At Brandeis, there is a long tradition of interesting experiments on the Belousov-Zhabostinsky reaction system, with the legendary Zhabotinsky himself having been a part of the fraternity. This reaction system shows interesting oscillatory and stable patterns (see videos on Youtube). In the Fraden lab, an oil emulsion of micron-sized water droplets containing the BZ reactions, was shown to show interesting synchronization properties and complex spatial patterns [Toiya et al, J. Phys. Chem. Lett. 1, 1241 (2010)]. A coupling between the droplets due to preferential diffusion of an inhibitory reactant (bromine) in the oil medium was seen to be responsible for these collective phenomena.
In a new paper titled “Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions” in Phys. Rev. E, Physics Ph. D student Michael Giver, postdoc Zahera Jabeen and Prof. Bulbul Chakraborty show that neighboring oscillators can be modeled as Kuramoto phase oscillators, coupled nonlinearly to its nearest neighbors. The form of the coupling chosen is repulsive, which favors out of phase synchronization. They show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. A linear chain of these repulsively coupled oscillators shows anti-phase synchronization, in which neighboring oscillators show a phase difference of π The phase difference between the neighboring oscillators when placed on a ring however depends on the number of oscillators. In such a case, the locally preferred phase difference of π is ruled out for an odd number of oscillators, as this may lead to frustration. When these oscillators are placed on a triangular lattice in two dimensions, the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2 π /3. Interestingly, domains with different helicities form in the lattice. In each domain, the phases of any three neighboring oscillators can vary continuously in either clockwise or an anti-clockwise direction. Hence, phase difference between the nearest neighbors are seen to be ±2π /3 in the two domains (See figure). A phase difference of π is seen at the interfaces of these domains. These domains can grow in time, resembling domain coarsening in other statistical studies. At large coupling strengths, the domains freeze in size due to frequency synchronization of all the oscillators. Hence, an interplay between frequency synchronization and phase synchronization was seen in this system. Ongoing studies in the BZ experimental setup at the Fraden Lab, find correlations with the above results. Hence, insights into a complex system like the BZ oscillators could be gained using the phase oscillator formalism.
The research was supported by the ACS Petroleum Research Fund and the Brandeis MRSEC. Michael Giver is a trainee in the Brandeis NSF-sponsored IGERT program Time, Space & Structure: Physics and Chemistry of BIological Systems