Pseudo-synchronous solutions for dissipative non-autonomous systems

Abstract

In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In the presence of dissipation, most of the quasi-periodic solutions disappear and one expects, at most, only a few of them to survive together with the periodic attractors. However, to prove that a quasi-periodic solution really exists, usually one assumes that the frequencies still satisfy a Diophantine condition and, furthermore, that some external parameters of the system are suitably tuned with them. In this paper we consider a class of systems on the one-dimensional torus, subject to a periodic perturbation and in the presence of dissipation, and show that, however small the dissipation, if the perturbation is a trigonometric polynomial in the angles and the unperturbed frequencies satisfy a non-resonance condition of finite order, depending on the size of the dissipation, then a quasi-periodic solution exists with slightly perturbed frequencies provided the size of the perturbation is small enough. If on the one hand the maximal size of the perturbation is not uniform in the degree of the trigonometric polynomial, on the other hand all but finitely many frequencies are allowed and there is no restriction arising from the tuning of the external parameters. A physically relevant case, where the result applies, is the spin-orbit model, which describes the rotation of a satellite around its own axis, while revolving on a Keplerian orbit around a planet, in the case in which the dissipation is taken into account through the MacDonald torque.

arXiv comment42 pages

References in corpus (11)

• Bodily tides near spin-orbit resonances
• Twistless KAM tori
• Tidal torques. A critical review of some techniques
• The free precession and libration of Mercury
• Spin-orbit evolution of Mercury revisited
• Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics
• How terrestrial planets traverse spin-orbit resonances: A camel goes through a needle's eye
• Bodily tides near the 1:1 spin-orbit resonance. Correction to Goldreich's dynamical model
• Attractiveness of periodic orbits in parametrically forced systemswith time-increasing friction
• The spin-orbit resonances of the Solar system: A mathematical treatment matching physical data
• The high-order Euler method and the spin-orbit model A fast algorithm for solving differential equations with small, smooth nonlinearity