The cubic autocatalator: The influence of degenerate singularities in a closed system

The appearance of apparently chaotic behaviour in this two-dimensional system is examined from an analytical point of view. The original two-parameter model exhibiting numerical solutions resembling chaos is unfolded to a three-parameter model. This enlarged model is shown to have a codimension-two degenerate Hopf bifurcation the unfolding of which contains phase portraits showing three concentric limit cycles. In some regions these limit cycles are so close to each other that numerical integration causes transitions across the unstable limit cycle, giving the appearance of chaotic behaviour. The region in parameter space of the ‘chaotic’ behaviour agrees well with the degenerate behaviour of the enlarged model.