In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator (x) over bar + x(3) = 0 persist when the differential equation is perturbed so as to become (x) over bar + x(3) + ax(3) cos t + yx = 0. For any frequency omega, there exists a threshold for the damping coefficient gamma, above which there is no periodic orbit with period 2 pi/omega. We conjecture that this threshold is infinitesimal in the perturbation parameter, with integer order depending on the frequency omega. Some rigorous analytical results towards the proof of this conjecture are given: these results would provide a complete proof if we could rule out the possibility that other periodic solutions arise besides subharmonic solutions. Moreover the relative size and shape of the basins of attraction of the existing stable periodic orbits are investigated numerically, showing that all attractors different from the origin are subharmonic solutions and hence giving further support to the validity of the conjecture. The method we use is different from those usually applied in bifurcation theory, such as Mel'nikov's method or that of Chow and Hale, and allows us to investigate situations in which the non-degeneracy assumptions on the perturbation are violated

Bartuccelli, M., Berretti, A., Deane, J., Gentile, G., Gourley, S. (2008). Selection rules for periodic orbits and scaling laws for a driven damped quartic oscillator. NONLINEAR ANALYSIS: REAL WORLD APPLICATIONS, 9(5), 1966-1988 [10.1016/j.nonrwa.2007.06.007].

Selection rules for periodic orbits and scaling laws for a driven damped quartic oscillator

BERRETTI, ALBERTO;
2008-01-01

Abstract

In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator (x) over bar + x(3) = 0 persist when the differential equation is perturbed so as to become (x) over bar + x(3) + ax(3) cos t + yx = 0. For any frequency omega, there exists a threshold for the damping coefficient gamma, above which there is no periodic orbit with period 2 pi/omega. We conjecture that this threshold is infinitesimal in the perturbation parameter, with integer order depending on the frequency omega. Some rigorous analytical results towards the proof of this conjecture are given: these results would provide a complete proof if we could rule out the possibility that other periodic solutions arise besides subharmonic solutions. Moreover the relative size and shape of the basins of attraction of the existing stable periodic orbits are investigated numerically, showing that all attractors different from the origin are subharmonic solutions and hence giving further support to the validity of the conjecture. The method we use is different from those usually applied in bifurcation theory, such as Mel'nikov's method or that of Chow and Hale, and allows us to investigate situations in which the non-degeneracy assumptions on the perturbation are violated
2008
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
Settore MAT/07 - FISICA MATEMATICA
Settore MAT/08 - ANALISI NUMERICA
English
Con Impact Factor ISI
asymptotic stability; basins of attraction; dissipative systems; elliptic functions; periodic motions; perturbation theory; resonances; scaling; subharmonic solutions
Bartuccelli, M., Berretti, A., Deane, J., Gentile, G., Gourley, S. (2008). Selection rules for periodic orbits and scaling laws for a driven damped quartic oscillator. NONLINEAR ANALYSIS: REAL WORLD APPLICATIONS, 9(5), 1966-1988 [10.1016/j.nonrwa.2007.06.007].
Bartuccelli, M; Berretti, A; Deane, J; Gentile, G; Gourley, S
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/48707
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