Belousov Zhabotinsky reaction systems: How “far” is far from equilibrium?

Understanding the behaviour of dynamical systems that are far from equilibrium has been a challenge for science and engineering. Belousov Zhabotinsky (BZ) reaction is a system that exhibits chemical oscillations due to periodic oxidation/reduction of the metal ion catalyst. In essence, the oxidized catalyst concentration varies with time crossing its steady state (SS) concentration with each oscillation. Mathematically, the Limit Cycles (LC), which represent self-sustained oscillations occur when the equilibrium (or SS) loses its stability via Hopf bifurcation. The LC resulting from HB, typically, surrounds the equilibrium, which is analogous to variation of oxidized catalyst concentration around SS. Using the Oregonator model, however, we demonstrate that the equilibrium point can be outside the LC under special conditions. Further, we use the nonlinear stability analyses and calculate Lyapunov coefficients to quantify how “far” the LCs are from the corresponding “equilibrium”. In addition, we predict the amplitude and frequency of oscillations, which are in good agreement with our simulation results. These findings can not only be used to characterise behaviours of nonlinear dynamical systems but can also be used to design smart functional materials.