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Periodically perturbed Hopf bifurcation. (English) Zbl 0625.70022
A general two-dimensional system of differential equations with periodic parametric excitation is considered with two real parameters one of them being the amplitude of the periodic excitation. As a matter of fact, the frequency of the excitation occurs also as an additional parameter, and in this respect the paper is related to the reviewer’s results [Acta Math. Acad. Sci. Hungar. 22, 337-348 (1971; Zbl 0239.34016); Studia Sci. Math. Hung. 7 (1972), 257-266 (1973; Zbl 0275.34038) and SIAM J. Math. Anal. 9, 876-890 (1978; Zbl 0387.34034)]. It is assumed that when the amplitude of the excitation is zero the trivial equilibrium of the system undergoes a Hopf bifurcation at some critical value of the other parameter. The method of averaging is applied and possible secondary bifurcations of the averaged equation are studied (one zero eigenvalue, resp. a pair of imaginary eigenvalues, resp. a double zero eigenvalue). The stability of the arising limit cycles is studied by different methods, numerical calculations are presented and bifurcation and stability charts drawn.
Reviewer: M.Farkas

MSC:
70K99 Nonlinear dynamics in mechanics
70-08 Computational methods for problems pertaining to mechanics of particles and systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
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