Bifurcations of nonlinear oscillations and frequency entrainment near resonance. (English) Zbl 0765.58018

Chicone, Carmen

SIAM J. Math. Anal. 23, No. 6, 1577-1608 (1992).

The author first gives a new proof for the theorem of Diliberto with a minor correction and then a better version for the application of the Poincaré-Andronov theorem on the global center bifurcation. However, the main part of the paper is to develop the theory of frequency entrainment for driven nonlinear oscillators when the period of the self- sustained free oscillation is nearly at resonance with a periodic external excitation. In the final section, artificial examples are analysed in detail via the above-mentioned theory. As the author points out there, he is not able at present to give a similar rigorous mathematical analysis for a model equation that arises from a physical problem, but he has been successful in applying the theory using numerical experiments.

Reviewer: Ding Tongren (Beijing)

MSC:

37G99	Local and nonlocal bifurcation theory for dynamical systems
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
34C23	Bifurcation theory for ordinary differential equations

Keywords:

limit cycles; center bifurcations; subharmonics; Melnikov method; forced oscillations; frequency entrainment; Diliberto theorem; Poincaré- Andronov theorem; bifurcation; resonance