Geodesic
Interaction of the folded Whitney umbrella and swallowtail in slow–fast systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 29-33 Cet article a éte moissonné depuis la source Math-Net.Ru

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The graph of a first integral of a smooth slow-fast system with two slow variables is a singular surface in the three-dimensional space; the variation of an external parameter on which the system depends gives rise to perestroikas ($=$transitions) of this surface. We find a normal form and present figures of the perestroika that describes the interaction between the swallowtail and folded Whitney umbrella on the graph of a first integral of a generic one-parameter family of such systems. 
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     author = {I. A. Bogaevsky},
     title = {Interaction of the folded {Whitney} umbrella and swallowtail in slow{\textendash}fast systems},
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I. A. Bogaevsky. Interaction of the folded Whitney umbrella and swallowtail in slow–fast systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 29-33. http://geodesic.mathdoc.fr/item/TM_2012_278_a2/

[1] Arnold V.I., “Kontaktnaya struktura, relaksatsionnye kolebaniya i osobye tochki neyavnykh differentsialnykh uravnenii”, Izbrannoe–60, Fazis, M., 1997, 391–396 | MR

[2] Arnold V.I., Teoriya katastrof, 3-e izd., Nauka, M., 1990 | MR

[3] Arnold V.I., Varchenko A.N., Gusein-Zade S.M., Osobennosti differentsiruemykh otobrazhenii. Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982 | MR

[4] Davydov A.A., “Normalnaya forma medlennykh dvizhenii uravneniya relaksatsionnogo tipa i rassloeniya binomialnykh poverkhnostei”, Mat. sb., 132:1 (1987), 131–139 | MR | Zbl

[5] Davydov A.A., “Normalnaya forma differentsialnogo uravneniya, ne razreshennogo otnositelno proizvodnoi, v okrestnosti ego osoboi tochki”, Funkts. analiz i ego pril., 19:2 (1985), 1–10 | MR | Zbl

[6] Zakalyukin V.M., Remizov A.O., “Lezhandrovy osobennosti v sistemakh neyavnykh obyknovennykh differentsialnykh uravnenii i bystro-medlennykh dinamicheskikh sistemakh”, Tr. MIAN, 261, 2008, 140–153 | MR | Zbl

[7] Bruce J.W., “A note on first order differential equations of degree greater than one and wavefront evolution”, Bull. London Math. Soc., 16:2 (1984), 139–144 | DOI | MR | Zbl

[8] Davydov A.A., “Whitney umbrella and slow-motion bifurcations of relaxation-type equations”, J. Math. Sci., 126:4 (2005), 1251–1258 | DOI | MR | Zbl

[9] Hayakawa A., Ishikawa G., Izumiya S., Yamaguchi K., “Classification of generic integral diagrams and first order ordinary differential equations”, Int. J. Math., 5:4 (1994), 447–489 | DOI | MR | Zbl