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

Periodic solutions of the Brillouin electron beam focusing equation

Maurizio Garrione 1, and  Manuel Zamora 2,

1. 

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 1, 60131, Ancona, Italy
2. 

Departamento de Matemática 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.
Keywords: Brillouin focusing system, Poincaré-Bohl theorem, singular nonlinearities, periodic solutions.
Mathematics Subject Classification: Primary: 34B15, 34C25; Secondary: 34B1.
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

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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

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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

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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

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