Relationships Between the Discriminant Curve and Other Bifurcation Diagrams

The parameter dependence of the number and type of the stationary points of an ODE is considered. The number of the stationary points is determined by the saddle-node (SN) bifurcation set and their type (e.g., stability) is given by another bifurcation diagram (e.g., Hopf bifurcation set). The relation between these bifurcation curves on the parmeter plane is investigated. It is shown that the cross-shaped diagram, when the Hopf bifurcation curve makes a loop around a cusp point of the SN curve, is typical in some sense. It is proved that the two bifurcation curves meet tangentially at their common points (Takens–Bogdanov point), and these common points persist as a third parameter is varied. An example is shown that exhibits two different types of 3-codimensional degenerate Takens–Bogdanov bifurcation.     

Dynamic instabilities of model electrochemical system with electrocatalytic oxidation and preceding chemical reaction

The method of impedance spectroscopy was used to study the regularities of behavior of a model electrochemical system with an electrocatalytic reaction on a planar electrode which is preceded by a homogeneous chemical reaction. It is shown that in the case of low rates of the homogeneous chemical reaction, two types of bifurcations may exist in the system at the chosen attraction constant value; namely, the Hopf bifurcation leading to spontaneous oscillations and saddle-node bifurcation resulting in bistability of the system. The Hopf bifurcation disappears from the system at medium and high rates of the homogeneous chemical reaction, while the saddle-node bifurcation is retained.     

Mathematical analysis of the smallest chemical reaction system with Hopf bifurcation

Recently we presented an up to now unstudied three-dimensional dynamical system which is, according to our given definition, the smallest chemical reaction system with Hopf bifurcation. We here study the Hopf bifurcation in detail and prove that near the bifurcation point a stable limit cycle arises. In the analysis we use the methods of local bifurcation theory, especially the center manifold and the normal form theorem. In a similar way we analyse the also occurring transcritical bifurcation. Besides studying local stability, we give the proofs for global stability of the trivial steady state in the whole positive phase space and for the nontrivial steady state in a closed domain containing the steady state point.     

Swallowtail model for predicting the global bifurcation behavior of CO oxidation reactions

Catalytic CO oxidation on platinum group metals exhibits nonlinear phenomena such as hysteresis and bifurcation.Elucidation of the nonlinear mechanisms is important for catalyst design and control of reaction routes.Previous studies have offered initial insights into the local bifurcation behavior of CO oxidation.However,since the bifurcation behavior of CO oxidation is determined by multiple parameters such as temperature,total flux,and CO fraction,it is difficult to predict the global bifurcation behavior...     

Smallest chemical reaction system with Hopf bifurcation

The smallest at most bimolecular chemical reaction system with Hopf bifurcation is presented. First the notion smallest reaction system is explained. Since the lowest number of intermediates has the highest priority in this characterization and since it has already been shown that three-component systems can have a Hopf bifurcation [1], the smallest reaction system must contain three intermediates. On the basis of a sufficient condition for a Hopf bifurcation in three-dimensional systems it is possible to find one reaction system which is according to the given characterization, the smallest one. In the first part of this paper it is shortly pictured and in the second part a more extensive proof that this system is really the searched smallest one is given.     

Dynamic behavior of the bray-liebhafsky oscillatory reaction controlled by sulfuric acid and temperature

The non-periodic, periodic and chaotic regimes in the Bray-Liebhafsky (BL) oscillatory reaction observed in a continuously fed well stirred tank reactor (CSTR) under isothermal conditions at various inflow concentrations of the sulfuric acid were experimentally studied. In each series (at any fixed temperature), termination of oscillatory behavior via saddle loop infinite period bifurcation (SNIPER) as well as some kind of the Andronov-Hopf bifurcation is presented. In addition, it was found that an increase of temperature, in different series of experiments resulted in the shift of bifurcation point towards higher values of sulfuric acid concentration.     

Complete set of Hopf bifurcation in an autocatalytic ring network

We investigate the Hopf bifurcation for a five species chemical ring network with an autocatalytic reaction. We show that the bifurcation hypersurface in the rate constants space is the boundary of a simply connected set. We use a numerical method to calculate this hypersurface.     

Role of ohmic losses in appearance of dynamic instabilities in model electrochemical system with cylindrical electrode under potentiostatic conditions

The impedance spectroscopy technique was used for theoretical studies of the conditions of appearance of Hopf instability and saddle-node instability in a model electrochemical system with a preceding homogeneous chemical reaction in the Nernst diffusion layer and electrocatalytic reaction on the surface of a cylindrical electrode under potentiostatic conditions. It is shown that the value of the ohmic resistance parameter in the system can affect the number of bifurcation points and bifurcation frequency. An increase in the ohmic resistance parameter results in the narrowing of the region of Hopf bifurcation giving rise to spontaneous periodical current oscillations and expansion of the region of saddle-node instability leading to bistability. There are threshold values of the ohmic resistance parameters critical for appearance and disappearance of the dynamic instabilities under consideration.     

Coexistence of two bifurcation regimes in a closed ferroin-catalyzed Belousov-Zhabotinsky reaction

The ferroin-catalyzed Belousov-Zhabotinsky (BZ) reaction was studied in a batch reactor under anaerobic conditions and was found to evolve through two separated regimes of complex oscillations. Significantly, the two bifurcation regimes exhibited qualitatively different dependence on compositions of the reaction mixture, i.e., initial concentrations of bromate, sulfuric acid, malonic acid, and ferroin. The reaction temperature also showed opposite effects on the two bifurcation regimes, in which complexities of the first bifurcation regime were enhanced while oscillations in the second bifurcation regime became simpler as a result of decreasing temperature. Numerical simulations with a 12-variable model developed specifically for the ferroin-BZ system were able to reproduce transient complex oscillations observed in experiments. These calculations further illustrated that reactions such as ferroin and HOBr, ferroin and HBrO2, and ferriin and Br- were not essential in describing complex dynamics of the ferroin-BZ reaction.     

Bifurcation analysis on a reactor model with combination of quadratic and cubic steps

In this paper we give a rigorous bifurcation analysis for a reactor model with combination of quadratic and cubic steps. We extend the analysis of Satnoianu et al. in two directions. First, we impose Dirichlet boundary conditions instead of the semi-infinite domain in one space dimension. Second, we consider a variety of different bifurcation phenomena of the corresponding steady state system, focusing on their parametric sensitivity. We show that the system exhibits subcritical pitchfork bifurcation, supercritical pitchfork bifurcation and transcritical bifurcation in the different regimes of control parameters. To compare the decades old work by Auchmuty et al., we show that the spatial structures formed by the parameter p ∈ [0, 1] are richer than those formed by the purely quadratic step for p = 0 and the purely cubic step for p = 1 respectively.     

Electrochemical control of bifurcation points for stochastic resonance in a model of the photosensitive belousov‐zhabotinsky reaction

We present a model of the Belousov‐Zhabotinsky reaction with external controls. The Oregonator model is modified to include three different control parameters, i.e., flow rate, light intensity, and electric current. Applying cathodic or anodic current shifts bifurcation points in an opposite direction in a two‐parameter space spanned by light intensity and flow rate. This external control enables steady states to be sensitive to noise in the two‐parameter space. The use of shifting bifurcation points is discussed in relation with stochastic resonance.     

用Lyapunov指数确定Brusselator和Oregonator的分岔

本文计算了Brusselator及Oregonator的Lyapunov指数随系统控制参量的变化,依据Lyapunov指数的稳定性理论,确定出上述两模型都是通过Hopf分岔产生极限环的。在Brusselator分岔行为上,Lyapunov指数理论与线性稳定性分析所得的结论是一致的。对Oregonator所代表的化学背景作了分析,指出影响B-Z反应发生分岔的主要因素是Br~-的再生能力与速度。     

Dynamic states of the Bray-Liebhafsky reaction when sulfuric acid is the control parameter

Dynamic behavior of hydrogen peroxide decomposition catalyzed by iodate and hydrogen ions (the Bray-Liebhafsky reaction), in a continuous stirred tank reactor is investigated. The experimental results are obtained at one operational point in concentration phase space by varying mixed inflow concentrations of the sulfuric acid. The experimental evidences for the onset and termination of oscillatory behavior via the saddle node infinite period bifurcation as well as some kind of the Andronov-Hopf bifurcation are presented. In addition, the possibility of excitability of a stable steady state by thiamine was observed. The article is published in the original.     

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.     

Experiments on temporal and spatial chemical patterns

Two examples of bifurcation sequences observed in experiments on nonequilibrium reactions in continuous flow stirred reactors are discussed: one experiment demonstrates that bifurcation diagrams can serve as a tool for elucidating chemical mechanisms, and another experiment illustrates that even very complex bifurcation sequences need not include chaotic states. A discussion of chemical spatial patterns emphasizes the need for the development of spatially extended reactors in which patterns can be maintained indefinitely. Three such recently developed reactors are described.     

Bifurcation and chaos of Tessiet type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of Tessiet type food chain chemostat with periodically varying 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. Furthermore, we numerically simulate a model with sinusoidal dilution rate, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halving.     

Bifurcation and domain walls in thin films of liquid crystals

A perturbation method is proposed to determine the bifurcation diagram for instabilities in thin liquid-crystalline layers which are subject to an external magnetic or electric field. Different types of continuous and discontinuous field-induced director reorientations can be classified by using elementary bifurcation theory. When two differently oriented director positions are stable, domains may appear. For weakly distorted samples the structure and velocity of walls between stable differently oriented domains is described by a solitary wavy solution of the non-linear director equation. The results of the perturbation method are applied to nematic and smectic C films.     

Critical threshold of noise-induced energy transduction in molecular machinery system

Responses of energy transduction of molecular machinery to random perturbation were investigated at the conditions where the system stayed near the bifurcation point. It was found that noise-induced oscillation (NIO) could occur. But how far from bifurcation point could one get the admissible region of NIO? We proposed and demonstrated numerically that there existed a critical threshold of NIO for each fixed noise intensity. Furthermore, it was found that noise intensity was a key factor for the determination of critical threshold. Finally, the detailed bifurcation diagram depending on noise intensity was replotted.     

Analysis of a Beddington–DeAngelis food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Beddington–DeAngelis type food chain chemostat with periodically varying 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. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

Influence of bifurcation position and length of side chains on the structure of isoindigo-based conjugated polymer thin films

We chose a series of isoindigo-based conjugated polymer(ⅡDDT,ⅡDDT-C3 and ⅡDDT-C4) with different length of side chains and bifurcation positions to investigate the relationship between the degree of alignment and the length of side chains and bifurcation positions.We found that the dichroic ratio was increased from 2.37 to 5.23 when the side chain was longer and the bifurcation position was away from the backbone.The π-π stacking distance was decreased from 3.67 A to 3.61 A when the bifurcation position was away from the backbone because of its smaller hindrance and the d-spacing of the(100)was increased from 20.06 A to 25.21 A when the side chain was longer.All the polymers were adopted an edge-on orientation with the backbone paralleled with the long axis of fibers.The weak interaction of side-chain in ⅡDDT-C4 was beneficial for the molecules being rearranged in parallel during the contact line receding and the strong n-n interaction could accelerate the interchain assembly of the parallel molecules through π-π interaction to form aligned fibers.