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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: We analyze a thermal engine in which the source of heat is the combustion reaction CH4+2O2→CO2+2H2O. The efficiency of the thermal engine is defined as the ratio of the power output of the engine to the power input. We compare the efficiency in two modes of operation, one in which the input and output of chemical species are constants and one in which their corresponding fluxes have an imposed oscillatory component of a given amplitude and frequency. The ratio of efficiency in the oscillatory mode to the stationary flux mode can be as high as 1.3 for an amplitude of variation of the input and output fluxes plus or minus 40% of the stationary value. At low amplitudes of variation (10%), a single peak is present in a plot of the ratio of efficiency vs frequency of the fluxes. For higher amplitudes there are two peaks in that plot. One peak is associated with a response of the system that has the same period as the fluxes and the other occurs in a frequency range for which the periodic response is twice that of the perturbation. We calculate plots of the oscillatory output flux and the power output divided by that flux, which is an analog of a thermodynamic force, as a function of time, to show the importance of both resonance effects and the phase relation between the flux and the thermodynamic force in determining the ratio of efficiency.

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: We investigate in detail the dynamics of a time-periodically forced chemical oscillator in the parameter plane of forcing amplitude and forcing period. In particular, we present computed bifurcation sets for two typical cases of a forced, autonomously oscillating continuous stirred tank reactor system. The total mass flow rate j is used as the forcing variable by varying it sinusoidally in time about the autonomous system's value. We find a wide variety of new nonlinear phenomena, including a global bifurcation structure—the skeletal bifurcation structure—that is common to the two cases presented and to other forced oscillator systems. The skeletal bifurcation structure is periodic along the forcing period axis and is mainly composed of the boundaries of Arnol'd tongues, which terminate at finite forcing amplitudes. In one of the cases studied, the invariant torus is destroyed between two critical curves and cascades of period doubling occur within the Arnol'd tongues; we relate this destruction of the torus to the discontinuous bifurcation that destroys the limit cycle in the autonomous system as j is varied.

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 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: A major limiting factor for percutaneous coronary interventions carried out via the femoral route is the time it takes to achieve femoral artery hemostasis and subsequent mobilization. Discharge from hospital usually occurs the following day. In this pilot study, we assessed the feasibility of mobilization at 4 hours and same-day discharge from hospital of selected elective patients undergoing intracoronary stenting using the Angio-Seal Vascular Closure device. Seventy-five patients (56 ± 10 years) with stable single-vessel coronary disease scheduled for elective coronary stenting were enrolled. All patients were mobilized at 4 hours and assessed at 10 hours postprocedure as to their suitability for hospital discharge. The first 50 patients remained in hospital overnight. The next 25 patients followed the same procedures but were discharged at 10 hours. The subjects were followed up at 48 hours and 30 days. Hemostasis was achieved in all patients following sheath removal and deployment of the Angio-Seal device. Twenty patients (27%) had minor groin oozing and two developed small hematoma. There were no major bleeding complications, pseudoaneurysm, vascular surgery, or groin infection. Groin oozing resulted in the delay of ambulation for 13 subjects but discharge was not delayed in any patient. All patients were reported to be suitable for hospital discharge at 10 hours postprocedure. There were no further complications at 30 days. The present study demonstrated that early mobilization and same-day discharge after coronary stenting using the Angio-Seal device is feasible in selected patients. Further studies are needed to determine the patient selection criteria and the potential cost-saving implications of this strategy.