Reaction-Diffusion Dissipative Systems—Detailed Stability Analysis-Pattern of Growth and Effect of Inhomogenity

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Abstract
A detailed stability analysis of the one dimensional steady state solutions for the Brusselator model under the conditions of diffusion of initial (non-autocatalytic) components has been performed both for zero flux as well as fixed boundary conditions. In addition to subcritical as well as supercritical bifurcations, situations have been observed where all solution branches at a bifurcation point are unstable. A case of degenerate steady state bifurcation (2 solutions emanating from the same bifurcation point) has also been noticed. A transient simulation of the system in growth reveals the importance of growth rate on the pattern selection process and suggests that the selection of branches at a bifurcation point may be influenced by perturbations/ fluctuations. It also indicates that a stability analysis of the bifurcation diagram alone cannot decide the state of the system in a transient process, and under certain situations complex behavior may be observed at limit points. Numerical calculations on coupled cells indicate that a heterogenity in the system can introduce multiple (two) time scales in the system. As the ratio of time scales increases, aperiodic or irregular oscillations are observed for the 'fast' variable. A combination of cells with one cell in a steady-state mode and the other in a periodic motion results in a combined motion of the entire system. For a distributed parameter system, a heterogenity can cause development of sharp local concentration gradients, alter the stability properties of steady state as well as periodic solutions and can cause partitioning of the system.

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