Feedback loops for chaos in activator-inhibitor systems

Previous investigations have revealed that special constellations of feedback loops in a network can give rise to saddle-node and Hopf bifurcations and can induce particular bifurcation diagrams including the occurrence of various codimension-two points. To elucidate the role of feedback loops in the generation of more complex dynamics, a minimal prototype for these networks will be taken as purely periodic starting model which will be extended by an additional species in different feedback loops. The dynamics of the resulting systems will be analyzed numerically for the occurrence of chaotic attractors. Especially, the consequences of codimension-two bifurcations and the role of homoclinic orbits in view of the emergence of Shil'nikov chaos will be discussed.     

Stoichiometric network analysis of the photochemical processes in the mesopause region

The mechanism of photochemistry in the mesopause region entails a chemical oscillator forced by solar short-wave radiation. A model with periodic forcing between day and night conditions produces nonlinear dynamics including period-doubling bifurcations and chaos. The photochemical mechanism represents a network involving positive and negative feedbacks that can be examined by methods of stoichiometric network analysis. We use these methods to decompose the network into irreducible subnetworks and then apply linear stability analysis to find all possible sources of oscillatory instabilities in the mesopause chemistry. These oscillators are classified according to topological features in their reaction networks and phase shifts of oscillating species. We subsequently compare phase shifts indicated by the network analysis with those from direct simulations to identify a specific subnetwork in the mechanism underlying the complex oscillatory dynamics observed in earlier simulations.     

Response analysis in biochemical chain reactions with negative feedforward and feedbackward loops

Finite difference equations can be used to study the responses of biochemical chain reactions at any step of the chain to an external stimulus. In this study, we developed mathematical models for two hypothetical chain reactions involving loops to study the responses in the chain as the length of the chain gets longer, so called transient and steady state responses. The first model is for a chain with a negative feedforward loop, and the second one is for a chain that has a negative feedback loop. Although both of the models have the same steady state equations and values, we showed that the chain with negative feedforward and negative feedback loops can produce significantly different behaviors. The former can bring the chain into oscillations with various periods and eventually chaos when the feedback is strong enough as the length of the reaction chain increases, whereas the latter is not capable of producing oscillations and more complicated dynamics.     

实验控制Belousov-Zhabotinsky-CSTR化学混沌反应

用两种方法来控制在连续流动釜式反应器(CSTR)中的Belousov-Zhabotinsky(BZ)化学混沌反应:(1)自适应延迟时间反馈控制方案;(2)线性自相互作用控制方案。第一种控制方案是一种微扰控制,并能将BZ-CSTR化学混沌稳定控制到其内嵌的不稳定周期轨道(UPO)上去。这种方法的优越之外可以由系统延迟时间的自适应调节得以显示。第二种控制方案可以驱动BZ-CSTR混沌动力学到达一系列规则的动力学状态,这些动力学状态可能并不是混沌轨道中内嵌的本征轨道。在控制BZ-CSTR化学混沌反应的实验中,这两种方案显示出了很大的灵活性和有效性。     

pH oscillations in the hemin-hydrogen peroxide-sulfite reaction

Oscillatory behaviour in the pH value has been observed during the oxidation of sulfite by hydrogen peroxide mediated by hemin, a well known enzyme model compound, in a continuous-flow stirred tank reactor. The dynamics of this reaction has been studied for a variety of flow rates of the reactants. As the flow rates increase, the oscillations evolve from relaxation oscillations to more complex shapes, displaying, among others, bursting behaviour. A reaction mechanism is proposed that involves the autocatalytic oxidation of HSO3- by H2O2, while slow equilibria between different pH-dependent forms of hemin account for the feedback loop which gives rise to oscillatory dynamics. It is shown in experiments that no participation of CO2 is required for oscillations to occur.     

Flow-induced control of chemical turbulence

We report spatiotemporal chaos in the Oregonator model of the Belousov-Zhabotinsky reaction. Spatiotemporal chaos spontaneously develops in a regime, where the underlying local dynamics show stable limit cycle oscillations (diffusion-induced turbulence). We show that spatiotemporal chaos can be suppressed by a unidirectional flow in the system. With increasing flow velocity, we observe a transition scenario from spatiotemporal chaos via a regime of travelling waves to a stationary steady state. At large flow velocities, we recover the known regime of flow distributed oscillations.     

化学自催化混沌反应模型中的耦合作用与混沌同步

选用混沌自催化反应作为子系统,构造了耦合自催化反应系统,研究了耦合变量、耦合系数对混沌动力学行为的影响,给出了不同耦合系数下系统的动力学特征,探讨了耦合作用机制.结果表明,耦合作用能明显地改变子系统的动力学行为,强化系统间的相关性.耦合后的混沌运动受到调整与抑制,耦合强度加大时,呈现出混沌运动轨线的周期化,耦合系数大于临界值,两子系统实现了完全的同步.不同变量的耦合时,影响最大的是第二种变量.对于三种物质均有耦合时,更容易出现混沌的抑制、运动状态的锁相与周期化和混沌的完全同步.     

On the Resolution Limit of Femtosecond Stimulated Raman Spectroscopy: Modelling Fifth‐Order Signals with Overlapping Pulses

Femtosecond stimulated Raman scattering (FSRS) spectroscopy is a powerful pump–probe technique that can track electronic and vibrational dynamics with high spectral and temporal resolution. The investigation of extremely short‐lived species, however, implies deciphering complex signals and is ultimately hampered by unwanted nonlinear effects once the time resolution limit is approached and the pulses overlap temporally. Using the loop diagrams formalism we calculate the fifth‐order response of a model system and address the limiting case where the relevant dynamics timescale is comparable to the pump–pulse duration and, consequently, the pump and the probe overlap temporally. We find that in this regime, additional diagrams that do not contribute for temporally well separated pulses need to be taken into account, giving rise to new time‐dependent features, even in the absence of photoinduced dynamics and for negative delays.     

Period-doubling and chaotic oscillations the ferroin-catalyzed Belousov-Zhabotinsky reaction in a CSTR

The ferroin-catalyzed Belousov-Zhabotinsky (BZ) reaction, the oxidation of malonic acid by acidic bromate, is the most commonly investigated chemical system for understanding spatial pattern formation. Various oscillatory behaviors were found from such as mixed-mode and simple period-doubling oscillations and chaos on both Pt electrode and Br-ISE at high flow rates to mixed-mode oscillations on Br-ISE only at Iow flow rates. The complex dynamic behaviors were qualitatively reproduced with a two-cycle coupling model proposed initially by Gy(o)rgyi and Field. This investigation offered a proper medium for studying pattern formation under complex temporal dynamics. In addition, it also shows that complex oscillations and chaos in the BZ reaction can be extended to other bromate-driven nonlinear reaction systems with different metal catalysts.     

Period-doubling and chaotic oscillations in the ferroin-catalyzed Belousov-Zhabotinsky reaction in a CSTR

The ferroin-catalyzed Belousov-Zhabotinsky(BZ) reaction,the oxidation of malonic acid by acidic bromate,is the most commonly investigated chemical system for understanding spatial pattern forma-tion. Various oscillatory behaviors were found from such as mixed-mode and simple period-doubling oscillations and chaos on both Pt electrode and Br-ISE at high flow rates to mixed-mode oscillations on Br-ISE only at low flow rates. The complex dynamic behaviors were qualitatively reproduced with a two-cycle coupling model proposed initially by Gy?rgyi and Field. This investigation offered a proper medium for studying pattern formation under complex temporal dynamics. In addition,it also shows that complex oscillations and chaos in the BZ reaction can be extended to other bromate-driven nonlinear reaction systems with different metal catalysts.     

Eliminating chaos in the Belousov–Zhabotinsky reaction by no-delay feedback and delayed feedback

We present numerical simulations of the three-variable model of the Belousov–Zhabotinsky reaction developed by Gyorgyi and Field, where chaos can occur in the free-running system. As the rate of in-flow in the continuous-flow stirred-tank reactor is controlled by the concentration of one of the reactor species through a feedback loop, we find two simple ways to bring the chaotic behavior to periodic behavior. One is to modify the feedback strength without time delay in the feedback loop, and the other is to modify the delay time of the feedback at constant feedback strength. The possible mechanisms for the two ways are discussed.     

闭系中硫(-II)氧化的非线性动力学一般特征

硫元素可出现有一2‘0‘＋2‘＋4、＋6等多个价态．－2价的硫化合物在氧化过程中涉及价态变化较多,反应机理十分复杂,有些中间体难于分离和检定．但在硫化学非线性动力学研究中使用不同的氧化剂（H20。、C10。一、Bio。一和104一等）发现了一些基本的共同特怔,应与S（－11）化合物氧化反应中许多复杂现象［‘］（振荡、双节律、混饨和反应扩散波等）有密切关系,本文就封闭体系中硫化学非线性机理特征进行探讨和研究．1封闭体系中反应动力学特征根据实验和文献,在硫化合物的氧化动力学曲线中Pt电位、pH及其一些氧化剂中间物的浓度（…     

闭系中硫(-Ⅱ)氧化的非线性动力学一般特征

The general dynamic features of the batch oxidation of Sulfur(-Ⅱ) compounds (S2- , thiosulfate,thiocynate and thiourea et al.) were discussed. The changes of Ph with reaction time consist of a rise and two separate drops. Dynamic model was proposed, further experimental studies on the batch oxidation of thiourea and thiocyanate by ClO2- were also carried out. Simulation curve was well consistent with experiment in a batch reactor. Non-catalytic oscillation of peroxide (hydrogen peroxide, persulfate et al.)- S(-Ⅱ) compound system resulted from the nonlinear dynamic of sulfur oxidation. In the oxidation of S(-Ⅱ) compounds by oxyhalogen compounds(ClO2- et al.), nonlinear dynamics of both oxyhalogen compound and sulfur compound should be considered simultaneously. So complex phenomena such as birhythmicity and chaos may be discovered.     

A molecular dynamics simulation study on polymer networks of end-linked flexible or rigid chains

The differences in formation and structural properties of polymer networks consisting of end-linked flexible or rigid chains were studied by molecular dynamics simulation. Networks were formed from monodisperse, linear, short, flexible or rigid chains with functional end groups and a stoichiometric ratio of trifunctional cross-linkers. The rigid chains had a rodlike shape defined by an angle potential, while the flexible chains had no angle potential. In order to understand the influence of chain rigidity, all parameters of precursor chains (length, reactivity, bond potential, nonbonding potential) were the same, with the exception of the angle potential. The system density rho, corresponding to the concentration of monomer in solvent, was varied from 0.01 to 0.11. Different network structures resulting from the different processes of network formation were observed. Simulations showed that the flexible chains created an inhomogeneous network on a large scale via microgel cluster formation, in agreement with experimental observations, whereas the rigid chains rapidly created a homogeneous network in the entire system volume without first generating microgel clusters, with the additional difference that they gave rise to mutually interpenetrating networks at the local scale.     

Structures of chaos in open reaction systems

By numerically simulating the Bray-Liebhafsky (BL) reaction (the hydrogen peroxide decomposition in the presence of hydrogen and iodate ions) in a continuously fed well stirred tank reactor (CSTR), we find "structured" types of chaos emerging in regular order with respect to flow rate as the control parameter. These chaotic "structures" appear between each two successive periodic states, and have forms and evolution resembling to the neighboring periodic dynamics. More precisely, in the transition from period-doubling route to chaos to the arising periodic mixture of different mixed-mode oscillations, we are able to recognize and qualitatively and quantitatively distinguish the sequence of "period-doubling" chaos and chaos consisted of mixed-mode oscillations (the "mixed-mode structured" chaos), both appearing in regular order between succeeding periodic states. Additionally, between these types of chaos, the chaos without such recognizable "structures" ("unstructured" chaos) is also distinguished. Furthermore, all transitions between two successive periodic states are realized through bifurcation of chaotic states. This scenario is a universal feature throughout the whole mixed-mode region, as well as throughout other mixed-mode regions obtained under different initial conditions.     

Linear conjugacy of chemical reaction networks

Under suitable assumptions, the dynamic behaviour of a chemical reaction network is governed by an autonomous set of polynomial ordinary differential equations over continuous variables representing the concentrations of the reactant species. It is known that two networks may possess the same governing mass-action dynamics despite disparate network structure. To date, however, there has only been limited work exploiting this phenomenon even for the cases where one network possesses known dynamics while the other does not. In this paper, we bring these known results into a broader unified theory which we call conjugate chemical reaction network theory. We present a theorem which gives conditions under which two networks with different governing mass-action dynamics may exhibit the same qualitative dynamics and use it to extend the scope of the well-known theory of weakly reversible systems.     

Chemical applications of topology and group theory

Earlier approaches to the analysis of chemical dynamic systems using kinetic logic are refined to deal more effectively with systems having the two or more feedback circuits required for chaos. The essential kinetic features of such a system can be represented by a directed graph (called aninfluence diagram) in which the vertices represent the internal species and the directed edges represent kinetic relationships between the internal species. Influence diagrams characteristic of chaotic chemical systems have the following additional features: (1) They are connected; (2) Each vertex has at least one edge directed towards it and one edge directed away from it; (3) There is at least one vertex, called a turbulent vertex, with at least two edges directed towards it. From such an influence diagram a state transition diagram representing the qualitative dynamics of the system can be obtained using the following 4-step procedure: (1) A logical relationship is assigned at each turbulent vertex; (2) A local truth table is generated for each circuit in the influence diagram; (3) The local truth tables are combined to give a global truth table using the logical relationships at the turbulent vertices; (4) The global truth table is used to determine the corresponding state transition diagram using previously described methods. This refined procedure leads to a more restricted set of influence diagrams having the interlocking cycle flow topology required for chaos than the procedure described earlier. Systems with 3 internal species are examined in detail using the refined procedure. All systems with 3 dynamic variables shown in the simulation studies of Rössler to give chaotic dynamics correspond to influence diagrams which give inter-locking cycle (chaotic) flow topologies by the refined procedure. In addition, two models for the Belousov-Zhabotinskii reaction are examined using the refined procedure. The results are potentially informative concerning possible mechanisms for the limitation of the accumulation of autocatalytically produced HBrO₂ (one of the internal species) during the course of this reaction.     

Hydrated electron production by reaction of hydrogen atoms with hydroxide ions: a first-principles molecular dynamics study

The solvated electron production by reaction between the H atom and the hydroxide anion was studied using Density Functional Theory based first-principles molecular dynamics. The simulation reveals a complex mechanism, controlled by proton transfers in the coordination sphere of the hydroxide and by the diffusion of the H atom in its solvent cavity. We formulate the hypothesis, based on a coupling between classical and first-principles molecular dynamics, that these two processes give rise to a lag time for the reaction that would explain the H atom extremely small reactivity compared to other radical species. Furthermore, the reaction observed gives an original insight in excess electron solvation.     

The onset of chaos in the vibrational dynamics of LiNC/LiCN

Recent advances in vibrational spectroscopy have greatly enhanced the possibilities of research of highly excited states in molecular systems of moderate size. At sufficiently high level of excitation the correspondence principle holds, and classical mechanical arguments constitute a useful interpretative tool. The corresponding dynamics often become very complex specially in systems with floppy degrees of freedom, and periodic motion plays an important role for its understanding. In this paper, we present a computational procedure to systematically calculate periodic orbits of LiNCLiCN with a given symmetry, that has the additional advantage of providing a useful insight into the onset of chaos in this system.     

硫脲氧化反应动力学研究进展

本文综述了硫脲氧化反应动力学的研究进展,根据氧化剂和氧化方式不同,将硫脲氧化体系分成含卤氧化体系和非卤氧化体系两大类,其中含卤氧化体系包括亚氯酸、碘酸、溴酸、卤素单质氧化硫脲的反应体系;非卤氧化体系包括双氧水、自由基、电化学和金属酸盐氧化硫脲的反应体系。总结了不同反应体系的动力学现象和反应机理研究状况,文中还介绍了在硫脲氧化反应动力学研究中光电磁及色谱方法的发展状况,提出硫脲氧化反应动力学机理研究突破可能途径。