American Institute of Mathematical Sciences

Advanced
July  2011, 16(1): 385-392. doi: 10.3934/dcdsb.2011.16.385

Positive periodic solution for Brillouin electron beam focusing system

Jingli Ren 1, , Zhibo Cheng 1, and  Stefan Siegmund 2,

1. 

Dept. of Math., Zhengzhou University, Zhengzhou 450001, China, China
2. 

Dept. of Math., Dresden University of Technology, Dresden 01062, Germany

Received  March 2010 Revised  July 2010 Published  April 2011

An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system $x''+a(1+\cos2t)x=\frac{1}{x}$ for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.
Keywords: Brillouin electron beam focusing system, singularity., Liénard equation.
Mathematics Subject Classification: Primary: 34B27; Secondary: 37C5.
Citation: Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
References:
[1]

V. Bevc, J. L. Palmer and C. Süsskind, On the design of the transition region of axisymmetric, magnetically focused beam valves, J. British Inst. Radio Engineer, 18 (1958), 696-708. 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, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31-38. Google Scholar

[4]

Weigao Ge, "Boundary Value Problems for Nonlinear Ordinary Differential Equations," Science Press, 2007. Google Scholar

[5]

J. Mawhin, Topological degree and boundary value problems for nonlinear differental equations, Topological Methods for Ordinary Differential Equations, 1537 (1993), 74-142. doi: 10.1007/BFb0085076.  Google Scholar

[6]

P. J. Torres, Existence and uniquenness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. Meth. Appl. Sci., 23 (2000), 1139-1143. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.  Google Scholar

[7]

Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory, Acta Math. Appl. Sinica, 1 (1978), 13-41.  Google Scholar

[8]

M. R. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl., 203 (1996), 254-269. doi: 10.1006/jmaa.1996.0378.  Google Scholar

[9]

M. R. Zhang, Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian, Nonlinear Analysis TMA, 29 (1997), 41-51. doi: 10.1016/S0362-546X(96)00037-5.  Google Scholar

show all references

References:
[1]

V. Bevc, J. L. Palmer and C. Süsskind, On the design of the transition region of axisymmetric, magnetically focused beam valves, J. British Inst. Radio Engineer, 18 (1958), 696-708. 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, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31-38. Google Scholar

[4]

Weigao Ge, "Boundary Value Problems for Nonlinear Ordinary Differential Equations," Science Press, 2007. Google Scholar

[5]

J. Mawhin, Topological degree and boundary value problems for nonlinear differental equations, Topological Methods for Ordinary Differential Equations, 1537 (1993), 74-142. doi: 10.1007/BFb0085076.  Google Scholar

[6]

P. J. Torres, Existence and uniquenness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. Meth. Appl. Sci., 23 (2000), 1139-1143. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.  Google Scholar

[7]

Y. Ye and X. Wang, Nonlinear differential equations in electron beam focusing theory, Acta Math. Appl. Sinica, 1 (1978), 13-41.  Google Scholar

[8]

M. R. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl., 203 (1996), 254-269. doi: 10.1006/jmaa.1996.0378.  Google Scholar

[9]

M. R. Zhang, Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian, Nonlinear Analysis TMA, 29 (1997), 41-51. doi: 10.1016/S0362-546X(96)00037-5.  Google Scholar

[1]

Maurizio Garrione, Manuel Zamora. Periodic solutions of the Brillouin electron beam focusing equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 961-975. doi: 10.3934/cpaa.2014.13.961
[2]

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
[3]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[4]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[5]

Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187
[6]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
[7]

Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[8]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
[9]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
[10]

Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137
[11]

Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441
[12]

Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783
[13]

Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
[14]

Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485
[15]

Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393
[16]

Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563
[17]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
[18]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[19]

Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955
[20]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences