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Multiple periodic solutions of Hamiltonian systems confined in a box
1. | Dipartimento di Matematica e Geoscienze, Universitá degli Studi di Trieste, P.le Europa 1, Ⅰ-34127 Trieste, Italy |
2. | Dipartimento di Ingegneria Industriale e Scienze Matematiche, Universitá Politecnica delle Marche, Via Brecce Bianche 12, Ⅰ-60131 Ancona, Italy |
We consider a nonautonomous Hamiltonian system, $T$-periodic in time, possibly defined on a bounded space region, the boundary of which consists of singularity points which can never be attained. Assuming that the system has an interior equilibrium point, we prove the existence of infinitely many $T$-periodic solutions, by the use of a generalized version of the Poincaré-Birkhoff theorem.
References:
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A. Boscaggin and R. Ortega,
Monotone twist maps and periodic solutions of systems of Duffing type, Math. Proc. Cambridge Philos. Soc., 157 (2014), 279-296.
doi: 10.1017/S0305004114000310. |
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A. Castro and A. C. Lazer,
On periodic solutions of weakly coupled systems of differential equations, Boll. Un. Mat. Ital., 18 (1981), 733-742.
|
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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. |
[4] |
A. Fonda, R. Manasevich and F. Zanolin,
Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311.
doi: 10.1137/0524074. |
[5] |
A. Fonda and A. Sfecci,
Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differential Integral Equations, 25 (2012), 993-1010.
|
[6] |
A. Fonda and A. Sfecci,
Periodic solutions of weakly coupled superlinear systems, J. Differential Equations, 260 (2016), 2150-2162.
doi: 10.1016/j.jde.2015.09.056. |
[7] |
A. Fonda and A. J. Ureña, A higher dimensional Poincaré Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, online first.
doi: 10.1016/j.anihpc.2016.04.002. |
[8] |
S. Fučík and V. Lovicar,
Periodic solutions of the equation $x"(t)+g(x(t))=p(t)$, Časopis Pěst. Mat., 100 (1975), 160-175.
|
[9] |
Ph. Hartman,
On boundary value problems for superlinear second order differential equations, J. Differential Equations, 26 (1977), 37-53.
doi: 10.1016/0022-0396(77)90097-3. |
[10] |
H. Jacobowitz,
Periodic solutions of $x"+f(x, t)=0$ via the Poincaré-Birkhoff theorem, J. Differential Equations, 20 (1976), 37-52.
doi: 10.1016/0022-0396(76)90094-2. |
[11] |
G. R. Morris,
An infinite class of periodic solutions of $x"+2x^3=p(t)$, Proc. Cambridge Philos. Soc., 61 (1965), 157-164.
doi: 10.1017/S0305004100038743. |
[12] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal., 13 (1982), 343-352.
doi: 10.1137/0513027. |
[13] |
A. Sfecci,
Positive periodic solutions for planar differential systems with repulsive singularities on the axes, J. Math. Anal. Appl., 415 (2014), 110-120.
doi: 10.1016/j.jmaa.2013.12.068. |
show all references
References:
[1] |
A. Boscaggin and R. Ortega,
Monotone twist maps and periodic solutions of systems of Duffing type, Math. Proc. Cambridge Philos. Soc., 157 (2014), 279-296.
doi: 10.1017/S0305004114000310. |
[2] |
A. Castro and A. C. Lazer,
On periodic solutions of weakly coupled systems of differential equations, Boll. Un. Mat. Ital., 18 (1981), 733-742.
|
[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. |
[4] |
A. Fonda, R. Manasevich and F. Zanolin,
Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311.
doi: 10.1137/0524074. |
[5] |
A. Fonda and A. Sfecci,
Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differential Integral Equations, 25 (2012), 993-1010.
|
[6] |
A. Fonda and A. Sfecci,
Periodic solutions of weakly coupled superlinear systems, J. Differential Equations, 260 (2016), 2150-2162.
doi: 10.1016/j.jde.2015.09.056. |
[7] |
A. Fonda and A. J. Ureña, A higher dimensional Poincaré Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, online first.
doi: 10.1016/j.anihpc.2016.04.002. |
[8] |
S. Fučík and V. Lovicar,
Periodic solutions of the equation $x"(t)+g(x(t))=p(t)$, Časopis Pěst. Mat., 100 (1975), 160-175.
|
[9] |
Ph. Hartman,
On boundary value problems for superlinear second order differential equations, J. Differential Equations, 26 (1977), 37-53.
doi: 10.1016/0022-0396(77)90097-3. |
[10] |
H. Jacobowitz,
Periodic solutions of $x"+f(x, t)=0$ via the Poincaré-Birkhoff theorem, J. Differential Equations, 20 (1976), 37-52.
doi: 10.1016/0022-0396(76)90094-2. |
[11] |
G. R. Morris,
An infinite class of periodic solutions of $x"+2x^3=p(t)$, Proc. Cambridge Philos. Soc., 61 (1965), 157-164.
doi: 10.1017/S0305004100038743. |
[12] |
P. H. Rabinowitz,
Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal., 13 (1982), 343-352.
doi: 10.1137/0513027. |
[13] |
A. Sfecci,
Positive periodic solutions for planar differential systems with repulsive singularities on the axes, J. Math. Anal. Appl., 415 (2014), 110-120.
doi: 10.1016/j.jmaa.2013.12.068. |


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