[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2012.49.6.1223

A NOTE ON QUASI-PERIODIC PERTURBATIONS OF ELLIPTIC EQUILIBRIUM POINTS

Zhao, Houyu (School of Mathematics Chongqing Normal University)

Publication Information

Bulletin of the Korean Mathematical Society / v.49, no.6, 2012 , pp. 1223-1240 More about this Journal

Abstract

The system $\dot{x}=(A+{\varepsilon}Q(t,{\varepsilon}))x+{\varepsilon}g(t,{\varepsilon})+h(x,t,{\varepsilon}),$ where A is elliptic whose eigenvalues are not necessarily simple and $h$ is $\mathcal{O}(x^2)$ . It is proved that, under suitable hypothesis of analyticity, for most values of the frequencies, the system is reducible.

Keywords

quasi-periodic perturbations; elliptic points; quasi-periodic solutions; nonresonant condition; small divisors; quasi-periodic Floquet theorem; KAM theory;

Citations & Related Records

Reference

1	N. N. Bogoljubov, Ju. A. Mitropoliski, and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976.
2	W. A. Coppel, Pseudo-autonomous linear system, Bull. Austral. Math. Soc. 16 (1977), no. 1, 61-65. DOI
3	E. I. Dinaburg and Y. G. Sinai, The one dimensional Schrodinger equation with a quasiperiodic potential, Funkcional. Anal. i Prilozen. 9 (1975), no. 1, 8-21. DOI
4	L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrodinger equation, Comm. Math. Phys. 146 (1992), no. 3, 447-482. DOI
5	H. Her and J. You, Full measure reducibility for generic one-parameter family of quasiperiodic linear systems, J. Dynam. Differential Equations 20 (2008), no. 4, 831-886. DOI
6	R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems, J. Differential Equations 41 (1981), no. 2, 262-288. DOI
7	A. Jorba, R. Ramrez-ros, and J. Villanueva, Effective reducibility of quasi-periodic linear equations close to constant coefficients, SIAM J. Math. Anal. 28 (1997), no. 1, 178-188. DOI ScienceOn
8	A. Jorba and C. Simo, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations 98 (1992), no. 1, 111-124. DOI
9	A. Jorba and C. Simo, On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal. 27 (1996), no. 6, 1704-1737. DOI ScienceOn
10	P. Lancaster, Theory of Matrices, Academic Press, New York, 1969.
11	J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136-176. DOI
12	J. Moser and J. Poschel, On the stationary Schrodinger equation with a quasiperiodic potential, Phys. A 124 (1984), no. 1-3, 535-542. DOI ScienceOn
13	X. P. Yuan and A. Nunes, A note on the reducibility of linear differential equations with quasi-periodic coefficients, Int. J. Math. Math. Sci. 2003 (2003), no. 64, 4071-4083. DOI
14	H. Russmann, On the one dimensional Schrodinger equation with a quasiperiodic potential, Nonlinear dynamics (Internat. Conf., New York, 1979), pp. 90-107, Ann. New York Acad. Sci., 357, New York Acad. Sci., New York, 1980.
15	J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coeffficients which are degenerate, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1445-1451. DOI ScienceOn
16	J. You, Perturbations of lower-dimensional tori for Hamiltonian systems, J. Differential Equations 152 (1999), no. 1, 1-29. DOI ScienceOn