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Rank deficiencies and bifurcation into affine subspaces for separable parameterized equations. (English) Zbl 1327.65086

Summary: Many applications lead to separable parameterized equations of the form \( F(y,\mu ,z) \equiv A(y, \mu )z+b(y, \mu )=0\), where \( y \in \mathbb{R}^n\), \( z \in \mathbb{R}^N\), \( A(y, \mu ) \in \mathbb{R}^{(N+n) \times N}\), \( b(y, \mu ) \in \mathbb{R}^{N+n}\) and \( \mu \in \mathbb{R}\) is a parameter. Typically \( N >>n\). Suppose bifurcation occurs at a solution point \( (y^*,\mu ^*,z^*)\) of this equation. If \( A(y^*, \mu ^*)\) is rank deficient, then the linear component \( z\) bifurcates into an affine subspace at this point. We show how to compute such a point \( (y,\mu ,z)\) by reducing the original system to a smaller separable system, while preserving the bifurcation, the rank deficiencies and a non-degeneracy condition. A numerical algorithm for solving the reduced system and examples illustrating the method are provided.

MSC:

65F30 Other matrix algorithms (MSC2010)
65H10 Numerical computation of solutions to systems of equations
37G10 Bifurcations of singular points in dynamical systems

Software:

AUTO; MATCONT
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Full Text: DOI

References:

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