The purpose of this paper is to give an account of recent developments in numerical methods for the solution of bifurcation problems. For readers not too familiar with our subject we shall first summarize important applications of bifurcation and dicuss some of the basic ideas, problems and tools of bifurcation theory.

Numerical Methods for Bifurcation Problems — A Survey and Classification

Evaluation of Hopf bifurcation points in parabolic equations describing heat and mass transfer in chemical reactors

A straightforward iteration technique is suggested for evaluation of Hopf’s (complex) bifurcation points in ordinary differential equations. Newton method is used to solve resulting system of nonlinear equations. Evaluation of eigenvalues of the Jacobian matrix of the right-hand sides is not requested. Numerical results for a problem taken from chemical engineering are presented.

Algorithm for Evaluation of Complex Bifurcation Points in Ordinary Differential Equations

The present paper deals with the computation of simple bifurcation points of nonlinear parameter dependent equations. At first, a minimally extended system of nonlinear equations is constructed by addition of one parameter and two equations. This augmented system has an isolated solution which yields to the simple bifurcation point directly. Using the structural properties of this auxiliary system an adapted Newton-like method is developed not requiring evaluations of second derivatives. Finally, the results of some computer experiments show the efficiency of theR-quadratically convergent method.

Milo Marek

Papers

Evaluation of limit and bifurcation points for algebraic equations and nonlinear boundary-value problems