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

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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

Huiping Jin ^1,

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

Received November 2009 Revised August 2010 Published November 2010

Abstract
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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 $n\geq 1, 0 \leq l \leq 2 n$.

Keywords: canonical transformation, Hamiltonian system, boundedness of solutions, Moser's twist theorem., oscillating potentials.

Mathematics Subject Classification: 37J40, 34C2.

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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