cxperimental and simulated normalized current. The simulated normalized current i / i 0 was generated with k, = 1 . O cm s-l, a = 0.5. and D = 2.4 X IO-' cm2 s-I.IoS l~l f i u r d Spcctrioir of N02BF, in Solution. In a Schlenk flask, a 0. I M solution of NO?+BFJ-in acetonitrile was made up under an argon atniosphcrc. The Schlenk flask was connected to a flowthrough AgCl ccll (0.1-mm Teflon spacer) using a 20-gauge Tcflon tubing. Another Schlenk flask was connected to the outlet (109) See also discussion by Bard and Faulkner in ref 33. The Fortran program is availablc on rcqucst. of the cell by using a 20-gauge Teflon tubing. The system was thoroughly flushed with argon. With the cell placed directly in the beam of infrared spectrometer, the sample flask was pressurized with argon, and the sample was slowly injected (20 m L min-I) into the IR cell. At a constant flow of the sample solution, the 1R spectrum was measured every 5 s o n a Nicolet IO DX FT spectrometer with 2 cm-l resolution. Acknowledgment.A general approach is presented briefly to the mechanistic characterization of chemical oscillatory reactions, based on a ncw operational classification of simple chemical oscillators, i.e., oscillators that contain only one source of instability (autocatalysis), and categorization of their species. We use stoichiometric network analysis to classify oscillatory reactions according to their basic unstable feature and the type of their dominant negative feedback loop. The species are first categorized into those essential and nonessential for the Occurrence of oscillations. The essential species are further divided into subcategories according to thcir rolcs in the mcchanism. The suggestcd proccdurc includes operational criteria for the assignment of a chemical oscillator to onc of the defined categories of mechanisms and for the identification of the roles of the species. Altogether 25 abstract modcls and realistic mechanisms of simple oscillators have been investigated; all fit into the four defined categories of incchanisms. Thc classification and proccdurcs prescntcd briefly here arc fully devcloped in an article to appear in Ado. C'hefH. PhYJ.
With delay feedback experiments on the minimal bromate oscillator, we show that chemical systems with delay display a variety of dynamical behavior. Using a nonlinear delayed feedback, we induce Hopf bifurcations, period doubling, bifurcations into chaos, and crisis (observed for the first time in a chemical system) into the system, which does not display this behavior without the delay. We test a conjecture [M. Le Berre, E. Ressayre, A. Tallet, H. M. Gibbs, D. L. Kaplan, and M. H. Rose, Phys. Rev. A 35, 4020 (1987)] that the dimension of a chaotic attractor is equal to τ/δf, where δf is the correlation time of the delayed feedback. Using the Grassberger–Procaccia algorithm [P. Grassberger and I. Procaccia, Phys. Status Solidi D 9, 189 (1983)] to calculate the dimensions of the chaotic attractors from the experimental system, we show that the calculated dimensions are less than those calculated by τ/δf. We compare numerical integrations of the proposed mechanism for the minimal bromate oscillator with the experimental results and find agreement of the predicted bifurcation sequence with the experimental observations. The results of this study indicate that with appropriate delay feedback functions, and a sufficiently nonlinear dynamical system, it is possible to ‘‘push’’ a dynamical system into further bifurcation regimes, of interest in themselves, which also yield information on the system without delay.
We examine the effects of interactive noise, i.e., noise which is processed by the system, on the Brusselator, a nonlinear oscillator. The Brusselator is investigated for three types of motion: periodic, quasiperiodic, and chaotic. Fluctuations are imposed on the system variables (Type V noise). The average fluctuation amplitudes are chosen between 10 and 10 000 ppm (1%) and they are Gaussian distributed. The simulated time series are analyzed by autocorrelation functions, Fourier spectra, Poincaré sections, one-dimensional maps, maximum Lyapunov exponents, and correlation dimensions. As a result, noisy periodic and quasiperiodic motion can be distinguished from deterministic chaos if the fluctuation amplitude is sufficiently small. The generic structure of the attractor can be recognized when Lyapunov exponents or correlation dimensions are extrapolated to zero fluctuation amplitude. Quasiperiodic attractors in the Brusselator are obscured even by small amounts of noise. Chaos in the Brusselator, on the other hand, is found to be robust against noise. For periodic motion we show that points close to a bifurcation exhibit a stronger sensitivity towards noise than points far away. In the log–log plots for the correlation dimension we observed break points for noisy periodic and quasiperiodic motion. They separate the noise from the purely deterministic part of the motion. For increasing noise levels the break points move to higher length scales of the attractor. Break points were not found for chaos in the Brusselator nor in the Lorenz and Rössler models. In the Brusselator very large noise levels beyond 1% obscure the deterministic structure even of a chaotic attractor so that any clear distinction between chaos and noise induced (statistical) aperiodicity is no longer possible. Implications on experimental systems are discussed.
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