Abstract
To motivate the problems to be discussed in this section, consider M: Λ × X → Z, M(0, 0) = 0, and suppose D x M(0, 0) is Fredholm of index zero. The method of Liapunov–Schmidt reduces the study of local bifurcation to the study of a system of nonlinear equations f(λ, u) = 0 for u ∈ ℝk, f ∈ ℝk, where k = dim N(D x M(0,0)). If k = 1 and ∂2 f(0,0)/∂u2 ≠ 0, we have seen in Section 6.2 that the local bifurcations are determined by a scalar function of λ. If ∂2f(0,0)/∂u2 = 0,∂3 f(0,0)/∂u3 ≠ 0, we have also seen in Section 6.4 that the local bifurcations are determined by two scalar functions of λ. In Section 6.8, we have also discussed the situation for arbitrary k—the basic result being that the local bifurcations are determined by a polynomial of degree k whose coefficients are functions of λ.
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© 1982 Springer-Verlag New York Inc.
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Chow, SN., Hale, J.K. (1982). Bifurcation with Higher Dimensional Null Spaces. In: Methods of Bifurcation Theory. Grundlehren der mathematischen Wissenschaften, vol 251. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8159-4_7
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DOI: https://doi.org/10.1007/978-1-4613-8159-4_7
Publisher Name: Springer, New York, NY
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