American Institute of Mathematical Sciences

Advanced
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.   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.  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.  doi: doi:10.1017/S0143385700006192.  Google Scholar

[4]

M. Levi, Quasi-periodic motions in superquadratic periodic potentials,, Comm. Math. Phys., 143 (1991), 43.  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.  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, (1992), 113.   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.  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, 8 (1992), 91.   Google Scholar

[9]

B. Liu, On Littlewood's boundedness problem for sublinear Duffing equations,, Transactions of the American mathematical society, 353 (2001), 1567.  doi: doi:10.1090/S0002-9947-00-02770-7.  Google Scholar

[10]

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

[11]

J. Littlewood, Unbounded solutions of $y''+g(y)=p(t)$,, Journal London Math. Soc., 41 (1966), 491.  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.  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.  doi: doi:10.1017/S0004972700024862.  Google Scholar

[14]

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

[15]

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

[18]

Y. Wang and J. You, Boundedness of solutions in polynomial potentials with $C^2$ coefficients,, ZAMP, 47 (1996), 943.  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.   Google Scholar

[20]

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

[21]

X. Yuan, Lagrange stability for Duffing-type equations,, J. Differential Equations, 160 (2000), 94.  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.   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.  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.  doi: doi:10.1017/S0143385700006192.  Google Scholar

[4]

M. Levi, Quasi-periodic motions in superquadratic periodic potentials,, Comm. Math. Phys., 143 (1991), 43.  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.  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, (1992), 113.   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.  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, 8 (1992), 91.   Google Scholar

[9]

B. Liu, On Littlewood's boundedness problem for sublinear Duffing equations,, Transactions of the American mathematical society, 353 (2001), 1567.  doi: doi:10.1090/S0002-9947-00-02770-7.  Google Scholar

[10]

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

[11]

J. Littlewood, Unbounded solutions of $y''+g(y)=p(t)$,, Journal London Math. Soc., 41 (1966), 491.  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.  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.  doi: doi:10.1017/S0004972700024862.  Google Scholar

[14]

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

[15]

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

[18]

Y. Wang and J. You, Boundedness of solutions in polynomial potentials with $C^2$ coefficients,, ZAMP, 47 (1996), 943.  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.   Google Scholar

[20]

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

[21]

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

[1]

Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
[2]

Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719
[3]

Daxiong Piao, Xiang Sun. Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 645-655. doi: 10.3934/cpaa.2014.13.645
[4]

Viktor L. Ginzburg and Basak Z. Gurel. The Generalized Weinstein--Moser Theorem. Electronic Research Announcements, 2007, 14: 20-29. doi: 10.3934/era.2007.14.20
[5]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[6]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
[7]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[8]

Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure & Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046
[9]

Yūki Naito, Takasi Senba. Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model. Conference Publications, 2011, 2011 (Special) : 1111-1118. doi: 10.3934/proc.2011.2011.1111
[10]

Alexander Sakhnovich. Dynamical canonical systems and their explicit solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1679-1689. doi: 10.3934/dcds.2017069
[11]

Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
[12]

Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078
[13]

Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317
[14]

István Győri, Ferenc Hartung, Nahed A. Mohamady. Boundedness of positive solutions of a system of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 809-836. doi: 10.3934/dcdsb.2018044
[15]

Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
[16]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032
[17]

Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464
[18]

Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173
[19]

Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431
[20]

Jiankai Xu, Song Jiang, Huoxiong Wu. Some properties of positive solutions for an integral system with the double weighted Riesz potentials. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2117-2134. doi: 10.3934/cpaa.2016030

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences