| In this thesis we use singularity theory, as developed by Golubitsky and co-workers (1979a, 1979b, 1981, 1985) to investigate multiple steady states and degenerate Hopf bifurcations which occur in the continuous stirred tank reactor with consecutive reactions A (--->) B (--->) C.; Earlier investigators had found up to five simultaneous steady states when they assumed equal activation energies for the two reactions and used the positive exponential approximation. We find that without these assumptions the maximum multiplicity increases to seven, but this happens only in a very small region in parameter space and only if the first reaction is exothermic and the second endothermic. Otherwise the maximum multiplicity remains five. We map out in parameter space the regions where butterfly singularities occur in the physical region. We also find that isolas of steady states and pitchfork bifurcations can occur in this model without using residence time formulation.; For the dynamics investigation we assume equal activation energies, use the positive exponential approximation, and concentrate on the case of two exothermic reactions. Building on the theory of Golubitsky and Langford (1981), we extend their Hopf bifurcation formulae to seventh order (in amplitude) and implement them numerically to study interactions of the four possible Hopf points as well as degeneracies leading to multiple periodic solutions. In the region of unique steady states we show how our results can be used to draw qualitative bifurcation diagrams, including the periodic branches, without having to calculate them explicitly. We uncover a complex structure relating interactions of periodic behavior with multiple steady states, involving degenerate double zero eigenvalue bifurcations of the Takens-Bogdanov type. These results indicate that in the multiple steady state region the predominant modes for termination of periodic solution branches are infinite period saddle loop or saddle-node loop bifurcations: this means that this system behaves very similarly to the single reaction model. |
