We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system’s variables are each car’s velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle–Takens–Newhouse scenario (limit cycles–two-tori–three-tori–chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum.

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic.

Experiments are performed in which a large number (∼104) of relaxation oscillators are globally coupled through the concentration of chemicals in the surrounding solution. Each oscillator consists of a microscopic catalyst-loaded particle that displays oscillations in the concentrations of chemical species when suspended in catalyst-free Belousov−Zhabotinsky (BZ) reaction solution. In the absence of stirring, the uncoupled particles display a range of oscillatory frequencies. In the well-stirred system, oscillations appear in the surrounding solution for greater than a critical number density of particles (ncrit). There is a growth in the amplitude of oscillations with increasing n, accompanied by a slight increase or no change in frequency. A model is proposed to account for the behavior, in which the transfer of activator and inhibitor to and from the bulk medium is considered for each particle. We demonstrate that the appearance and subsequent growth in the amplitude of oscillations may be associated w...

Collective behavior of a population of chemically coupled oscillators.

Mechanistic classification of chemical oscillators and the role of species

Detailed experimental and numerical results are presented about the pattern formation mechanism of spatially organized partially synchronized states in a networked chemical system with oscillatory metal dissolution. Numerical simulations of the reaction system are used to identify experimental conditions (heterogeneity, network topology, and coupling time-scale) under which the chemical reactions, which take place in a network, are split into coexisting coherent and incoherent domains through the chimera mechanism. Experiments are carried out with a network of twenty electrodes arranged in a ring with seven nearest neighbor couplings in both directions along the ring. The patterns are characterized by analyzing the oscillation frequencies and entrainments to the mean field of the phases of oscillations. The chimera state forms from two domains of elements: the chimera core in which the elements have identical frequencies and are entrained to their corresponding mean field and the chimera shell where the elements exhibit desynchrony with each other and the mean field. The experiments point out the importance of low level of heterogeneities (e.g., surface conditions) and optimal level of coupling strength and time-scale as necessary components for the realization of the chimera state. For systems with large heterogeneities, a ‘remnant’ chimera state is identified where the pattern is strongly affected by the presence of frequency clusters. The exploration of dynamical features with networked reactions could open up ways for identification of novel types of patterns that cannot be observed with reaction diffusion systems (with localized interactions) or with reactions under global constraints, coupling, or feedback.

Spatially organized partial synchronization through the chimera mechanism in a network of electrochemical reactions

By using MO calculations based on DFT, absorption, and fluorescence spectroscopy, we have comprehensively studied the low-lying excited singlet states of α,ω-diphenylpolyynes (DPY) having 1−6 triple bonds. The ag vibrational modes of the C⋮C stretching and of the phenyl ring motion were observed in the fluorescence spectra of diphenylacetylene and 1,4-diphenylbutadiyne. On the other hand, in the fluorescence spectra of the long DPY with the triple-bond number (N) more than two, the phenyl ring motion with ag symmetry disappeared and the b1g modes of the phenyl ring twisting (∼400 cm-1) and of the C−H bending (∼900 cm-1) were detected. The observed fluorescent states of DPY with N ≤ 2 and N ≥ 3 are assigned to the 11B1u (πxπx*) and 11Au (πxπy* and/or πyπx*) states, respectively, based on the vibronic structures, the relatively short lifetimes, and the solvatochromic shifts of the fluorescence spectra. Not only the allowed transition of 11B1u ← S0 but also the forbidden transition of 11Au ← S0 was detected ...

Symmetry Switching of the Fluorescent Excited State in α,ω-Diphenylpolyynes

We present a system of two identical chemical oscillators which are coupled by means of their flow rates. The individual flow rate of one oscillator is controlled by the output of the other oscillator and vice versa. A delay time is applied between the output of one oscillator and the input to the other one. This delay has been investigated both theoretically and experimentally for the minimal bromate oscillator. By variation of the delay time both oscillators dramatically change their periods and amplitudes. At lower coupling strengths, the periods of both oscillators increase and decrease in a sawtooth fashion with increasing delay times. At high coupling strengths multimode patterns were obtained at high delay times. No quasiperiodic or aperiodic behavior was observed in this coupled system. The experimental results are in general agreement with those calculated with the Noyes-Field-Thompson mechanism. 38 refs., 8 figs.

F. W. Schneider

Papers

Mutually coupled oscillators with time delay

Delayed-output-controlled chemical oscillations