Oscillatory reactive dynamics on surfaces: A lattice limit cycle model

Abstract

Complex reactive dynamics on low-dimensional lattices is studied using mean-field models and Monte Carlo simulations. A lattice-compatible reactive scheme that gives rise to limit cycle behavior is constructed, involving a quadrimolecular reaction step and bimolecular adsorption and desorption steps. The resulting lattice limit cycle model is dissipative and, in the mean-field limit, exhibits sustained oscillations of the species concentrations for a wide range of parameter values. Lattice Monte Carlo simulations of the lattice limit cycle model show locally the emergence of sustained oscillations of the species concentrations. Random fluctuations of the concentrations, clustering between homologous species, and competition between the various clusters/species cause the in-phase oscillations of neighboring sites. Distant regions oscillate out of phase and spatial correlations decay exponentially with the distance. The amplitude and period of the local oscillations depend on the system parameters.

Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities

“…In the parameter space (p 1 , p 2 ) for fixed p 3 , Q 4 is either a stable node or a stable focus which becomes unstable through a supercritical Hopf bifurcation [48]. Because of the physical condition x, y, s ≥ 0 the flow is always directed to the inside of the reaction simplex x > 0, y > 0, x + y < 1 which follows from mass-action kinetics.…”

“…Initially, it was implemented on a square lattice with single occupancy per lattice site using Kinetic Monte Carlo (KMC) simulations. Direct KMC realizations on the 2D square lattice with nearest neighbor interactions produced intricate fractal patterns and local oscillations of the species concentrations [48]. Later on, longdistance diffusion was introduced as a mixing mechanism allowing the species to react with all particles within a specific range, thus giving them the possibility to change their places in the lattice at finite or infinite distances [59].…”

“…(2)) has four fixed points: one saddle-point Q 1 = (0, 0), two other fixed points Q 2 = (0, 1) with an unstable eigenvector direction along the y-axis and Q 3 = (1, 0) with a stable eigenvector direction along the x-axis (each having one zero eigenvalue [48]), and one nontrivial fixed point whose coordinates depend on the system parameters:…”

“…Originally, the LLC scheme [48] was devised as a highly nonlinear model for a cyclic reaction-diffusion process with predator-prey interactions among three particles X, Y , and S. The reaction scheme reads:…”

“…A toy model which exhibits well defined oscillations in the form of a limit cycle and is relevant in ecological, chemical, and social dynamics is the Lattice Limit Cycle (LLC) model, proposed in [48]. The LLC model is a nonlinear system that describes reaction, birth, and death processes and studies how the concentrations of the specific species change with time under the influence of the control parameters.…”