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March  2017, 37(3): 1425-1436. doi: 10.3934/dcds.2017059

Multiple periodic solutions of Hamiltonian systems confined in a box

Alessandro Fonda 1,, and  Andrea Sfecci 2,

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

* Corresponding author: Alessandro Fonda

Received  June 2015 Revised  October 2016 Published  December 2016

Fund Project: The authors were partially supported by INdAM-GNAMPA.

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.

Keywords: Periodic solutions, Poincaré-Birkhoff theorem, singularities, Hamiltonian systems, rotation number.
Mathematics Subject Classification: 34C25.
Citation: Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal., 13 (1982), 343-352.  doi: 10.1137/0513027.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems: A survey, SIAM J. Math. Anal., 13 (1982), 343-352.  doi: 10.1137/0513027.  Google Scholar

[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.  Google Scholar

Figure 1.  The regions where we estimate the angular velocity of the solutions, in the three cases (a), (b) and (c)
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Figure 2.  The construction of the first lap of the guiding curve, outside the rectangle $\mathcal{R}(p_2)$, using the level curves of the energy functions
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