# American Institute of Mathematical Sciences

March  2014, 13(2): 961-975. doi: 10.3934/cpaa.2014.13.961

## Periodic solutions of the Brillouin electron beam focusing equation

 1 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 1, 60131, Ancona, Italy 2 Departamento de Matemática Aplicada, Universidad de Granada, 18071, Granada, Spain

Received  February 2013 Revised  September 2013 Published  October 2013

Quite unexpectedly with respect to the numerical and analytical results found in literature, we establish a new range for the real parameter $b$ for which the existence of $2\pi-$periodic solutions of the Brillouin focusing beam equation \begin{eqnarray} \ddot{x}+b(1+\cos t)x=\frac{1}{x} \end{eqnarray} is guaranteed. This is possible thanks to suitable nonresonance conditions acting on the rotation number of the solutions in the phase plane.
Citation: 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
##### 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] D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillators at resonance, Discrete Contin. Dyn. Syst., 8 (2002), 907-930. doi: 10.3934/dcds.2002.8.907.  Google Scholar [3] A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincaré-Birkhoff theorem, Nonlinear Anal., 74 (2011), 4166-4185. doi: 10.1016/j.na.2011.03.051.  Google Scholar [4] H. Broer and M. Levi, Geometrical aspects of stability theory for Hill's equations, Arch. Rational Mech. Anal., 131 (1995), 225-240. doi: 10.1007/BF00382887.  Google Scholar [5] A. Cabada and J. A. Cid, On comparison principles for the periodic Hill's equation, J. Lond. Math. Soc., 86 (2012), 272-290. doi: 10.1112/jlms/jds001.  Google Scholar [6] M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.  Google Scholar [7] T. Ding, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natur. Univ. Pekinensis, 11 (1965), 31-38 (Chinese). Google Scholar [8] C. Fabry, Periodic solutions of the equation $x'' + f(t, x)=0$, Séminaire de Mathématique, 117 (1987), Louvain-la-Neuve. Google Scholar [9] C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math. (Basel), 60 (1993), 266-276. doi: 10.1007/BF01198811.  Google Scholar [10] A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential systems in the plane, J. Differential Equations, 252 (2012), 1369-1391. doi: 10.1016/j.jde.2011.08.005.  Google Scholar [11] A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Anal., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.  Google Scholar [12] W. Magnus and S. Winkler, "Hill's Equation," corrected reprint of 1966 edition, Dover, New York, 1979.  Google Scholar [13] J. Ren, Z. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing systems, Discrete Cont. Dyn. Syst. Ser. B, 16 (2011), 385-392. doi: 10.3934/dcdsb.2011.16.385.  Google Scholar [14] P.J. Torres, Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. Methods Appl. Sci., 23 (2000), 1139-1143. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.  Google Scholar [15] P.J. Torres, Twist solutions of a Hill's equation with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287.  Google Scholar [16] P.J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662. doi: 10.1016/S0022-0396(02)00152-3.  Google Scholar [17] Q. Yao, Periodic positive solution to a class of singular second-order ordinary differential equations, Acta Appl. Math., 110 (2010), 871-883. doi: 10.1007/s10440-009-9482-9.  Google Scholar [18] Y. Ye and X. Wang, Nonlinear differential equations arising in the theory of electron beam focusing, Acta Math. Appl. Sinica, 1 (1978), 13-41.  Google Scholar [19] M. 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 [20] M. Zhang, A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1099-1114. doi: 10.1017/S0308210500030080.  Google Scholar [21] M. Zhang, Periodic solutions of equations of Emarkov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67.  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] D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillators at resonance, Discrete Contin. Dyn. Syst., 8 (2002), 907-930. doi: 10.3934/dcds.2002.8.907.  Google Scholar [3] A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincaré-Birkhoff theorem, Nonlinear Anal., 74 (2011), 4166-4185. doi: 10.1016/j.na.2011.03.051.  Google Scholar [4] H. Broer and M. Levi, Geometrical aspects of stability theory for Hill's equations, Arch. Rational Mech. Anal., 131 (1995), 225-240. doi: 10.1007/BF00382887.  Google Scholar [5] A. Cabada and J. A. Cid, On comparison principles for the periodic Hill's equation, J. Lond. Math. Soc., 86 (2012), 272-290. doi: 10.1112/jlms/jds001.  Google Scholar [6] M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.  Google Scholar [7] T. Ding, A boundary value problem for the periodic Brillouin focusing system, Acta Sci. Natur. Univ. Pekinensis, 11 (1965), 31-38 (Chinese). Google Scholar [8] C. Fabry, Periodic solutions of the equation $x'' + f(t, x)=0$, Séminaire de Mathématique, 117 (1987), Louvain-la-Neuve. Google Scholar [9] C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math. (Basel), 60 (1993), 266-276. doi: 10.1007/BF01198811.  Google Scholar [10] A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential systems in the plane, J. Differential Equations, 252 (2012), 1369-1391. doi: 10.1016/j.jde.2011.08.005.  Google Scholar [11] A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Anal., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.  Google Scholar [12] W. Magnus and S. Winkler, "Hill's Equation," corrected reprint of 1966 edition, Dover, New York, 1979.  Google Scholar [13] J. Ren, Z. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing systems, Discrete Cont. Dyn. Syst. Ser. B, 16 (2011), 385-392. doi: 10.3934/dcdsb.2011.16.385.  Google Scholar [14] P.J. Torres, Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system, Math. Methods Appl. Sci., 23 (2000), 1139-1143. doi: 10.1002/1099-1476(20000910)23:13<1139::AID-MMA155>3.0.CO;2-J.  Google Scholar [15] P.J. Torres, Twist solutions of a Hill's equation with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287.  Google Scholar [16] P.J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662. doi: 10.1016/S0022-0396(02)00152-3.  Google Scholar [17] Q. Yao, Periodic positive solution to a class of singular second-order ordinary differential equations, Acta Appl. Math., 110 (2010), 871-883. doi: 10.1007/s10440-009-9482-9.  Google Scholar [18] Y. Ye and X. Wang, Nonlinear differential equations arising in the theory of electron beam focusing, Acta Math. Appl. Sinica, 1 (1978), 13-41.  Google Scholar [19] M. 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 [20] M. Zhang, A relationship between the periodic and the Dirichlet BVPs of singular differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1099-1114. doi: 10.1017/S0308210500030080.  Google Scholar [21] M. Zhang, Periodic solutions of equations of Emarkov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67.  Google Scholar
 [1] 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 [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] 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 [4] Jean Mawhin. Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4015-4026. doi: 10.3934/dcds.2012.32.4015 [5] Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 [6] Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051 [7] N. D. Cong, T. S. Doan, S. Siegmund. A Bohl-Perron type theorem for random dynamical systems. Conference Publications, 2011, 2011 (Special) : 322-331. doi: 10.3934/proc.2011.2011.322 [8] 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 [9] Pablo Amster, Mónica Clapp. Periodic solutions of resonant systems with rapidly rotating nonlinearities. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 373-383. doi: 10.3934/dcds.2011.31.373 [10] Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009 [11] 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 [12] 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 [13] Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006 [14] Markus Banagl. Singular spaces and generalized Poincaré complexes. Electronic Research Announcements, 2009, 16: 63-73. doi: 10.3934/era.2009.16.63 [15] C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 25-34. doi: 10.3934/dcds.1997.3.25 [16] V. Mastropietro, Michela Procesi. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. Communications on Pure & Applied Analysis, 2006, 5 (1) : 1-28. doi: 10.3934/cpaa.2006.5.1 [17] Zaihong Wang. Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 751-770. doi: 10.3934/dcds.2003.9.751 [18] Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure & Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029 [19] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 [20] Marcos L. M. Carvalho, Edcarlos D. Silva, Claudiney Goulart, Carlos A. Santos. Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4401-4432. doi: 10.3934/cpaa.2020201

2020 Impact Factor: 1.916