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Volume 14,     Number 3,     Fall 2006

 

A UNIFIED APPROACH TO COMPUTING
DYNAMICAL EQUILIBRIA
YUN-QIU SHEN AND TJALLING J. YPMA

Abstract. Different types of equilibria of dynamical systems, such as turning points and bifurcation points, are typically computed in different ways. We present a unified approach to computing any such point and show how to adapt the general approach to specific cases. The concept of a dynamical characterization matrix, which has the relevant type of singularity at the equilibrium point sought, is introduced. This matrix is embedded within a larger nonsingular matrix by using a bordering computed from one singular value decomposition at a point near the solution. The bordered matrix is used to define an extended system of nonlinear equations, for which Newton's method converges quadratically to the desired point. We give a general algorithm and provide numerical examples illustrating the technique.

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© 2006, Canadian Applied Mathematics Quarterly (CAMQ)