Abstract
A point (x*,λ*) is called apitchfork bifurcation point of multiplicityp≥1 of the nonlinear systemF(x, λ)=0,F:ℝn×ℝ1→ℝn, if rank∂ xF(x*, λ*)=n−1, and if the Ljapunov-Schmidt reduced equation has the normal formg(ξ, μ)=±ξ 2+ p±μξ=0. It is shown that such points satisfy a minimally extended systemG(y)=0,G:ℝn+2→ℝn+2 the dimensionn+2 of which is independent ofp. For solving this system, a two-stage Newton-type method is proposed. Some numerical tests show the influence of the starting point and of the bordering vectors used in the definition of the extended system on the behavior of the iteration.
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Pönisch, G., Schnabel, U. & Schwetlick, H. Computing multiple pitchfork bifurcation points. Computing 59, 209–222 (1997). https://doi.org/10.1007/BF02684441
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DOI: https://doi.org/10.1007/BF02684441