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Bifurcation of subharmonic solutions. A generic approach. (English) Zbl 0557.34040

Differential equations, Proc. Conf., Birmingham/Ala. 1983, North-Holland Math. Stud. 92, 545-552 (1984).
Summary: [For the entire collection see Zbl 0539.00010.]
In this contribution we consider abstract bifurcation problems having a trivial branch of solutions, and equivariant under a finite cyclic group action. An example is the bifurcation of subharmonic solutions. We obtain, under a generically satisfied transversality condition, an abstract version of the Arnol’d tongues [V. I. Arnol’d, Chapitres supplémentaires de la théorie des équations différentielles ordinaires (1980; Zbl 0455.34001)]; when restricting to one-parameter problems one obtains results similar to those of G. Iooss and D. Joseph [Elementary stability and bifurcation theory (1980; Zbl 0443.34001)].

MSC:

34C25 Periodic solutions to ordinary differential equations