Periodic orbits in the case of a zero eigenvalue. (English) Zbl 1131.34034

The paper deals with a special case for investigating periodic orbits of a dynamic system in the case of zero eigenvalue of the linearized system. It could be considered as a supplement of some results of Moser (1976) and Weinstein (1973) related to this problem. Certain conditions for the existence of periodic solutions are formulated and proved. Two examples are given.


34C25 Periodic solutions to ordinary differential equations
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
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[1] Dubrovin, B.; Krichever, I.; Novikov, S., (Integrable Systems. Integrable Systems, Encyclopedia of Math. Sci., vol. 4 (1990), Springer-Verlag: Springer-Verlag Berlin), 173-280
[2] Lyapunov, M. A., Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 2, 203-474 (1907) · JFM 38.0738.07
[3] Moser, J., Periodic orbits and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29, 727-747 (1976) · Zbl 0346.34024
[4] Weinstein, A., Normal modes for non-linear Hamiltonian systems, Invent. Math., 20, 47-57 (1973) · Zbl 0264.70020
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