Versal matrix families, normal forms and higher order bifurcations in dynamic chemical systems

Abstract

Characterization of the possible dynamic phenomena which arise when a steady state loses stability due to parametric variations is important in process and control system design and is useful for dynamic model identification and evaluation. The linearized system Jacobian Jordan block structure at bifurcation implies corresponding possibilities for nonlinear dynamic phenomena near bifurcation. The linearized system characteristic equation and the theory of versal representation of matrix families are applied to identify the number of system parameters which must be varied simultaneously to achieve different eigenvalue configurations. The theory of normal forms is used to illustrate the topological equivalence, near bifurcation, of the original system and the normal form representation which contains a relatively small number of nonlinear differential equations. This theory is used to organize and interpret studies of dynamics in two chemical reaction systems: (i) consecutive-competitive reactions in an isothermal CSTR with multivariable proportional feedback control; and (ii) coupled oscillations in two interacting CSTR's with autocatalytic reactions.