American Institute of Mathematical Sciences

Advanced
  • 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

Chao Wang 1,, , Qihuai Liu 2, and  Zhiguo Wang 3,

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  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.

Keywords: Impact oscillator, sublinear Hill's type equation, phase-plane analysis method, subharmonic bouncing solution, Poincaré-Birkhoff twist theorem.
Mathematics Subject Classification: Primary:34C15;Secondary:34C25.
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. Ding, D. Qian, C. 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. Wang, C. 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. Wang, Q. 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. Wang, D. 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. Ding, D. Qian, C. 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. Wang, C. 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. Wang, Q. 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. Wang, D. 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]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[2]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[3]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[4]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[5]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[6]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[7]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[8]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
[9]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[10]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[11]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035
[12]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[13]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[14]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[15]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[16]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[17]

Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024
[18]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[19]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[20]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (88)
  • HTML views (58)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences