Analysis of the bifurcation phenomena for thecontinuous stirred tank tractor with consecutive reactions

Abstract

A method is suggested about how to apply the singularity theory not to steady state equation but to the static bifurcation for finding the maximum number of possible steady states of lumped parameter system. Since the static bifurcation occurs when the Jacobian determinant vanishes at the steady states, the singularity theory is applied to the Jacobian determinant. This scheme is applied to the system of the stirred tank reactor with two first order consecutive reactions when the two reaction rate constants are equal. From the results of this analysis, the cusp singularity is obtained for the equation which represents the static bifurcation (especially, the turning point bifurcation). In the neighborhood of this singularity, six different shapes of stable bifurcation diagrams and three unstable transition bifurcation diagrams are obtained. These results are verified by the eight possible different types of bifurcation diagrams for steady states. This new scheme has advantages that the order of singularity is reduced by one and the stability of steady states can be noticed at the first glance through the diagrams. When the two reaction rate constants are not equal, the existence of fifth order singularity is confirmed. One singular point of the fifth order is found. Unfortunately, this point is physically beyond the feasible region but the most part of its neighborhood belongs to the feasible region since it is close to the boundary of feasible region. From the analysis for these proper regions, it is found that the parameter space is divided into the regions with different bifurcation diagrams and has the natures of butterfly catastrophe. For the Hopf bifurcation, a new scheme is proposed that the singularity theory can be applied to the Hopf bifurcation theory for finding the multiple Hopf bifurcation points. When the two reaction rates are equal, the two third order swallow-tail catastrophes are obtained. As the combined results from two new schemes (for the ...