American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Levi, Quasi-periodic motions in superquadratic periodic potentials, Comm. Math. Phys., 143 (1991), 43-83. doi: doi:10.1007/BF02100285.  Google Scholar

[5]

M. Levi, KAM theory for particles in periodic potentials, Ergod. Th. and Dynam. Sys., 10 (1990), 777-785. doi: doi:10.1017/S0143385700005897.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Liu, Boundedness in asymmetric oscillations, JMAA, 231 (1999), 355-373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss, Gottingen Math. -Phys., Kl. II (1962), 1-20.  Google Scholar

[15]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  Google Scholar

[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.  Google Scholar

[17]

H. Rüssman, On the existence of invariant curves of twist mapping of an annulus, Lecture Notes in Math., 1007 (1981), 677-718.  Google Scholar

[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.  Google Scholar

[19]

J. You, Boundedness for solutions of superlinear Duffing equations via the twist theorem, Sci. China Ser. A, 35 (1992), 399-412.  Google Scholar

[20]

X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-262. doi: doi:10.1006/jdeq.1997.3356.  Google Scholar

[21]

X. Yuan, Lagrange stability for Duffing-type equations, J. Differential Equations, 160 (2000), 94-117. doi: doi:10.1006/jdeq.1999.3663.  Google Scholar

show all references

References:
[1]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Levi, Quasi-periodic motions in superquadratic periodic potentials, Comm. Math. Phys., 143 (1991), 43-83. doi: doi:10.1007/BF02100285.  Google Scholar

[5]

M. Levi, KAM theory for particles in periodic potentials, Ergod. Th. and Dynam. Sys., 10 (1990), 777-785. doi: doi:10.1017/S0143385700005897.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Liu, Boundedness in asymmetric oscillations, JMAA, 231 (1999), 355-373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss, Gottingen Math. -Phys., Kl. II (1962), 1-20.  Google Scholar

[15]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  Google Scholar

[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.  Google Scholar

[17]

H. Rüssman, On the existence of invariant curves of twist mapping of an annulus, Lecture Notes in Math., 1007 (1981), 677-718.  Google Scholar

[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.  Google Scholar

[19]

J. You, Boundedness for solutions of superlinear Duffing equations via the twist theorem, Sci. China Ser. A, 35 (1992), 399-412.  Google Scholar

[20]

X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-262. doi: doi:10.1006/jdeq.1997.3356.  Google Scholar

[21]

X. Yuan, Lagrange stability for Duffing-type equations, J. Differential Equations, 160 (2000), 94-117. doi: doi:10.1006/jdeq.1999.3663.  Google Scholar

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