Numerical Hopf bifurcation analysis in nonlinear ordinary and partial differential systems from chemical reactor theory

Abstract

A code has been developed which will automatically locate and analyze points of Hopf bifurcation in autonomous ordinary differential systems. The code first locates critical value(s) v c of a user-specified parameter v (the bifurcation parameter) such that a stationary (equilibrium) solution x *( v) loses linear stability by virtue of a complex conjugate pair of eigenvalues. The code computes x *( v) during the location of v c . Then the code computes the various coefficients in a local approximation to the family of periodic solutions which arise, a process which involves computation of second and third partial derivatives by numerical differencing of the user-supplied Jacobian matrix. The current version of the code, called BIFOR2, is fully described in Hassard, Kazarinoff, and Wan, Theory and Applications of Hopf Bifurcation, Cambridge U.P., 1981. In this paper we demonstrate the code in applications to two systems drawn from chemical reactor theory. The first application is to a 4th order ordinary differential system modeling a coupled tank reactor. The second application is to a partial differential system modeling a catalyst particle system. These represent the first applications of the code to chemically reacting systems other than the Brusselator. The second application demonstrates how collocation methods may be used in conjunction with BIFOR2 to perform Hopf bifurcation analysis of partial differential systems.