-
Previous Article
Asymptotic behavior for solutions of some integral equations
- CPAA Home
- This Issue
-
Next Article
Blowing up at zero points of potential for an initial boundary value problem
Boundedness in a class of duffing equations with oscillating potentials via the twist theorem
1. | Yiwu Industrial and Commercial College, Yiwu Zhejiang 322000, China |
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. |
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. |
[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. |
[1] |
Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017 |
[2] |
Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719 |
[3] |
Xuefeng Zhao, Yong Li. A Moser theorem for multiscale mappings. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022037 |
[4] |
Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022045 |
[5] |
Daxiong Piao, Xiang Sun. Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials. Communications on Pure and Applied Analysis, 2014, 13 (2) : 645-655. doi: 10.3934/cpaa.2014.13.645 |
[6] |
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 |
[7] |
Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 |
[8] |
Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 |
[9] |
Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure and Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046 |
[10] |
Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4531-4543. doi: 10.3934/dcds.2021047 |
[11] |
V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 |
[12] |
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 |
[13] |
Alexander Sakhnovich. Dynamical canonical systems and their explicit solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1679-1689. doi: 10.3934/dcds.2017069 |
[14] |
Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure and Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599 |
[15] |
Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032 |
[16] |
Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078 |
[17] |
Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 |
[18] |
Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737 |
[19] |
István Győri, Ferenc Hartung, Nahed A. Mohamady. Boundedness of positive solutions of a system of nonlinear delay differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 809-836. doi: 10.3934/dcdsb.2018044 |
[20] |
Xiangdong Zhao. Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5095-5100. doi: 10.3934/dcdsb.2020334 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]