Engineering of synchronization and clustering of a population of chaotic chemical oscillators.

Description, control, and design of weakly interacting dynamical units are challenging tasks that play a central role in many physical, chemical, and biological systems. Control of temporal and spatial variations of reaction rates is especially daunting when the dynamical units exhibit deterministic chaotic oscillations that are sensitive to initial conditions and are long-term unpredictable. Control of spatiotemporal chaos by means of suppression of spiralwave turbulence to standing waves, cluster patterns, and uniform periodic oscillations, in the catalytic CO oxidation on a Pt (110) single-crystal surface has been successfully achieved with linear delayed feedback of the carbon monoxide partial pressure. The same delayed global feedback methodology has also been applied to induce various dynamical clusters in electrochemical corrosion processes and in the light-sensitive Belousov–Zhabotinsky (BZ) reaction. A fundamental problem of tuning the complex structure of chaotic systems is choosing a feedback scheme and appropriate control parameters to steer the system to a desired structure of spatiotemporal reactivity. This objective requires the use of simple yet accurate models of nonlinear processes, their interactions, and responses to external perturbations. When the individual units exhibit periodic oscillatory behavior, the effect of weak feedback and coupling can be described by phase models in which the state of each unit is represented by a single variable, the phase of oscillations. Typical phase-synchronized behavior of a population of periodic oscillators is dynamical differentiation (clustering) where the elements form groups (clusters) in which the phases are identical at all times but different from the phases of the elements in the other groups. Phase models successfully described synchronization and dynamical differentiation (clustering) of oscillatory electrochemical and BZ bead experiments. Experiment-based phase models can also be used for synchronization engineering of periodic oscillators where a carefully designed external feedback is applied to dial-up complex dynamical structures such as stable and sequentially visited dynamical cluster patterns and desynchronization. Herein, we consider populations of chaotic oscillators that exhibit strong cycle-to-cycle period variations. We show that the feedback scheme can be designed to obtain dynamical synchronization states of phase coherent chaotic oscillators in a quantitative manner. The method relies on the flexibility and versatility of synchronization engineering (originally developed for periodic oscillators) and on the observation that the long-term phase dynamics of chaotic oscillators is often similar to that of noisy periodic oscillators. An experimental system of uncoupled, phase-coherent, chaotic oscillators was constructed by using an electrochemical cell consisting of 64 Ni working electrodes (99.98% pure), a Pt mesh counter electrode, and a Hg/Hg2SO4/K2SO4 (saturated) reference electrode, with a 4.5m H2SO4 electrolyte (Figure 1a) at (11! 0.5) 8C. A potentiostat (EG&G Princeton Applied Research) was used to set the circuit potential (Vo) of the cell such that the electrodes undergo transpassive dissolution. With a 906W resistor (Rp) attached to each electrode chaotic current oscillations were observed at Vo= 1.131 V (Figure 1b). For each cycle of the chaotic current oscillations (e.g., in Figure 1b) the peak-to-peak periods were determined with a standard peak-finding algorithm. The observed period distribution of these elements is illustrated in Figure 1c while the structure of the dynamical attractor can be seen in Figure 1d. The period distribution of the low-dimensional chaotic attractor is relatively broad and exhibits a multipeak structure, which is characteristic of chaotic behavior obtained from a period-doubling bifurcation route to chaos. We use order parameters R1 and R2 to describe the extent of relative organization of oneand two-cluster states [Eq. (1)],