Volume 14, Number 3, Fall 2006
A UNIFIED APPROACH TO COMPUTING
DYNAMICAL EQUILIBRIA
YUNQIU 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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