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.     

Global dynamics and transition state theories: Comparative study of reaction rate constants for gas-phase chemical reactions

In this review article, we present a systematic comparison of the theoretical rate constants for a range of bimolecular reactions that are calculated by using three different classes of theoretical methods: quantum dynamics (QD), quasi-classical trajectory (QCT), and transition state theory (TST) approaches. The study shows that the difference of rate constants between TST results and those of the global dynamics methods (QD and QCT) are seen to be related to a number of factors including the number of degrees-of-freedom (DOF), the density of states at transition state (TS), etc. For reactions with more DOF and higher density of states at the TS, it is found that the rate constants from TST calculations are systematically higher than those obtained from global dynamics calculations, indicating large recrossing effect for these systems. The physical insight of this phenomenon is elucidated in the present review.     

化学反应过渡态的结构和动力学

化学反应过渡态决定了包括反应速率和微观反应动力学在内的化学反应的基本特性,而无论是从理论还是实验上研究和观测化学反应过渡态都是极具挑战性的课题.近年来,我国科学家们利用交叉分子束-里德堡氢原子飞行时间谱仪,结合高精度的量子动力学计算,对H＋H2和F＋H2这两个教科书式的典型反应体系进行了全量子态分辨的反应动力学研究,从中得出了关于这两个反应体系的过渡态的结构和动力学性质的结论性的研究成果.     

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.     

Coriolis coupling and nonadiabaticity in chemical reaction dynamics

The nonadiabatic quantum dynamics and Coriolis coupling effect in chemical reaction have been reviewed, with emphasis on recent progress in using the time-dependent wave packet approach to study the Coriolis coupling and nonadiabatic effects, which was done by K. L. Han and his group. Several typical chemical reactions, for example, H+D(2), F+H(2)/D(2)/HD, D(+)+H(2), O+H(2), and He+H(2)(+), have been discussed. One can find that there is a significant role of Coriolis coupling in reaction dynamics for the ion-molecule collisions of D(+)+H(2), Ne+H(2)(+), and He+H(2)(+) in both adiabatic and nonadiabatic context.     

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.     

Stochastic path-integral method for chemical reaction dynamics: Application to the full 3D H3 system

A stochastic path-integral (SPI) technique for chemical reaction dynamics is explored. It is shown that this technique enables the direct computation of the transition amplitude with a finite space-time range, by generating a set of classical paths subject to simultaneous stochastic differential equations. The numerical values of the Boltzmann matrix elements for a harmonic potential are in good agreement with the analytical ones. Within the quantum transition state theory, the flux-flux autocorrelation function is also evaluated at 630 K for the H + H₂ exchange reaction and is found to give a satisfactory agreement with the previous studies. To appraise the influence of the dimensionality, both one-dimensional Eckart potential and a full three-dimensional (3D) Liu-Siegbahn-Truhlar-Horowitz (LSTH) potential calculations have been performed. The calculated values of the Boltzmann matrix elements for the colinear and the full 3D cases are found to deviate slightly from each other in the lower temperature range. The 3D thermal rate constant is in very good agreement with the previous one. © 1996 John Wiley & Sons, Inc.     

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.