# A NOTE ON QUASI-PERIODIC PERTURBATIONS OF ELLIPTIC EQUILIBRIUM POINTS

• Zhao, Houyu (School of Mathematics Chongqing Normal University)
• Received : 2010.09.28
• Published : 2012.11.30

#### 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.

#### References

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. https://doi.org/10.1017/S0004972700023005
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. https://doi.org/10.1007/BF01078168
4. L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrodinger equation, Comm. Math. Phys. 146 (1992), no. 3, 447-482. https://doi.org/10.1007/BF02097013
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. https://doi.org/10.1007/s10884-008-9113-6
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. https://doi.org/10.1016/0022-0396(81)90062-0
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. https://doi.org/10.1137/S0036141095280967
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. https://doi.org/10.1016/0022-0396(92)90107-X
9. A. Jorba and C. Simo, On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal. 27 (1996), no. 6, 1704-1737. https://doi.org/10.1137/S0036141094276913
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. https://doi.org/10.1007/BF01399536
12. J. Moser and J. Poschel, On the stationary Schrodinger equation with a quasiperiodic potential, Phys. A 124 (1984), no. 1-3, 535-542. https://doi.org/10.1016/0378-4371(84)90269-3
13. 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.
14. 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. https://doi.org/10.1090/S0002-9939-98-04523-7
15. J. You, Perturbations of lower-dimensional tori for Hamiltonian systems, J. Differential Equations 152 (1999), no. 1, 1-29. https://doi.org/10.1006/jdeq.1998.3515
16. 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. https://doi.org/10.1155/S0161171203206025