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.     

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.     

Hopf bifurcation and multiple limit cycles in bio-chemical reaction of the morphogenesis process

Morphogenetic process is an interesting but very hard bio-chemical problem. In this paper, we consider a bio-chemical model in temporal morphogenesis which is a generalization of the model studied by Gierer–Meinhardt. By using the theory of ordinary differential equations, it is shown that the model undergoes a Hopf bifurcation if the parameters in the model satisfy the following relationship: λ = 2/(ρ ₂ + x*)−1. It is also proved that the close orbit created by the Hopf bifurcation is stable. The conditions that guarantee the system has three closely nested limit cycles are also obtained in the paper.     

Entropy production in a mesoscopic chemical reaction system with oscillatory and excitable dynamics

Stochastic thermodynamics of chemical reaction systems has recently gained much attention. In the present paper, we consider such an issue for a system with both oscillatory and excitable dynamics, using catalytic oxidation of carbon monoxide on the surface of platinum crystal as an example. Starting from the chemical Langevin equations, we are able to calculate the stochastic entropy production P along a random trajectory in the concentration state space. Particular attention is paid to the dependence of the time-averaged entropy production P on the system size N in a parameter region close to the deterministic Hopf bifurcation (HB). In the large system size (weak noise) limit, we find that P ～ N(β) with β = 0 or 1, when the system is below or above the HB, respectively. In the small system size (strong noise) limit, P always increases linearly with N regardless of the bifurcation parameter. More interestingly, P could even reach a maximum for some intermediate system size in a parameter region where the corresponding deterministic system shows steady state or small amplitude oscillation. The maximum value of P decreases as the system parameter approaches the so-called CANARD point where the maximum disappears. This phenomenon could be qualitatively understood by partitioning the total entropy production into the contributions of spikes and of small amplitude oscillations.     

Molecular potential-energy surfaces for chemical reaction dynamics

This paper reviews the construction of molecular potential-energy surfaces by an interpolation method which has been developed over the last several years. The method uses ab initio quantum chemistry calculations of the molecular electronic energy in an automated procedure to construct global potential- energy surfaces which can be used to simulate chemical reactions with either classical or quantum dynamics. The methodology is explained and several applications are presented to illustrate the approach. Received: 22 February 2002 / Accepted: 2 May 2002 / Published online: 6 November 2002 Correspondence to: M. A. Collins e-mail: collins@rsc.anu.edu.au Acknowledgements. The methods described in this overview are the result of collaborations with former members of my group, in particular with Josef Ischtwan, Meredith Jordon, Keiran Thompson and Ryan Bettens. I am also indebted for inspiration gained from many discussions with my colleagues Leo Radom and Donghui Zhang (National University of Singapore). This work has been supported by the Supercomputer Facility of the Australian National University and the Australian Partnership for Advanced Computing.     

Effect of Coriolis coupling in chemical reaction dynamics

It is essential to evaluate the role of Coriolis coupling effect in molecular reaction dynamics. Here we consider Coriolis coupling effect in quantum reactive scattering calculations in the context of both adiabaticity and nonadiabaticity, with particular emphasis on examining the role of Coriolis coupling effect in reaction dynamics of triatomic molecular systems. We present the results of our own calculations by the time-dependent quantum wave packet approach for H + D2 and F(2P3/2,2P1/2) + H2 as well as for the ion-molecule collisions of He + H2 +, D(-) + H2, H(-) + D2, and D+ + H2, after reviewing in detail other related research efforts on this issue.     

On the study of nonlinear dynamics of complex chemical reaction systems

When driven far from equilibrium,nonlinear chemical reactions often show a variety of self-organization behavior,including chemical oscillations,waves,chaos and patterns[1].Recently,the study of such nonlinear phenomena in‘complex’systems,such as the li…     

Sample dispersion with chemical reaction in a flow-injection system

Computer simulations of signals based on a dispersion equation involving diffusion, convection and chemical reaction terms are reported for systems with and without chemica reaction. 2-(2-Thiazolylazo)-4-methyl-5-(sulfomethylamino)benzoic acid (TAMSMB) is used as the reagent because it reacts very quickly with copper(II) and much more slowly with nickel(II). Comparison of experimental signal profiles with the simulated ones enables sample dispersion from the reaction rates to be elucidated. Concetration profiles of the reaction product in a straight, narrow tube were also stimulated; they explain satisfactorily the signal profiles obtained experimentally.     

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.     

Investigating the chemical dynamics of the reaction of ground-state carbon atoms with acetylene and its isotopomers

We investigated the multichannel reaction of ground-state carbon atoms with acetylene, C2H2 (X1Sigmag+), to form the linear and cyclic C3H isomers (atomic hydrogen elimination pathway) as well as tricarbon plus molecular hydrogen. The experiments were conducted under single-collision conditions at three different collision energies between 8.0 kJ mol-1 and 31.0 kJ mol-1. Our studies were complemented by crossed molecular beam experiments of carbon with three isotopomers C2D2(X1Sigmag+), C2HD (X1Sigma+), and 13C2H2 (X1Sigmag+) to clarify a potential intersystem crossing (ISC), the effect of the symmetry of the reaction intermediates on the center-of-mass angular distributions, the collision energy-dependent branching ratios of the atomic versus molecular hydrogen elimination pathways, and deuterium-enrichment processes. The results are discussed in light of recent electronic structure and dynamics calculations.     

Quantum chemical reaction dynamics on a highly parallel supercomputer

Summary In this paper we describe the solution of the quantum mechanical equation for the scattering of an atom by a diatomic molecule on a high-performance distributed-memory parallel supercomputer, using the method of symmetrized hyperspherical coordinates and local hyperspherical surface functions. We first cast the problem in a format whose inherent parallelism can be exploited effectively. We next discuss the practical implementation of the parallel programs that were used to solve the problem. The benchmark results and timing obtained from the Caltech/JPL Mark IIIfp hypercube are competitive with the CRAY X-MP, CRAY 2 and CRAY Y-MP supercomputers. These results demonstrate that such highly parallel architectures permit quantum scattering calculations with high efficiency in parallel fashion and should allow us to study larger, more complicated chemical systems. Future extensions to this approach are discussed.Work performed in partial fulfillment of the requirements for the Ph.D. degree in Chemistry at the California Institute of Technology.Contribution number 8209     

Molecular dynamics of the hydrogeniodine reaction system

The molecular dynamics of the hydrogeniodine exchange reaction, H₂ + I₂ ? 2 HI, have been examined in a classical trajectory analysis with trajectories selected within the transition region. The results yield the overall mechanism, the rate and the energy distributions of reactants and products for reaction at 700°K.     

Characterization of the flow dynamics of an enzyme reaction system

The characterization of the flow dynamics of an automated enzyme reaction system is discussed. Parametric and non-parametric characterizations are constructed and then tested against actual experiments. The analytical model provides a more compact representation than the numerical model. A second-order lag-plus-dead-time model gives a good fit to the experimental data.     

A refined ring polymer molecular dynamics theory of chemical reaction rates

We further develop the ring polymer molecular dynamics (RPMD) method for calculating chemical reaction rates [I. R. Craig and D. E. Manolopoulos, J. Chem. Phys. 122, 084106 (2005)]. We begin by showing how the rate coefficient we obtained before can be calculated in a more efficient way by considering the side functions of the ring-polymer centroids, rather than averaging over the side functions of the individual ring-polymer beads. This has two distinct advantages. First, the statistics of the phase-space average over the ring-polymer coordinates and momenta are greatly improved. Second, the resulting flux-side correlation function converges to its long-time limit much more rapidly. Indeed the short-time limit of this flux-side correlation function already provides a "quantum transition state theory" approximation to the final rate coefficient. In cases where transition state recrossing effects are negligible, and the transition state dividing surface is put in the right place, the RPMD rate is therefore obtained almost instantly. We then go on to show that the long-time limit of the new flux-side correlation function, and hence the fully converged RPMD reaction rate, is rigorously independent of the choice of the transition state dividing surface. This is especially significant because the optimum dividing surface can often be very difficult to determine for reactions in complex chemical systems.     

Calculation of the mass transfer coefficient with a chemical reaction in a gas-liquid system

A solution of the problem of mass transfer in a turbulent boundary layer has been obtained with a first-order chemical reaction occurring in the liquid phase. The dependence of the enhancement factor for absorption and of the mass transfer coefficient on the model parameters can be used for building up a hierarchic model of the gas-liquid reactor.

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