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.     

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。     

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.     

Spatiotemporal complexity in a diffusive Brusselator model

Journal of Mathematical Chemistry - Turing–Hopf bifurcation of the diffusive Brusselator model with homogeneous Neumann boundary conditions is considered in this paper. By stability analysis,...     

Complex and chaotic oscillations in a model for the catalytic hydrogen peroxide decomposition under open reactor conditions

Numerous periodic and aperiodic dynamic states obtained in a model for hydrogen peroxide decomposition in the presence of iodate and hydrogen ions (the Bray-Liebhafsky reaction) realized in an open reactor (CSTR), where the flow rate was the control parameter, have been investigated numerically. Between two Hopf bifurcation points, different simple and complex oscillations and different routes to chaos were observed. In the region of the mixed-mode evolution of the system, the transitions between two successive mixed-mode simple states are realized by period-doubling of the initial state leading to a chaotic window in which the next dynamic state emerges mixed with the initial one. It appears in increasing proportions in concatenated patterns until total domination. Thus, with increasing flow rate the period-doubling route to chaos was obtained, whereas with decreasing flow rate the peak-adding route to chaos was obtained. Moreover, in very narrow regions of flow rates, chaotic mixtures of mixed-mode patterns were observed. This evolution of patterns repeats until the end of the mixed-mode region at high flow rates that corresponds to chaotic mixtures of one large and many small amplitude oscillations. Starting from the reverse Hopf bifurcation point and decreasing the flow rate, simple small amplitude sinusoidal oscillations were encountered and then the period-doubling route to chaos. With a further decreasing flow rate, the mixed-mode oscillations emerge inside the chaotic window.     

Delay induced oscillations in a turbidostat with feedback control

A model of competition between two species in a turbidostat with delayed feedback control is investigated. By choosing the delay in the measurement of the optical sensor to the turbidity of the fluid as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Computer simulations illustrate the results.     

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 diagrams of the heterogeneous catalytic oxidation of methanol

The experimental bifurcation diagrams of the heterogeneous catalytic oxidation of methanol on Pd have been determined by changing the concentration of methanol or oxygen in the feed. Hopf bifurcations have been obtained.     

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.     

Ruelle-Takens-Newhouse scenario in reaction-diffusion-convection system

Direct numerical simulations of the transition process from periodic to chaotic dynamics are presented for two variable Oregonator-diffusion model coupled with convection. Numerical solutions to the corresponding reaction-diffusion-convection system of equations show that natural convection can change in a qualitative way, the evolution of concentration distribution, as compared with convectionless conditions. The numerical experiments reveal distinct bifurcations as the Grashof number is increased. A transition to chaos similar to Ruelle-Takens-Newhouse scenario is observed. Numerical results are in agreement with the experiments.     

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.     

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.     

Multistage homotopy perturbation method for nonlinear reaction networks

In this paper, we present the multistage homotopy perturbation method for finding the solution of the chemical kinetics with nonlinear reactions. We develop a general scheme for finding the analytic solution of chemical reaction networks and apply it to motivating chemical examples such as the enzyme kinetics model and the Brusselator model. We illustrate the numerical result for the models and show the accuracy of the method.     

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.     

Fluctuation theorem and mesoscopic chemical clocks

The fluctuation theorems for dissipation and the currents are applied to the stochastic version of the reversible Brusselator model of nonequilibrium oscillating reactions. It is verified that the symmetry of these theorems holds far from equilibrium in the regimes of noisy oscillations. Moreover, the fluctuation theorem for the currents is also verified for a truncated Brusselator model.     

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.     

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.     

Uncertainty analysis of correlated parameters in automated reaction mechanism generation

Uncertainty analysis is a useful tool for inspecting and improving detailed kinetic mechanisms because it can identify the greatest sources of model output error. Owing to the very nonlinear relationship between kinetic and thermodynamic parameters and computed concentrations, model predictions can be extremely sensitive to uncertainties in some parameters while uncertainties in other parameters can be irrelevant. Error propagation becomes even more convoluted in automatically generated kinetic models, where input uncertainties are correlated through kinetic rate rules and thermodynamic group values. Local and global uncertainty analyses were implemented and used to analyze error propagation in Reaction Mechanism Generator (RMG), an open-source software for generating kinetic models. A framework for automatically assigning parameter uncertainties to estimated thermodynamics and kinetics was created, enabling tracking of correlated uncertainties. Local first-order uncertainty propagation was implemented using sensitivities computed natively within RMG. Global uncertainty analysis was implemented using adaptive Smolyak pseudospectral approximations as implemented in the MIT Uncertainty Quantification Library to efficiently compute and construct polynomial chaos expansions to approximate the dependence of outputs on a subset of uncertain inputs. Cantera was used as a backend for simulating the reactor system in the global analysis. Analyses were performed for a phenyldodecane pyrolysis model. Local and global methods demonstrated similar trends; however, many uncertainties were significantly overestimated by the local analysis. Both local and global analyses show that correlated uncertainties based on kinetic rate rules and thermochemical groups drastically reduce a model's degrees of freedom and have a large impact on the determination of the most influential input parameters. These results highlight the necessity of incorporating uncertainty analysis in the mechanism generation workflow.     

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.