atrial fibrillation and suppressing arrhythmias in cardiac models.

Effect of excitability on partially pinned scroll waves in excitable chemical media.

We present an investigation of excitability effects on the dynamics of scroll waves partially pinned to inert cylindrical obstacles in three-dimensional Belousov-Zhabotinsky excitable media. We also report on corresponding numerical simulations with the Oregonator model. The excitability varies according to the concentration of sulfuric acid [H_{2}SO_{4}] in the Belousov-Zhabotinsky (BZ) reaction and the parameter ɛ^{-1} in the Oregonator model. Initially, the freely rotating scroll segment rotates faster than the pinned one. The difference in the frequency of the two parts results in a transition from a straight pinned scroll wave to a twisted one, which helically wraps around the entire obstacle. The wave frequency in the whole volume is equal to that of the freely rotating scroll wave. When the excitability is increased, the time for the transition to the twisted wave structure decreases while the average speed s of the development increases. After the transition, the twisted wave remains stable. In media with higher excitability, the helical pitch is shorter but the twist rate ω is higher. Analysis presented in this study together with our previous findings of the effect of the cylindrical obstacle diameter on the wave dynamics results in common features: The average speed s and the twist rate ω of both studies fit well to functions of the difference in the initial frequency Δf of the freely rotating and untwisted pinned waves. We also demonstrate the robustness of the partially pinned scroll waves against perturbations from spontaneous waves emerging during the wave generation in the BZ medium with high [H_{2}SO_{4}]. Even though the scroll wave is partly disturbed at the beginning of the experiment, the spontaneous waves are gradually suppressed and the typical wave structure is finally developed.

Kinetics, Mechanism, and Reactivity of the Cerium(IV)-Oxalatereaction in a Sulphate Medium

This work presents an approach for studying the kinetics, mechanism, and reactivity of intermediates in a wide class of the redox reactions for which the rate-limiting step is the redox-decomposition of an intermediate complex. This approach is applied to investigate the oxalic acid (H2Ox) oxidation by cerium(IV) in sulfuric acid medium, which is an integral part of the cerium-catalyzed oscillatory Belousov–Zhabotinsky (BZ) reaction. Using experimental, mathematical and computational techniques commonly used to study metal complexes in a stable oxidation state (OS), kinetically generalized by the authors for studying variable-valence metal complexes, the characteristics of intermediate complexes of the cerium(IV)-oxalate reaction were studied, the general rate law was derived on the basis of a set of equations describing the rapid establishment of preequilibria in the system and the subsequent nonequilibrium process. A quantitative reaction model is proposed that includes two parallel reaction pathways, for which two different intermediate cerium(IV)-oxalate complexes with close reactivity have been identified and characterized that may be due to the similarity in the structure of their inner coordination spheres and an inner sphere mechanism of electron transfer in the complexes. Based on the developed model, the distribution diagram was also constructed for the computed fractions of all the detectable cerium(IV) species under conditions of the BZ reaction, which testify to the necessity to take into account under these conditions the formation of intermediate complexes CeOHOx (n = 1, 2) during oxidation of oxalic acid. The main difference of the proposed model of the cerium(IV)-oxalate reaction as part of the BZ-reaction from the previous models is the explicit accounting of the formation of intermediate cerium(IV) complexes with anions of oxalic acid and sulfate background.

The excitable nature of polymerizing actin and the Belousov-Zhabotinsky reaction.

The intricate regulatory processes behind actin polymerization play a crucial role in cellular biology, including essential mechanisms such as cell migration or cell division. However, the self-organizing principles governing actin polymerization are still poorly understood. In this perspective article, we compare the Belousov-Zhabotinsky (BZ) reaction, a classic and well understood chemical oscillator known for its self-organizing spatiotemporal dynamics, with the excitable dynamics of polymerizing actin. While the BZ reaction originates from the domain of inorganic chemistry, it shares remarkable similarities with actin polymerization, including the characteristic propagating waves, which are influenced by geometry and external fields, and the emergent collective behavior. Starting with a general description of emerging patterns, we elaborate on single droplets or cell-level dynamics, the influence of geometric confinements and conclude with collective interactions. Comparing these two systems sheds light on the universal nature of self-organization principles in both living and inanimate systems.

Role of Fast and Slow Inhibitors in Oscillatory Rhythm Design.

In biological or abiotic systems, rhythms occur, owing to the coupling between positive and negative feedback loops in a reaction network. Using the Semenov-Whitesides oscillatory network for thioester hydrolysis as a prototype, we experimentally and theoretically analyzed the role of fast and slow inhibitors in oscillatory reaction networks. In the presence of positive feedback, a single fast inhibitor generates a time delay, resulting in two saddle-node bifurcations and bistability in a continuously stirred tank reactor. A slow inhibitor produces a node-focus bifurcation, resulting in damped oscillations. With both fast and slow inhibitors present, the node-focus bifurcation repeatedly modulates the saddle-node bifurcations, producing stable periodic oscillations. These fast and slow inhibitions result in a pair of time delays between steeply ascending and descending dynamics, which originate from the positive and negative feedbacks, respectively. This pattern can be identified in many chemical relaxation oscillators and oscillatory models, e.g., the bromate-sulfite pH oscillatory system, the Belousov-Zhabotinsky reaction, the trypsin oscillatory system, and the Boissonade-De Kepper model. This study provides a novel understanding of chemical and biochemical rhythms and suggests an approach to designing such behavior.

Empirically developed model of the stirring-controlled Belousov–Zhabotinsky reaction

The Belousov–Zhabotinsky chemical reaction is famous for its oscillatory behavior occurring due to inherent physicochemical processes. Oscillations of the Belousov–Zhabotinsky reaction may be periodic or chaotic, depending on parameters, and these dynamics can be controlled by external impacts such as stirring. The complex behavior, the variability of the oscillatory modes, and controllability make the Belousov–Zhabotinsky reaction an important research subject in many fields of science, including various applications such as chemical computing or smart materials. One of the key limitations for the practical application of this oscillatory reaction is the absence of its exact mathematical model which can reproduce dynamics close to the real chemical process. This work aims to fill this gap by designing a novel model of the Belousov–Zhabotinsky reaction based on experimental data. To solve this task, we designed a complex experimental setup based on a photometric station with controllable stirring mechanisms. We performed two experiments in batch reactors (cuvettes). In the first experiment, we investigated how constant stirring with different rates affects the reaction behavior, and in the second experiment, we periodically stirred the reaction and observed its response. As a result, we found that chemical oscillations in the Belousov–Zhabotinsky reaction are in both cases chaotic, their amplitude depends on stirring more than their period does, and they are highly non-stationary: in the course of the reaction, their sensitivity to stirring decreases, so the oscillations remain controllable only during the first hour. Using a novel two-stage differential equation reconstruction approach, we managed to develop a complex of local dynamical models for the stirring-controlled BZ reaction with good correspondence to the empirical data. The presented models are more consistent with experimentally observed Belousov–Zhabotinsky dynamics than earlier models. The developed model can be used in various applications and extended to a wide class of controlled oscillatory reactions.

Two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky system with spatiotemporal interaction

We prove the existence of the minimal speed of propagation c⁎(r,b,K)∈[21−r,2] for wavefronts in the Belousov-Zhabotinsky system with a spatiotemporal interaction defined by the convolution with (possibly, “fat-tailed”) kernel K. The model is assumed to be monostable non-degenerate, i.e. r∈(0,1). The slowest wavefront is termed pushed or nonlinearly determined if its velocity c⁎(r,b,K)>21−r. We show that c⁎(r,b,K) is close to 2 if i) positive system's parameter b is sufficiently large or ii) K is spatially asymmetric to one side (e.g. to the left: in such a case, the influence of the right side concentration of the bromide ion on the dynamics is more significant than the influence of the left side). Consequently, this reveals two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky reaction.

Fe(bathophen)2(phen)-based self-oscillating gel driven by the Belousov–Zhabotinsky reaction

The synthesis of a new Fe(bathophen)2(phen)-based self-oscillating gel driven by the Belousov–Zhabotinsky reaction is described. The gel undergoes spontaneous chemomechanical oscillations without any on–off switching by external stimuli and demonstrates the dependence of chemomechanical behavior on the concentration of the catalyst in it.

Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator

This paper introduces the optimal auxiliary function method (OAFM) for solving a nonlinear system of Belousov–Zhabotinsky equations. The system is characterized by its complex dynamics and is treated using the Caputo operator and concepts from fractional calculus. The OAFM provides a systematic approach to obtain approximate analytical solutions by constructing an auxiliary function. By optimizing the parameters of the auxiliary function, an approximate solution is derived that closely matches the behavior of the original system. The effectiveness and accuracy of the OAFM are demonstrated through numerical simulations and comparisons with existing methods. Fractional calculus enhances the understanding and modeling of the nonlinear dynamics in the Belousov–Zhabotinsky system. This study contributes to fractional calculus and nonlinear dynamics, offering a powerful tool for analyzing and solving complex systems such as the Belousov–Zhabotinsky equation.

Quality evaluation of Keteling capsules based on fingerprinting, multicomponent quantification, and quantitative prediction

The Keteling capsule (KC) is a traditional Chinese medicine (TCM) made from the dried extract of Ficus microphylla and an appropriate amount of chlorpheniramine maleate. It is widely used to treat cough and relieve asthma. Despite its extensive usage, a rapid and comprehensive quality evaluation strategy for KC remains a challenge. This study introduces an electrochemical fingerprint analysis technique, in addition to the commonly employed HPLC fingerprints, for efficient and convenient quality evaluation. Moreover, a cost-effective, rapid, and accurate multi-component quantification technique known as the “Multi-markers assay by the monolinear method (MAML)” and the “FT-IR quantitative model” were explored. The HPLC fingerprints were evaluated using a systematically quantified fingerprint method, while the electrochemical fingerprints, based on the Belousov-Zhabotinsky oscillation reaction principle, were effectively analyzed and characterized using oxidation induction times and oscillation lifetimes. Multi-component quantitative analysis was carried out through the MAML and FT-IR quantitative models. The HPLC fingerprint successfully classified the 22 samples into eight grades with excellent discrimination. Active ingredient content analysis was achieved using reliable parameters obtained from electrochemical fingerprinting. The no significant difference in the quantitative results proves the accuracy of the MAML method. Additionally, successful FT-IR quantitative prediction models were developed for chlorogenic acid, isovitexin, and chlorpheniramine maleate. This study offers a dependable and effective approach for enhancing the quality control of KC, and it can provide new insights for improving the quality analysis methods in the field of TCM.

Towards Soliton Computer Based on Solitary Wave Solution of Maxwell-Dirac Equation: A Plausible Alternative to Manakov System

In recent years, there are a number of proposals to consider collision-based soliton computer based on certain chemical reactions, namely Belousov-Zhabotinsky reaction, which leads to soliton solutions of coupled Nonlinear Schroedinger equations. They are called Manakov System. But it seems to us that such a soliton computer model can also be based on solitary wave solution of Maxwell-Dirac equation, which reduces to Choquard equation. And soliton solution of Choquard equation has been investigated by many researchers, therefore it seems more profound from physics perspective. However, we consider both schemes of soliton computer are equally possible. More researches are needed to verify our proposition.

Theory and experiments of spiral unpinning in the Belousov-Zhabotinsky reaction using a circularly polarized electric field.

We present the first experimental study of unpinning an excitation wave using a circularly polarized electric field. The experiments are conducted using the excitable chemical medium, the Belousov-Zhabotinsky (BZ) reaction, which is modeled with the Oregenator model. The excitation wave in the chemical medium is charged so that it can directly interact with the electric field. This is a unique feature of the chemical excitation wave. The mechanism of wave unpinning in the BZ reaction with a circularly polarized electric field is investigated by varying the pacing ratio, the initial phase of the wave, and field strength. The chemical wave in the BZ reaction unpins when the electric force opposite the direction of the spiral is equal to or above a threshold. We developed an analytical relation of the unpinning phase with the initial phase, the pacing ratio, and the field strength. This is then verified in experiments and simulations.

Pressure and Light Multi-Physics Fields Comodulate the Multi-Length Scales of Chemical Wave Propagation in Diffusion-Fed Gels

During plant growth, the combined effects of multi-physics fields such as light, temperature, and material concentration create a complex multi-length scales phenomenon. However, the mechanism of multi-physical field interactions on biological multi-length scales has not been thoroughly investigated. In this paper, an open diffusion-fed system is constructed by coupling gels with a Belousov-Zhabotinsky (BZ) chemical reaction system. The multi-length scales propagation of chemical waves in the gel system under the combined effect of multi-physical fields such as light (I) and pressure (ΔP) is investigated. It is found that when 85 Pa ≤ ΔP ≤ 100 Pa or 200 μW·cm-2 ≤ I ≤ 300 μW·cm-2, the complexity of the multi-length scales periodic structure of chemical waves shows a nonlinear change with increasing light intensity or pressure. Beyond this range, the complexity of the chemical wave multi-length scales periodic structure decreases linearly on enhancing the light intensity or increasing the pressure.

Transitions of Collective Motions Driven by Phase Resetting.

Abrupt (i. e. step) environmental changes, such as natural disasters or the intervention of predators, can alter the internal dynamics of groups with active units, leading to the rapid destruction and/or restructuring of the group, with the emergence of new collective structures that endow the system with adaptability. Few studies, to date, have considered the influence of abrupt environmental changes on emergent behavior. Here, we use a model of active matter, the Belousov-Zhabotinsky (BZ) self-oscillating gel, to study the mechanism of formation and transition between modes of collective locomotion caused by changes of illumination intensity in arrays of interacting photosensitive active units. New forms of collective motion can be generated by step changes of illumination intensity. These transformations arise from the phase resetting and wave-signal regeneration induced by the abrupt parameter variation, while gradual change results in different evolution of collective motion. Our results not only suggest a novel mechanism for emergence, but also imply that new collective behaviors could be accessible via discontinuous parameter changes.

A Comparative Study of the Fractional-Order Belousov–Zhabotinsky System

In this article, we present a modified strategy that combines the residual power series method with the Laplace transformation and a novel iterative technique for generating a series solution to the fractional nonlinear Belousov–Zhabotinsky (BZ) system. The proposed techniques use the Laurent series in their development. The new procedures’ advantages include the accuracy and speed in obtaining exact/approximate solutions. The suggested approach examines the fractional nonlinear BZ system that describes flow motion in a pipe.

A new outlook for analysis of Noyes-Field model for the nonlinear Belousov-Zhabotinsky reaction using operator splitting method

A new outlook for analysis of Noyes-Field model for the nonlinear Belousov-Zhabotinsky reaction using operator splitting method

Numerical Analysis of the Fractional-Order Belousov–Zhabotinsky System

This paper presents a new approach for finding analytic solutions to the Belousov–Zhabotinsky system by combining the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) with the Elzaki transform. The ADM and HPM are both powerful techniques for solving nonlinear differential equations, and their combination allows for a more efficient and accurate solution. The Elzaki transform, on the other hand, is a mathematical tool that transforms the system into a simpler form, making it easier to solve. The proposed method is applied to the Belousov–Zhabotinsky system, which is a well-known model for studying nonlinear chemical reactions. The results show that the combined method is capable of providing accurate analytic solutions to the system. Furthermore, the method is also able to capture the complex behavior of the system, such as the formation of oscillatory patterns. Overall, the proposed method offers a promising approach for solving complex nonlinear differential equations, such as those encountered in the field of chemical kinetics. The combination of ADM, HPM, and the Elzaki transform allows for a more efficient and accurate solution, which can provide valuable insights into the behavior of nonlinear systems.

Nonplanar traveling fronts of the diffusion system with Belousov-Zhabotinskii reaction in [formula omitted

This paper studies nonplanar traveling fronts for a diffusion system with Belousov-Zhabotinskii (BZ for short) reaction in R3. First, we show the existence and stability results of pyramidal traveling fronts, and investigate their qualitative properties. By taking the limit of a pyramidal traveling front sequence, we obtain the conical traveling front and study its qualitative properties. Although both the pyramidal fronts and the conical fronts have larger traveling speeds than the planar front, their global mean speed equals that of the planar front in R3.

From Turing patterns to chimera states in the 2D Brusselator model.

The Brusselator has been used as a prototype model for autocatalytic reactions and, in particular, for the Belousov-Zhabotinsky reaction. When coupled at the diffusive limit, the Brusselator undergoes a Turing bifurcation resulting in the formation of classical Turing patterns, such as spots, stripes, and spirals in two spatial dimensions. In the present study, we use generic nonlocally coupled Brusselators and show that in the limit of the coupling range R→1 (diffusive limit), the classical Turing patterns are recovered, while for intermediate coupling ranges and appropriate parameter values, chimera states are produced. This study demonstrates how the parameters of a typical nonlinear oscillator can be tuned so that the coupled system passes from spatially stable Turing structures to dynamical spatiotemporal chimera states.

Richard “Dick” Field

Richard “Dick” Field, 81, died Dec. 3, 2022, in Missoula, Montana. “Dick and his wife Judith journeyed from the east coast to the University of Oregon and an opportunity to work with Richard Noyes in 1968. A fortunate introduction to the Belousov–Zhabotinsky reaction in 1969 helped him become a pioneer in the area of nonlinear dynamics in chemistry. Moving to the University of Montana in 1975, Dick focused the rest of his life on his family and professing chemistry while developing an internationally visible research program. His personal and professional influence on family, friends, and colleagues will be missed.”—Eli Field, son Most recent title: Professor of chemistry, University of Montana Education: BS, chemistry, University of Massachusetts Amherst, 1963; MS, chemistry, College of the Holy Cross, 1964; PhD, physical chemistry, University of Rhode Island, 1968 Survivors: Son, Eli; daughter, Sara Carlson; five grandchildren To recognize your late loved one or colleague,