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

Positive periodic solution for Brillouin electron beam focusing system

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.
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.

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations,", Higher Education Press, (2004).

[3]

T. R. Ding, A boundary value problem for the periodic Brillouin focusing system,, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31.

[4]

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

[5]

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

[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. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.

[7]

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

[8]

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

[9]

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

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.

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations,", Higher Education Press, (2004).

[3]

T. R. Ding, A boundary value problem for the periodic Brillouin focusing system,, Acta Sci. Natru. Univ. Pekinensis, 11 (1965), 31.

[4]

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

[5]

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

[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. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.

[7]

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

[8]

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

[9]

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

[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 - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic & Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251

[20]

Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]