American Institute of Mathematical Sciences

Advanced
May  2012, 32(5): 1557-1574. doi: 10.3934/dcds.2012.32.1557

Periodic and subharmonic solutions for duffing equation with a singularity

Zhibo Cheng 1, and  Jingli Ren 1,

1. 

Dept. of Math., Zhengzhou University, Zhengzhou 450001

Received  December 2010 Revised  April 2011 Published  January 2012

This paper is devoted to the existence and multiplicity of periodic and subharmonic solutions for a superlinear Duffing equation with a singularity. In this manner, various preceding theorems are improved and sharpened. Our proof is based on a generalized version of the Poincaré-Birkhoff twist theorem.
Keywords: superlinear; singularity, harmonic and subharmonic solution, Poincaré-Birkhoff twist theorem., Duffing equation.
Mathematics Subject Classification: 34C15, 34C2.
Citation: Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557
References:
[1]

P. Le Calvez and J. Wang, Some remarks on the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 138 (2010), 703-715. doi: 10.1090/S0002-9939-09-10105-3.  Google Scholar

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations," Higher Education Press, Beijing, 2004. Google Scholar

[3]

T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378. doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

[4]

T. R. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solution of semilinear Duffing equation, J. Differential Equations, 105 (1993), 364-409. doi: 10.1006/jdeq.1993.1093.  Google Scholar

[5]

W. Y. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346. doi: 10.1090/S0002-9939-1983-0695272-2.  Google Scholar

[6]

A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second-order differential equatins with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311. doi: 10.1137/0524074.  Google Scholar

[7]

A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Analy., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.  Google Scholar

[8]

P. Habets and L. Sanchez, Periodic solution of some Liénard equations with singularities, Proc. Amer. Math. Soc., 109 (1990), 1035-1044. doi: 10.2307/2048134.  Google Scholar

[9]

D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 211 (2005), 282-302. doi: 10.1016/j.jde.2004.10.031.  Google Scholar

[10]

Z. Opial, Sur les périodes des solutions de l'équation différentielle $ x''+g(x)= 0$, Ann. Polon. Math., 10 (1961), 49-72.  Google Scholar

[11]

M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. R. Soc. Edinb. Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.  Google Scholar

[12]

M. del Pino and R. Manásevich, Infinitely many $T$-periodic solutions for a problem ariding in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277. doi: 10.1006/jdeq.1993.1050.  Google Scholar

[13]

J. L. Ren, Z. B. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system,, Discrete Continuous Dynam. Systems B, ().   Google Scholar

[14]

S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal., 3 (1979), 897-904. doi: 10.1016/0362-546X(79)90057-9.  Google Scholar

[15]

P. J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007), 277-284. doi: 10.1016/j.jde.2006.08.006.  Google Scholar

[16]

Z.-H. Wang, Periodic solutions of the second-order differential equations with singularity, Nonlinear Anal., 58 (2004), 319-331. doi: 10.1016/j.na.2004.05.006.  Google Scholar

[17]

J. Xia and Z.-H. Wang, Existence and multiplicity of periodic solutions for the Duffing equation with singularity, Proc. R. Soc. Edinb. Sect. A, 137 (2007), 625-645.  Google Scholar

show all references

References:
[1]

P. Le Calvez and J. Wang, Some remarks on the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 138 (2010), 703-715. doi: 10.1090/S0002-9939-09-10105-3.  Google Scholar

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations," Higher Education Press, Beijing, 2004. Google Scholar

[3]

T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378. doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

[4]

T. R. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solution of semilinear Duffing equation, J. Differential Equations, 105 (1993), 364-409. doi: 10.1006/jdeq.1993.1093.  Google Scholar

[5]

W. Y. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346. doi: 10.1090/S0002-9939-1983-0695272-2.  Google Scholar

[6]

A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second-order differential equatins with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311. doi: 10.1137/0524074.  Google Scholar

[7]

A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Analy., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.  Google Scholar

[8]

P. Habets and L. Sanchez, Periodic solution of some Liénard equations with singularities, Proc. Amer. Math. Soc., 109 (1990), 1035-1044. doi: 10.2307/2048134.  Google Scholar

[9]

D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 211 (2005), 282-302. doi: 10.1016/j.jde.2004.10.031.  Google Scholar

[10]

Z. Opial, Sur les périodes des solutions de l'équation différentielle $ x''+g(x)= 0$, Ann. Polon. Math., 10 (1961), 49-72.  Google Scholar

[11]

M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. R. Soc. Edinb. Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.  Google Scholar

[12]

M. del Pino and R. Manásevich, Infinitely many $T$-periodic solutions for a problem ariding in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277. doi: 10.1006/jdeq.1993.1050.  Google Scholar

[13]

J. L. Ren, Z. B. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system,, Discrete Continuous Dynam. Systems B, ().   Google Scholar

[14]

S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal., 3 (1979), 897-904. doi: 10.1016/0362-546X(79)90057-9.  Google Scholar

[15]

P. J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007), 277-284. doi: 10.1016/j.jde.2006.08.006.  Google Scholar

[16]

Z.-H. Wang, Periodic solutions of the second-order differential equations with singularity, Nonlinear Anal., 58 (2004), 319-331. doi: 10.1016/j.na.2004.05.006.  Google Scholar

[17]

J. Xia and Z.-H. Wang, Existence and multiplicity of periodic solutions for the Duffing equation with singularity, Proc. R. Soc. Edinb. Sect. A, 137 (2007), 625-645.  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]

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
[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]

Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014
[6]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[7]

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
[8]

Anja Randecker, Giulio Tiozzo. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. Journal of Modern Dynamics, 2021, 17: 183-211. doi: 10.3934/jmd.2021006
[9]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
[10]

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
[11]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857
[12]

Zhaosheng Feng, Goong Chen, Sze-Bi Hsu. A qualitative study of the damped duffing equation and applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1097-1112. doi: 10.3934/dcdsb.2006.6.1097
[13]

S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660
[14]

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
[15]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[16]

Vassilis Rothos. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Conference Publications, 2005, 2005 (Special) : 756-767. doi: 10.3934/proc.2005.2005.756
[17]

Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259
[18]

D. P. Demuner, M. Federson, C. Gutierrez. The Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 495-509. doi: 10.3934/dcds.2009.25.495
[19]

Seppo Granlund, Niko Marola. Phragmén--Lindelöf theorem for infinity harmonic functions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 127-132. doi: 10.3934/cpaa.2015.14.127
[20]

Lizhi Zhang, Congming Li, Wenxiong Chen, Tingzhi Cheng. A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1721-1736. doi: 10.3934/dcds.2016.36.1721

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (106)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences