The irreversible Brusselator: the influence of degenerate singularities in a closed system

The appearance of apparently chaotic behaviour in numerical treatment of this two-dimensional system is examined from an analytical point of view. The original two-parameter model exhibiting chaotic-like solutions is unfolded to a three-parameter model. This enlarged model is shown to have a condimension-two degenerate Hopf bifurcation, the unfolding of which contains phase portraits with three concentric limit cycles. Some segments of these limit cycles are so close to each other that numerical integration causes transitions across the unstable limit cycle, thus giving the appearance of chaotic behaviour. The region in parameter space where this occurs is quite significant and it includes part of the plane of the original two-parameter model.