# American Institute of Mathematical Sciences

January  2011, 10(1): 179-192. doi: 10.3934/cpaa.2011.10.179

## Boundedness in a class of duffing equations with oscillating potentials via the twist theorem

 1 Yiwu Industrial and Commercial College, Yiwu Zhejiang 322000, China

Received  November 2009 Revised  August 2010 Published  November 2010

In this paper, we prove the boundedness of all solutions and the existence of periodic and quasi-periodic solutions for the equation $\ddot{x}+x^{2n+1}+\sum_{j=0}^l x^j p_j (x,t)=0$, where $p_j (x,t)$ are smooth 1-periodic functions in both $x$ and $t$ with $n\geq 1, 0 \leq l \leq 2 n$.
Citation: Huiping Jin. Boundedness in a class of duffing equations with oscillating potentials via the twist theorem. Communications on Pure and Applied Analysis, 2011, 10 (1) : 179-192. doi: 10.3934/cpaa.2011.10.179
##### References:
 [1] R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95. [2] T. Kupper and J. You, Existence of quasiperiodic solutions and Littlewood's boundedness problem of Duffing equations with subquadratic potentials, Nonlinear Analysis, 35 (1999), 549-559. doi: doi:10.1016/S0362-546X(97)00709-8. [3] S. Laederich and M. Levi, Invariant curves and time-dependent potentials, Ergod. Th. and Dynam. Sys., 11 (1991), 365-378. doi: doi:10.1017/S0143385700006192. [4] M. Levi, Quasi-periodic motions in superquadratic periodic potentials, Comm. Math. Phys., 143 (1991), 43-83. doi: doi:10.1007/BF02100285. [5] M. Levi, KAM theory for particles in periodic potentials, Ergod. Th. and Dynam. Sys., 10 (1990), 777-785. doi: doi:10.1017/S0143385700005897. [6] M. Levi, On Littlewood's counterexample on unbounded motion in superquadratic potentials, Dynamics Reported I (ed. C.K.R.T. Jones, U. Kirchgraber and H. O. Walther, Springer, Berlin, 1992), 113-124. [7] B. Liu, Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem, J. Differential Equations, 79 (1989), 304-315. doi: doi:10.1016/0022-0396(89)90105-8. [8] B. Liu, Boundedness of solutions of nonlinear periodic differential equations via Moser's twist theorem, Acta. Mathematica Sinca, New Series, 8 (1992), 91-98. [9] B. Liu, On Littlewood's boundedness problem for sublinear Duffing equations, Transactions of the American mathematical society, 353 (2001), 1567-1585. doi: doi:10.1090/S0002-9947-00-02770-7. [10] B. Liu, Boundedness in asymmetric oscillations, JMAA, 231 (1999), 355-373. [11] J. Littlewood, Unbounded solutions of $y''+g(y)=p(t)$, Journal London Math. Soc., 41 (1966), 491-496. doi: doi:10.1112/jlms/s1-41.1.491. [12] Y. Long, An unbounded solution of a superlinear Duffing's equation, Acta Mathematica in Sinica, 7 (1991), 360-369. doi: doi:10.1007/BF02594893. [13] G. Morris, A case of boundedness in Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14 (1976), 71-93. doi: doi:10.1017/S0004972700024862. [14] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss, Gottingen Math. -Phys., Kl. II (1962), 1-20. [15] R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342. [16] R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proceeding London Math. Soc., 79 (1999), 381-413. doi: doi:10.1112/S0024611599012034. [17] H. Rüssman, On the existence of invariant curves of twist mapping of an annulus, Lecture Notes in Math., 1007 (1981), 677-718. [18] Y. Wang and J. You, Boundedness of solutions in polynomial potentials with $C^2$ coefficients, ZAMP, 47 (1996), 943-952. doi: doi:10.1007/BF00920044. [19] J. You, Boundedness for solutions of superlinear Duffing equations via the twist theorem, Sci. China Ser. A, 35 (1992), 399-412. [20] X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-262. doi: doi:10.1006/jdeq.1997.3356. [21] X. Yuan, Lagrange stability for Duffing-type equations, J. Differential Equations, 160 (2000), 94-117. doi: doi:10.1006/jdeq.1999.3663.

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##### References:
 [1] R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95. [2] T. Kupper and J. You, Existence of quasiperiodic solutions and Littlewood's boundedness problem of Duffing equations with subquadratic potentials, Nonlinear Analysis, 35 (1999), 549-559. doi: doi:10.1016/S0362-546X(97)00709-8. [3] S. Laederich and M. Levi, Invariant curves and time-dependent potentials, Ergod. Th. and Dynam. Sys., 11 (1991), 365-378. doi: doi:10.1017/S0143385700006192. [4] M. Levi, Quasi-periodic motions in superquadratic periodic potentials, Comm. Math. Phys., 143 (1991), 43-83. doi: doi:10.1007/BF02100285. [5] M. Levi, KAM theory for particles in periodic potentials, Ergod. Th. and Dynam. Sys., 10 (1990), 777-785. doi: doi:10.1017/S0143385700005897. [6] M. Levi, On Littlewood's counterexample on unbounded motion in superquadratic potentials, Dynamics Reported I (ed. C.K.R.T. Jones, U. Kirchgraber and H. O. Walther, Springer, Berlin, 1992), 113-124. [7] B. Liu, Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem, J. Differential Equations, 79 (1989), 304-315. doi: doi:10.1016/0022-0396(89)90105-8. [8] B. Liu, Boundedness of solutions of nonlinear periodic differential equations via Moser's twist theorem, Acta. Mathematica Sinca, New Series, 8 (1992), 91-98. [9] B. Liu, On Littlewood's boundedness problem for sublinear Duffing equations, Transactions of the American mathematical society, 353 (2001), 1567-1585. doi: doi:10.1090/S0002-9947-00-02770-7. [10] B. Liu, Boundedness in asymmetric oscillations, JMAA, 231 (1999), 355-373. [11] J. Littlewood, Unbounded solutions of $y''+g(y)=p(t)$, Journal London Math. Soc., 41 (1966), 491-496. doi: doi:10.1112/jlms/s1-41.1.491. [12] Y. Long, An unbounded solution of a superlinear Duffing's equation, Acta Mathematica in Sinica, 7 (1991), 360-369. doi: doi:10.1007/BF02594893. [13] G. Morris, A case of boundedness in Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14 (1976), 71-93. doi: doi:10.1017/S0004972700024862. [14] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss, Gottingen Math. -Phys., Kl. II (1962), 1-20. [15] R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342. [16] R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proceeding London Math. Soc., 79 (1999), 381-413. doi: doi:10.1112/S0024611599012034. [17] H. Rüssman, On the existence of invariant curves of twist mapping of an annulus, Lecture Notes in Math., 1007 (1981), 677-718. [18] Y. Wang and J. You, Boundedness of solutions in polynomial potentials with $C^2$ coefficients, ZAMP, 47 (1996), 943-952. doi: doi:10.1007/BF00920044. [19] J. You, Boundedness for solutions of superlinear Duffing equations via the twist theorem, Sci. China Ser. A, 35 (1992), 399-412. [20] X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-262. doi: doi:10.1006/jdeq.1997.3356. [21] X. Yuan, Lagrange stability for Duffing-type equations, J. Differential Equations, 160 (2000), 94-117. doi: doi:10.1006/jdeq.1999.3663.
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