Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

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<jats:title>Abstract</jats:title><jats:p>A cusp type germ of vector fields is a <jats:italic>C</jats:italic><jats:sup>∞</jats:sup> germ at 0∈ℝ<jats:sup>2</jats:sup>, whose 2-jet is <jats:italic>C</jats:italic><jats:sup>∞</jats:sup> conjugate to</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0143385700004119_eqnU1" /></jats:disp-formula></jats:p><jats:p>We define a submanifold of codimension 5 in the space of germs <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0143385700004119_inline1" /> consisting of germs of cusp type whose 4-jet is <jats:italic>C</jats:italic><jats:sup>0</jats:sup> equivalent to</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0143385700004119_eqnU2" /></jats:disp-formula></jats:p><jats:p>Our main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ<jats:sup>2</jats:sup> × ℝ<jats:sup>3</jats:sup> cutting <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0143385700004119_inline2" /> transversally in (0, 0) is fibre-<jats:italic>C</jats:italic><jats:sup>0</jats:sup> equivalent to</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0143385700004119_eqnU3" /></jats:disp-formula></jats:p>

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