• Previous Article
    Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions
  • CPAA Home
  • This Issue
  • Next Article
    High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $
January  2021, 20(1): 281-300. doi: 10.3934/cpaa.2020266

Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles

1. 

School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, China

2. 

School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, 541004, China

3. 

School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

* Corresponding author

Received  May 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: The first author is supported by Natural Science Foundation of Jiangsu Province (Grant Nos. BK20171275, BK20181058) and NSFC (Grant No. 12071410). The second author is supported by NSFC (Grant No. 11771105)

In this paper, we prove the existence and multiplicity of subharmonic bouncing motions for a Hill's type sublinear oscillator with an obstacle. Furthermore, we also consider the existence, multiplicity and dense distribution of symmetric periodic bouncing solutions when the weight function is even. Based on an appropriate coordinate transformation and the method of phase-plane analysis, we can study our main results via Poincar$ \acute{e} $ map by applying some suitable fixed point theorems.

Citation: Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266
References:
[1]

D. Bonheure and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bounces, Nonlinearity, 15 (2002), 1281-1297.  doi: 10.1088/0951-7715/15/4/314.  Google Scholar

[2]

W. Ding and D. Qian, Infinitesimal periodic solutions of impact Hamiltonian systems, Science China: Math., 40 (2010), 563-574.   Google Scholar

[3]

W. DingD. QianC. Wang and Z. Wang, Existence of periodic solutions of sublinear hamiltonian systems, Acta Math. Appl. Sin., 32 (2016), 621-632.  doi: 10.1007/s10114-016-4162-y.  Google Scholar

[4]

C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math., 60 (1993), 266-276.  doi: 10.1007/BF01198811.  Google Scholar

[5]

A. Fonda and A. Sfecci, Periodic bouncing solutions for nonlinear impact oscillators, Adv. Nonlinear Stud., 13 (2013), 179-189.  doi: 10.1515/ans-2013-0110.  Google Scholar

[6]

M. Jiang, Periodic solutions of second order differential equations with an obstacle, Nonlinearity, 19 (2006), 1165. doi: 10.1088/0951-7715/19/5/007.  Google Scholar

[7]

A. C. Lazer and P J. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differ. Integral Equ., 5 (1992), 165-172.   Google Scholar

[8]

F. Nakajima, Even and periodic solutions of the equation $\ddot{u}+g(u) = e(t)$, J. Differ. Equ., 83 (1990), 277-299.  doi: 10.1016/0022-0396(90)90059-X.  Google Scholar

[9]

R. Ortega, Variational and Topological Methods in the Study of Nonlinear Phenomena, Springer, New York, 2002.  Google Scholar

[10]

R. Ortega, Asymmetric Oscillators and Twist Mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.  Google Scholar

[11]

D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 201-213.  doi: 10.1017/S0308210500003164.  Google Scholar

[12]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.  doi: 10.1137/S003614100343771X.  Google Scholar

[13]

D. Qian, Large amplitude periodic bouncing for impact oscillators with damping, Proc. Am. Math. Soc., 133 (2005), 1797-1804.  doi: 10.1090/S0002-9939-04-07759-7.  Google Scholar

[14]

D. Qian and X. Sun, Invariant tori for asymptotically linear impact oscillators, Sci. China Math., 49 (2006), 669-687.  doi: 10.1007/s11425-006-0669-5.  Google Scholar

[15]

X. Sun and D. Qian, Periodic bouncing solutions for attractive singular second-order equations, Nonlinear Anal., 71 (2009), 4751-4757.  doi: 10.1016/j.na.2009.03.049.  Google Scholar

[16]

Z. WangC. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators, Nonlinear Anal., 27 (2010), 17-30.   Google Scholar

[17]

Z. WangQ. Liu and D. Qian, Existence of quasi-periodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators, Nonlinear Anal., 76 (2011), 5606-5617.  doi: 10.1016/j.na.2011.05.046.  Google Scholar

[18]

C. WangD. Qian and Q. Liu, Impact oscillators of Hill'S type with indefinite weight: periodic and chaotic dynamics, Discrete Contin. Dyn. Syst., 36 (2016), 2305-2328.  doi: 10.3934/dcds.2016.36.2305.  Google Scholar

show all references

References:
[1]

D. Bonheure and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bounces, Nonlinearity, 15 (2002), 1281-1297.  doi: 10.1088/0951-7715/15/4/314.  Google Scholar

[2]

W. Ding and D. Qian, Infinitesimal periodic solutions of impact Hamiltonian systems, Science China: Math., 40 (2010), 563-574.   Google Scholar

[3]

W. DingD. QianC. Wang and Z. Wang, Existence of periodic solutions of sublinear hamiltonian systems, Acta Math. Appl. Sin., 32 (2016), 621-632.  doi: 10.1007/s10114-016-4162-y.  Google Scholar

[4]

C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math., 60 (1993), 266-276.  doi: 10.1007/BF01198811.  Google Scholar

[5]

A. Fonda and A. Sfecci, Periodic bouncing solutions for nonlinear impact oscillators, Adv. Nonlinear Stud., 13 (2013), 179-189.  doi: 10.1515/ans-2013-0110.  Google Scholar

[6]

M. Jiang, Periodic solutions of second order differential equations with an obstacle, Nonlinearity, 19 (2006), 1165. doi: 10.1088/0951-7715/19/5/007.  Google Scholar

[7]

A. C. Lazer and P J. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differ. Integral Equ., 5 (1992), 165-172.   Google Scholar

[8]

F. Nakajima, Even and periodic solutions of the equation $\ddot{u}+g(u) = e(t)$, J. Differ. Equ., 83 (1990), 277-299.  doi: 10.1016/0022-0396(90)90059-X.  Google Scholar

[9]

R. Ortega, Variational and Topological Methods in the Study of Nonlinear Phenomena, Springer, New York, 2002.  Google Scholar

[10]

R. Ortega, Asymmetric Oscillators and Twist Mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.  Google Scholar

[11]

D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 201-213.  doi: 10.1017/S0308210500003164.  Google Scholar

[12]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.  doi: 10.1137/S003614100343771X.  Google Scholar

[13]

D. Qian, Large amplitude periodic bouncing for impact oscillators with damping, Proc. Am. Math. Soc., 133 (2005), 1797-1804.  doi: 10.1090/S0002-9939-04-07759-7.  Google Scholar

[14]

D. Qian and X. Sun, Invariant tori for asymptotically linear impact oscillators, Sci. China Math., 49 (2006), 669-687.  doi: 10.1007/s11425-006-0669-5.  Google Scholar

[15]

X. Sun and D. Qian, Periodic bouncing solutions for attractive singular second-order equations, Nonlinear Anal., 71 (2009), 4751-4757.  doi: 10.1016/j.na.2009.03.049.  Google Scholar

[16]

Z. WangC. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators, Nonlinear Anal., 27 (2010), 17-30.   Google Scholar

[17]

Z. WangQ. Liu and D. Qian, Existence of quasi-periodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators, Nonlinear Anal., 76 (2011), 5606-5617.  doi: 10.1016/j.na.2011.05.046.  Google Scholar

[18]

C. WangD. Qian and Q. Liu, Impact oscillators of Hill'S type with indefinite weight: periodic and chaotic dynamics, Discrete Contin. Dyn. Syst., 36 (2016), 2305-2328.  doi: 10.3934/dcds.2016.36.2305.  Google Scholar

[1]

Julián López-Gómez, Eduardo Muñoz-Hernández, Fabio Zanolin. On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2393-2419. doi: 10.3934/dcds.2020119

[2]

Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305

[3]

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

[4]

Denis Blackmore, Jyoti Champanerkar, Chengwen Wang. A generalized Poincaré-Birkhoff theorem with applications to coaxial vortex ring motion. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 15-33. doi: 10.3934/dcdsb.2005.5.15

[5]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173

[6]

Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303

[7]

Yongsheng Song. Stein’s method for the law of large numbers under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 199-212. doi: 10.3934/puqr.2021010

[8]

Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763

[9]

Luca Biasco, Laura Di Gregorio. Periodic solutions of Birkhoff-Lewis type for the nonlinear wave equation. Conference Publications, 2007, 2007 (Special) : 102-109. doi: 10.3934/proc.2007.2007.102

[10]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455

[11]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607

[12]

William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control & Related Fields, 2020, 10 (1) : 27-45. doi: 10.3934/mcrf.2019028

[13]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185

[14]

David Brander. Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane. Electronic Research Announcements, 2008, 15: 8-16. doi: 10.3934/era.2008.15.8

[15]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990

[16]

Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055

[17]

Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011

[18]

Ferdinand Verhulst. Henri Poincaré's neglected ideas. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1411-1427. doi: 10.3934/dcdss.2020079

[19]

Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115

[20]

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

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (178)
  • HTML views (62)
  • Cited by (0)

Other articles
by authors

[Back to Top]