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March  2014, 13(2): 645-655. doi: 10.3934/cpaa.2014.13.645

Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials

Daxiong Piao 1, and  Xiang Sun 1,

1. 

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China, China

Received  January 2013 Revised  July 2013 Published  October 2013

In this paper, we consider the boundedness of solutions for a class of impact oscillators with time dependent polynomial potentials, \begin{eqnarray} \ddot{x}+x^{2n+1}+\sum_{i=0}^{2n}p_{i}(t)x^{i}=0, \quad for\ x(t)> 0,\\ x(t)\geq 0,\\ \dot{x}(t_{0}^{+})=-\dot{x}(t_{0}^{-}), \quad if\ x(t_{0})=0, \end{eqnarray} where $n\in N^+$, $p_i(t+1)=p_i(t)$ and $p_i(t)\in C^5(R/Z).$
Keywords: Canonical transformation, Time-dependent polynomial potentials, Moser's small twist theorem., Boundedness of solutions, Impactor oscillators.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: 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
References:
[1]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the Twist Theorem,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 79.   Google Scholar

[2]

S. Laederich and M. Levi, Invariant curves and time-dependent potentials,, Ergo.Th. and Dynam. Syst., 11 (1991), 365.  doi: 10.1017/S0143385700006192.  Google Scholar

[3]

X. Yuan, Invariant tori of Duffing-type equations,, Adv. in Math. (China), 24 (1995), 375.   Google Scholar

[4]

X. Yuan, Invariant tori of Duffing-type equations,, J. Differential Equations, 142 (1998), 231.   Google Scholar

[5]

M. Kunze, "Non-Smooth Dynamical Systems,", in: Lecture Notes in Math., (2000).   Google Scholar

[6]

H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator,, Physica D, 82 (1995), 117.  doi: 10.1016/0167-2789(94)00222-C.  Google Scholar

[7]

P. Boyland, Dual billiards, twist maps and impact oscillators,, Nonlinearity, 9 (1996), 1411.  doi: 10.1088/0951-7715/9/6/002.  Google Scholar

[8]

M. Corbera and J. Llibre, Periodic orbits of a collinear restricted three body problem,, Celestial Mech. Dynam. Astronom., 86 (2003), 163.  doi: 10.1023/A:1024183003251.  Google Scholar

[9]

D. Qian and P. J. Torres, Periodic motions of linear impact oscilltors via successor map,, SIAM J. Math. Anal., 36 (2005), 1707.  doi: 10.1137/S003614100343771X.  Google Scholar

[10]

D. Qian and X. Sun, Inariant tori for asymptotically linear impact oscillators,, Sci. China: Ser. A Math., 49 (2006), 669.  doi: 10.1007/s11425-006-0669-5.  Google Scholar

[11]

V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts,, Comm. Math. Phys., 211 (2000), 289.  doi: 10.1007/s002200050813.  Google Scholar

[12]

Z. Wang and Y. Wang, Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators,, Applied Mathematics and Computation, 217 (2011), 6417.  doi: 10.1016/j.amc.2011.01.037.  Google Scholar

[13]

Z. Wang, Q. Liu and D. Qian, Existence of quasiperiodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators,, Nonlinear Analysis, 74 (2011), 5606.  doi: 10.1016/j.na.2011.05.046.  Google Scholar

[14]

D. Qian, Large amplitude periodic bouncing in impact oscillators with damping,, Proc. Amer. Math. Soc., 133 (2005), 1797.  doi: 10.1090/S0002-9939-04-07759-7.  Google Scholar

[15]

D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity,, Proc. Roy. Soc. Edinburgh, 134 (2004), 201.  doi: 10.1017/S0308210500003164.  Google Scholar

[16]

Z. Wang, C. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators,, J. Nanjing Univ. Math. Biquart., 27 (2010), 17.  doi: 10.3969/j.issn.0469-5097.2010.01.003.  Google Scholar

[17]

R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem,, Proc. London. Math. Soc., 79 (1999), 381.  doi: 10.1112/S0024611599012034.  Google Scholar

[18]

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

[19]

B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.  doi: 10.1006/jdeq.1998.3553.  Google Scholar

[20]

L. Jiao, D. Piao and Y. Wang, Boundedness for the general semilinear Duffing equation via the twist theorem,, J. Differential Equations, 252 (2012), 91.  doi: 10.1016/j.jde.2011.09.019.  Google Scholar

[21]

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

[22]

H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes Math., 1007 (1983), 677.  doi: 10.1007/BFb0061441.  Google Scholar

show all references

References:
[1]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the Twist Theorem,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 79.   Google Scholar

[2]

S. Laederich and M. Levi, Invariant curves and time-dependent potentials,, Ergo.Th. and Dynam. Syst., 11 (1991), 365.  doi: 10.1017/S0143385700006192.  Google Scholar

[3]

X. Yuan, Invariant tori of Duffing-type equations,, Adv. in Math. (China), 24 (1995), 375.   Google Scholar

[4]

X. Yuan, Invariant tori of Duffing-type equations,, J. Differential Equations, 142 (1998), 231.   Google Scholar

[5]

M. Kunze, "Non-Smooth Dynamical Systems,", in: Lecture Notes in Math., (2000).   Google Scholar

[6]

H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator,, Physica D, 82 (1995), 117.  doi: 10.1016/0167-2789(94)00222-C.  Google Scholar

[7]

P. Boyland, Dual billiards, twist maps and impact oscillators,, Nonlinearity, 9 (1996), 1411.  doi: 10.1088/0951-7715/9/6/002.  Google Scholar

[8]

M. Corbera and J. Llibre, Periodic orbits of a collinear restricted three body problem,, Celestial Mech. Dynam. Astronom., 86 (2003), 163.  doi: 10.1023/A:1024183003251.  Google Scholar

[9]

D. Qian and P. J. Torres, Periodic motions of linear impact oscilltors via successor map,, SIAM J. Math. Anal., 36 (2005), 1707.  doi: 10.1137/S003614100343771X.  Google Scholar

[10]

D. Qian and X. Sun, Inariant tori for asymptotically linear impact oscillators,, Sci. China: Ser. A Math., 49 (2006), 669.  doi: 10.1007/s11425-006-0669-5.  Google Scholar

[11]

V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts,, Comm. Math. Phys., 211 (2000), 289.  doi: 10.1007/s002200050813.  Google Scholar

[12]

Z. Wang and Y. Wang, Existence of quasiperiodic solutions and Littlewood's boundedness problem of super-linear impact oscillators,, Applied Mathematics and Computation, 217 (2011), 6417.  doi: 10.1016/j.amc.2011.01.037.  Google Scholar

[13]

Z. Wang, Q. Liu and D. Qian, Existence of quasiperiodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators,, Nonlinear Analysis, 74 (2011), 5606.  doi: 10.1016/j.na.2011.05.046.  Google Scholar

[14]

D. Qian, Large amplitude periodic bouncing in impact oscillators with damping,, Proc. Amer. Math. Soc., 133 (2005), 1797.  doi: 10.1090/S0002-9939-04-07759-7.  Google Scholar

[15]

D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity,, Proc. Roy. Soc. Edinburgh, 134 (2004), 201.  doi: 10.1017/S0308210500003164.  Google Scholar

[16]

Z. Wang, C. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators,, J. Nanjing Univ. Math. Biquart., 27 (2010), 17.  doi: 10.3969/j.issn.0469-5097.2010.01.003.  Google Scholar

[17]

R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem,, Proc. London. Math. Soc., 79 (1999), 381.  doi: 10.1112/S0024611599012034.  Google Scholar

[18]

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

[19]

B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.  doi: 10.1006/jdeq.1998.3553.  Google Scholar

[20]

L. Jiao, D. Piao and Y. Wang, Boundedness for the general semilinear Duffing equation via the twist theorem,, J. Differential Equations, 252 (2012), 91.  doi: 10.1016/j.jde.2011.09.019.  Google Scholar

[21]

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

[22]

H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes Math., 1007 (1983), 677.  doi: 10.1007/BFb0061441.  Google Scholar

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