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The generalized homoclinic bifurcation. (English) Zbl 0801.34044
The author considers a family \(X_ \lambda\) of vector fields that has at \(\lambda= 0\) a homoclinic loop of multiplicity \(n\). The aim of the paper is to present conditions of the versality of \(X\) in a neighborhood of the loop.
For this, the author uses the representation of the displacement function given by R. Roussarie [Bol. Soc. Bras. Mat. 17, 67-101 (1986; Zbl 0628.34032)]. The needed result is a consequence of a theorem stating the existence of a homeomorphism \(\alpha(\lambda)= (\alpha_ 0(\lambda),\alpha_ 1(\lambda),\dots,\alpha_{n-1}(\lambda))\) such that for \(\lambda\) and \(\varepsilon\) small enough zeros of the displacement function coincide in \([0,\varepsilon]\) with these of the polynomial \(\alpha_ 0+ \alpha_ 1 x+\cdots+ \alpha_{n-1} x^{n-1}+ x^ n\).
For the proof, the author uses the notion of the Chebyshev system of functions that are \(C^ 0\) at the end points of the interval of their existence and are \(C^ n\) inside it.

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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