Analytical analysis of the Gavrilov-Guckenheimer bifurcation unfolding in the case of a proportional-integral controlled CSTR

In the field of reactor modeling a single ideal nonisothermal continuous stirred tank reactor is considered as an example of a proportional-integral controlled dynamic system. The aim is to analyze the normal form theory in the case of a nongeneric Gavrilov-Guckenheimer bifurcation and to make a validity check, at least in the related unfolding. The observed nongeneric nature of the considered bifurcation is strictly dependent upon the integral controller mode introduction. Infinitesimal analysis, coupled with the study of implicitly defined functions, is required to successfully handle such a bifurcation. Special attention has also been paid to the study of complex dynamic phenomena that may typically originate in the neighborhood of the present bifurcation: the presence and the asymptotical stability properties of the rising family of invariant tori have been analytically proved. Last, the quoted analysis is generalized to any three-dimensional PI controlled dynamic system, thereby showing the intrinsic nature of the observed normal form degeneracy and the generality of the performed asymptotical analysis.