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Notes: Periodic variations of an external parameter or constraint of open chemical systems have been shown to induce changes in time averaged kinetic and thermodynamic quantities. We examine the effects of the analytic form of the periodic variation on the time averaged quantities and find the maximum changes obtainable through periodic variations. A variational procedure is proposed, based on a Fourier expansion of the form of the periodic perturbation, the laws of thermodynamics, conservation of matter, and the kinetics. The efficiency of power production in a combustion system is examined with this method in a numerical example. A unique maximum in the efficiency is found, with the gains achievable for more complex functions exceeding those for a sinusoidal perturbation. We interpret the changes in efficiency in terms of the magnitude of the response of the system (resonance) and phase shifts between the periodic perturbations and the response of the system. We illustrate the mechanisms of efficiency changes in this system with two examples; one in which the periodic perturbation affects the phase relations and one in which the periodic perturbation affects the magnitude of the response. Finally, we note that multiple attractors may coexist in this system for certain forms of the periodic perturbation, each with a distinct efficiency.

Notes: Periodic variations are applied to the influxes of oxygen and methane entering a reaction vessel in which takes place a combustion reaction and, in the absence of these variations, the system is in a stable focus. We measure the reaction of the system to these variations and find a resonant response, and changes of phase relations between the forcing and response of the system, near the autonomous frequency. We calculate the enthalpy content of the gases and using a simple model of a Carnot engine we study the power output of the system and find increases (∼7% in power) in these quantities near the autonomous frequency. The experimental results are compared to the predictions of a numerical model specific to our system and to the analytic solution of a linear set of differential equations with a stable focus. We find good agreement with both, but there is an aspect of the experimental results which requires additional hypotheses.

Notes: As in a prior article (Ref. 1), we consider an oscillatory dissipative system driven by external sinusoidal perturbations of given amplitude Q and frequency ω. The kinetic equations are transformed to normal form and solved for small Q near a Hopf bifurcation to oscillations in the autonomous system. Whereas before we chose irrational ratios of the frequency of the autonomous system ωn to ω, with quasiperiodic response of the system to the perturbation, we now choose rational coprime ratios, with periodic response (entrainment). The dissipative system has either two variables or is adequately described by two variables near the bifurcation. We obtain explicit solutions and develop these in detail for ωn/ω=1; 1:2; 2:1; 1:3; 3:1. We choose a specific dissipative model (Brusselator) and test the theory by comparison with full numerical solutions. The analytic solutions of the theory give an excellent approximation for the autonomous system near the bifurcation. The theoretically predicted and calculated entrainment bands agree very well for small Q in the vicinity of the bifurcation (small μ); deviations increase with increasing Q and μ. The theory is applicable to one or two external periodic perturbations.

Notes: We study experimentally continuous transitions from quasiperiodic to periodic states for a time-periodically forced chemical oscillator. The chemical reaction is the hydration of 2,3-epoxy-1-propanol, and is carried out in a continuous stirred tank reactor (CSTR). Periodic oscillatory states are observed to arise in the autonomous system through supercritical Hopf bifurcations as either the total flow rate or the cooling coil temperature is changed. Under conditions of oscillation for the autonomous system, small-amplitude periodic variation of the total flow rate generates an attracting two-torus from the stable limit cycle. From the experiments we determine the structure of the toroidal flow, stroboscopic phase portraits, and circle maps as a function of the forcing amplitude and period. A continuous transition from the quasiperiodic to a periodic state, in which the two-torus contracts to a closed curve (Neimark–Sacker torus bifurcation), is observed as the forcing amplitude is increased at a constant forcing period, or as the forcing period is changed at a constant moderate forcing amplitude. Qualitative theoretical predictions compare well with the experimental observations. This paper presents the first experimental observation of a Neimark–Sacker torus bifurcation in a forced chemical oscillator system, and relates the bifurcation diagram of the unforced system to that of the forced system.

Notes: Many-body effects in reaction rates depend on the ratio ε of a rate coefficient to the product of a diffusion coefficient and a radius, and on the reduced volume fraction φ0 of one or more reactants. We present a statistical-mechanical theory of the macroscopic kinetics (deterministic rates) of reactions in solutions, and fluctuations therefrom, for arbitrary ε and φ0, by deriving expressions for effective forward and reverse rate coefficients and their dependence on ε, φ0 to lowest order. We use an enzyme-catalyzed reaction as an example. There are two corrections to rate coefficients (for ε=0, φ0=0) at a given ε, φ0≠0, and both are proportional to φ1/20 (the square root of the total enzyme density in the example). The first is an uncorrelated screening term described by the single enzyme distribution function, which increases the rate; and the second a term described by correlations among enzymes, which decreases the rate. In the limit of very fast reactions the correlation term is negligible, and the screening term reduces to that previously obtained for diffusion controlled reactions. For other cases both terms contribute: for example, in the range φ0∼10−2 to 10−1 and ε∼1–10 the corrections vary from a few percent to 30%, as obtained from numerical solutions of the corrections for the enzyme example. We discuss a quasistationary state of the example and derive a generalization of the Michaelis–Menten equation for all ε, φ0. Fluctuations from the deterministic motion are shown to be small for three-dimensional systems.

Notes: We prove that Liapunov functions for a single reactive intermediate evolving toward a nonequilibrium steady state can be obtained from a global potential φ. We consider reactions occurring in a chamber containing a reactant, the intermediate, and a product. Reservoirs connected to the chamber serve to hold the reactant and product concentrations constant, in nonequilibrium proportions. The Liapunov property of φ is significant because of the role it plays in the thermodynamic and stochastic analysis of nonequilibrium systems: φ is defined in terms of the reactive flux to produce the intermediate and the flux to remove the intermediate. The derivative of φ with respect to the concentration of the intermediate yields an effective chemical driving force that is specific to the intermediate, and its time derivative yields a species-specific component of the dissipation that is minimized at steady states. These results hold both near to equilibrium and far from equilibrium for systems with one intermediate, independent of the number of steady states. Local Liapunov functions are also provided by the "excess dissipation,'' the second variation in the entropy or in the Helmholtz free energy for the reaction chamber, and quadratic functions introduced in Keizer's fluctuation–dissipation theory. Linearization of the force and flux expansions for nonequilibrium systems yields an idealized model in which the dissipation decreases monotonically in time and thus provides a Liapunov function for evolution to steady states. This result does not hold for a chemical system approaching a steady state with an arbitrarily small, but macroscopic displacement from equilibrium, even though the series expansions of the reactive fluxes and conjugate thermodynamic forces are closely approximated by truncation at the linear terms. There are always small regions in the immediate vicinity of nonequilibrium steady states where the dissipation increases in time while the system relaxes.

Notes: Aperiodic dynamics are observed experimentally in the cool flame combustion region of acetaldehyde (ACH) in a continuous stirred tank reactor (CSTR). A gradual transition is seen, with variation of exit orifice size, from limit cycle oscillation to aperiodic variations in light emission, and then back to near periodic oscillations. We analyze this transition by calculating power spectra, autocorrelation functions, phase portraits, period distributions, and Poincaré sections. The variation in peak amplitude and peak-to-peak period of the temporal variations of light emission increases during the transition. There are many initial indications of a transition to chaos. However, after an in-depth analysis, given in the following article, we ascribe the transition to the presence of a Hopf bifurcation and noise: the path traced out in the constraint space by the change in exit orifice size is nearly tangent to a Hopf bifurcation set but does not cross this set.

Notes: We examine an experimental transition from periodic to aperiodic and back to periodic dynamics in the combustion of acetaldehyde(ACH) in a continuous stirred tank reactor (CSTR) with power spectra, autocorrelation functions, phase portraits, Poincar´e sections, the Wolf–Swift–Swinney–Vastano (WSSV) method for determining the largest Lyapounov exponent, and the Grassberger–Procaccia (GP) method for determining correlation dimension. Each technique gives some indications of a transition to chaos, but there are discrepancies in that the largest Lyapounov exponent is positive but does not converge and the GP method results in a correlation dimension between one and two for two aperiodic data sets. We explore in instructive detail possible explanations for false indications of chaos by comparing our results with calculations on the Rössler chaotic attractor and the van der Pol periodic attractor modified to examine the effects of uneven point distribution and three types of experimental noise. An uneven distribution of points results in a decreased range of length scales for convergence and a larger required embedding dimension for the GP method, but does not explain our experimental results. Observation noise (a Gaussian noise added to each term in the time series but not entering in the equations of motion) and constraint shift (the motion relaxes to an attractor but a constraint changes monotonically during the course of measurement) added to a periodic attractor both result in a low length scale cutoff below which the attractor dimension does not converge with embedding dimension, and above which it converges to 1. Constraint variation noise (a Gaussian noise is added to each term in the time series and enters into the equations of motion as a stochastic perturbation) does yield correlation dimensions between 1 and 2. The experimental transition shows many similarities to a Hopf bifurcation found in another experiment on the same system and to a theoretical Hopf bifurcation with constraint variation noise. A modification of the WSSV Lyapounov exponent analysis for this experimental transition shows the random walk separation of trajectories expected for constraint variation noiseadded to the dynamics of a periodic attractor with a Hopf bifurcation. We therefore identify the experimental transition as an arc in constraint space which does not cross but is nearly tangent to a Hopf bifurcation set.

Notes: The rate of entropy production due to chemical reaction is calculated for a variety of parameter values in the reversible Oregonator model. The average values over cycles of oscillation are compared to those in the coexisting stationary states. The present work corrects an earlier calculation [A. K. Dutt, J. Chem. Phys. 86, 3959 (1987)]. Contrary to previous impressions and claims, there is no consistent relationship between the magnitudes of the entropy production in coexisting stationary and oscillatory states, in this case.