Geodesic
On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 3, pp. 37-53.

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We suggest a method for studying asymptotically linear vector fields with a parameter. The method permits one to prove theorems on asymptotic bifurcation points (bifurcation points at infinity) for the case of double degeneration of the principal linear part. We single out a class of fields that have more than two unbounded branches of singular points in a neighborhood of a bifurcation point. Some applications of the general theorems to bifurcations of periodic solutions and subharmonics as well as to the two-point boundary value problem are given.
Keywords: asymptotic bifurcation point, asymptotically homogeneous operator, periodic oscillations, subharmonic.
Mots-clés : solution branch 
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A. M. Krasnosel'skii; D. I. Rachinskii. On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 3, pp. 37-53. http://geodesic.mathdoc.fr/item/FAA_2005_39_3_a3/

[1] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhizdat, M., 1956 | MR

[2] Schmitt K., Wang Z. Q., “On bifurcation from infinity for potential operators”, Differential Integral Equations, 4:5 (1991), 933–943 | DOI | MR | Zbl

[3] Krasnosel'skii A. M., “Asymptotic homogeneity of hysteresis operators”, Z. Angew. Math. Mech., 76,Suppl. 2 (1996), 313–316

[4] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR

[5] Lazer A. C., Leach D. E., “Bounded perturbations of forced harmonic oscillators at resonance”, Ann. Mat. Pura Appl. (4), 82 (1969), 46–68 | DOI | MR

[6] Krasnosel'skii A. M., Mawhin J., “Periodic solutions of equations with oscillating nonlinearities. Nonlinear operator theory”, Math. Comput. Modelling, 32 (2000), 1445–1455 | DOI | MR

[7] Bliman P.-A., Vladimirov A. A., Krasnoselskii A. M., Sorin M., “Vynuzhdennye kolebaniya v sistemakh upravleniya s gisterezisom”, Dokl. RAN, 347:4 (1996), 458–461 | MR | Zbl

[8] Krasnosel'skii A. M., Kuznetsov N. A., Rachinskii D. I., “On resonant differential equations with unbounded non-linearities”, Z. Anal. Anwendungen, 21:3 (2002), 639–668 | DOI | MR

[9] Diamond P., Kloeden P. E., Krasnosel'skii A. M., Pokrovskii A. V., “Bifurcations at infinity for equations in spaces of vector-valued functions”, J. Austral. Math. Soc. Ser. A, 63:2 (1997), 263–280 | DOI | MR | Zbl

[10] Krasnosel'skii A. M., Mawhin J., “The index at infinity of some twice degenerate compact vector fields”, Discrete Contin. Dynam. Systems, 1:2 (1995), 207–216 | DOI | MR

[11] Arnold V. I., Zadachi Arnolda, Fazis, M., 2000 | MR