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.     

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.     

Periodic, Mixed-Mode, and Chaotic Regimes in the Belousov–Zhabotinskii Reaction Catalyzed by Ferroin

The periodic, mixed-mode, and chaotic regimes in the ferroin-catalyzed Belousov–Zhabotinskii (BZ) reaction observed in a continuous stirred tank reactor (CSTR) reactor at various flow rates were experimentally studied. It was found that an increase in the flow rate resulted in the appearance of various complex oscillations. The possibility of the numerical simulation of experimentally observed asymptotic mixed-mode oscillations and chaotic regimes with the use of a kinetic scheme that includes experimental rate constants of each step of the ferroin-catalyzed BZ reaction was first demonstrated. The reaction scheme adequately describes the bifurcation sequence of experimentally observed oscillating regimes.     

Temperature-induced route to chaos in the H2O2-HSO3(-)-S2O3(2-) flow reaction system

Low-frequency, high-amplitude pH-oscillations observed experimentally in the H2O2-HSO3(-)-S2O3(-) flow reaction system at 21.0 degrees C undergo period-doubling cascades to chemical chaos upon decreasing the temperature to 19.0 degrees C in small steps. Period-4 oscillations are observed at 20.0 degrees C and can be calculated on the basis of a simple model. A reverse transition from chaos to high-frequency limit cycle oscillations is also observable in the reaction system upon decreasing further the temperature step by step to 15.0 degrees C. Period-2 oscillations are measured at 18.0 degrees C. Such a temperature-change-induced transition between periodic and chaotic oscillatory states can be understood by taking into account the different effects of temperature on the rates of composite reactions in the oscillatory system. Small differences in the activation energies of the composite reactions are responsible for the observed transitions. Temperature-change-induced period doubling is suggested as a simple tool for determining whether an experimentally observed random behavior in chemical systems is of deterministic origin or due to experimental noise.     

Complex oscillations in a simple model for the Briggs-Rauscher reaction

Complex oscillations in a simple model of the Briggs-Rauscher reaction mechanism in a continuously stirred tank reactor proposed by Kim et al. [J. Chem. Phys. 117, 2710 (2002)] are investigated numerically. The k(0)-[CH(2)(COOH)(2)](0) phase diagram is constructed first where k(0) is the flow rate and [...](0) is the input concentration. Within the region surrounded by the Hopf bifurcation curve, we find complex oscillation regions which are again separated from the regular oscillation region by the secondary Hopf bifurcation curves. Mixed mode oscillations with an incomplete Farey sequence, periodic-chaotic (or nonperiodic) sequence, and various types of burst oscillations are observed in complex oscillation regions. Also, chaotic burst oscillations, which are due to the transition from one kind of burst to another kind, are reported.     

Conditions for Mixed Mode Oscillations and Deterministic Chaos in Nonlinear Chemical Systems

An approach has been proposed for finding the conditions for the existence of mixed-mode oscillations and deterministic chaos in a kinetic scheme after reduction to a simple system of equations. Analysis of the position and stability of the steady states of this system suggested simple conditions for the existence of mixed-mode oscillations and deterministic chaos. The boundaries for monostability, bistability, and oscillations were also found. The results obtained were completely confirmed by numerical modelling.     

Study of Lotka-volterra food chain chemostat with periodically varying dilution rate

In this paper, we introduce and study a model of Lotka-volterra chemostat food chain chemostat with periodically varying dilution rate, which contains with predator, prey, and substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, we numerically simulate a model with sinusoidal dilution rate, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the system experiences following process: periodic solution → periodic doubling cascade → chaos.     

Study of a Monod–Haldene type food chain chemostat with pulsed substrate

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

Influence of metal ions on oscillatory behavior of zinc anode in nitric acid-potassium dichromate media

A zinc anode in acidic media is a new oscillatory electrochemical system which manifests interesting behaviors, from steady states to simple oscillations and chaos. This paper presents an experimental study of the influence of metal ions on the shape, amplitude and duration of the cell potential oscillations, and gives a qualitative explanation of the system’s behavior. A small quantity of Cu²⁺, Zn²⁺ or Fe³⁺ ions added to the system change dramatically the potential oscillations from chaotic behavior to simple oscillations. The method may be used for chaos attenuation.     

Analysis of Monod type food chain chemostat with k-times’ periodically pulsed input

In this paper, we introduce and study a model of a predator-prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

Analysis of a Tessiet type food chain chemostat with k-times’ periodically pulsed input

In this paper, we introduce and study a Tessiet type food chain chemostat, which contains with predator, prey and k-times’ periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing. When impulsive period is small, there exists quasiperiodic oscillation in the impulsive system.     

亚氯酸盐-硫代硫酸盐非缓冲体系的动力学

研究了亚氯酸盐-硫代硫酸盐反应体系在非缓冲条件下的复杂动力学行为.结果发现,在开放体系中反应的pH值和Pt电位存在准周期振荡分叉和混合模式振荡分叉通向混沌的过程，且pH峰与Pt电位峰反相位.当与起始浓度比相对较小时，随着流速的逐渐升高，体系的pH值和Pt电位从简单的小振幅振荡(S)经过准周期振荡分叉到混沌，最后回到简单大振幅振荡(L)；而当与起始浓度比相对较高时，随着流速的降低，体系的pH值和Pt电位出现LS1、LS2、LS3…LSn的混合模式振荡,并在每对(LSn、LSn＋1)振荡区间发现了LSn、LSn＋1随机出现的非周期振荡行为.运用硫价态变化的一般动力学模型，模拟出了反应体系的混合模式振荡及非周期振荡.     

Analysis of a Monod–Haldene type food chain chemostat with seasonally variably pulsed input and washout

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with seasonally variably pulsed input and washout. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, bifurcation diagrams have shown that there exists complexity for the pulsed system including periodic doubling cascade, periodic halving cascade and Pitchfork bifurcations and tangent bifurcations.      

Bifurcation analysis and chaos control in discrete-time glycolysis models

In this paper, the qualitative behavior of two discrete-time glycolysis models is discussed. The discrete-time models are obtained by implementing forward Euler’s scheme and nonstandard finite difference method. The parametric conditions for local asymptotic stability of positive steady-states are investigated. Moreover, we discuss the existence and directions of period-doubling and Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. OGY feedback control and hybrid control methods are implemented in order to control chaos in discrete-time glycolysis model due to emergence of period-doubling and Neimark–Sacker bifurcations. Numerical simulations are provided to illustrate theoretical discussion.     

Effect of temperature on bifurcation of self-oscillating modes in the Belousov-Zhabotinskii reaction in a continuous-flow stirred-tank reactor

The effect of the temperature on self-oscillating modes in the Belousov-Zhabotinskii reaction in a continuous-flow stirred-tank reactor (CSTR) at T=288–333 K was investigated. It was found that changing the temperature leads to a cascade of different bifurcations: the steady state changes into sinusoidal oscillations, then chaotic and complex periodic oscillations appear in the system and turn into regular oscillations with a further increase in the temperature. The regular oscillations disappear at T=329 K as a result of a degenerate supercritical Hopf bifurcation.Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 31, No. 2, pp. 69–75, March–April, 1995.     

Experimental study of embedding of attractors in concentration space

The embedding of attractors and their stable and unstable manifolds can be studied experimentally by controlled addition of chemical species to bring about a particular response. For stable small amplitude oscillations near a Hopf bifurcation from a steady state the embedding can be completely determined even in systems where two of the species are not observable. A quenching of the oscillations by dilution candetermine the steady state concentrations. A species that cannot quench the oscillations almost certainly cannot be an essential component of the oscillation. The method can be extended to a study of attractor associated with subharmonic and quasiperiodic bifurcations and of attractors corresponding to nonperiodic motion. We present preliminary results for a subharmonic bifurcation.     

钙离子体系的振动双共振控制

研究了钙离子振荡体系在高、低两种不同频率信号作用下所产生的振动双共振(VBR)及其控制方法.结果表明:系统对低频信号响应的幅值随高频信号振幅的变化产生了振动双共振现象,并且低频信号的频率越低,振幅越大,系统通过振动双共振对微弱低频信号的放大倍数越大.体系离霍普夫(Hopf)分岔点的距离越近(控制参数域值越小),体系发生振动双共振所需要的最大高频信号幅值越往大的方向漂移,同时体系振动双共振的强度越小.细胞内钙波形成过程中的反馈机制对体系振动双共振的增强和减弱起着重要的作用,即正反馈机制对体系振动双共振强度起增强的作用,而负反馈机制却起减弱的作用.另外,体系中引入噪音所产生的随机共振不仅削弱振动双共振的强度而且还影响振动峰的个数,也发现存在极限噪音强度使体系产生不同的振荡行为,极限噪音强度之下,体系产生VBR现象,而极限噪音强度之上,体系则发生单峰振荡共振现象.     

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.     

Bromide ions induced chaotic behavior in H2O2–H2SO4–Pt electrochemical system

We have found a new chaotic current oscillation in the H₂O₂–H₂SO₄–Pt electrochemical system due to the addition of small amounts of bromide ions. In the system with bromide ions, an oscillation, called oscillation D, appears near the potential where another oscillation, called oscillation A, appears. The chaotic oscillation is observed in a potential region where both oscillations A and D simultaneously appear. When the electrode potential is stepped to a potential in the above region from the rest potential, a period-1 oscillation first appears for a while. A period-doubling bifurcation cascade then occurs, which is followed by a chaos. The appearance of the chaotic oscillations is explained on the basis of the reported mechanisms for oscillations A and D.Dedicated to Professor György Horányi to celebrate his 70th birthday in recognition of many contributions to electrochemistry.