Multiple periodic solutions of Hamiltonian systems confined in a box

Alessandro Fonda; Andrea Sfecci

doi:10.3934/dcds.2017059

Article Contents

2017, Volume 37, Issue 3: 1425-1436. Doi: 10.3934/dcds.2017059

This issue Previous Article Minimal subshifts of arbitrary mean topological dimension Next Article Homogenization of second order discrete model with local perturbation and application to traffic flow

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
* Corresponding author: Alessandro Fonda

Received: May 31, 2015

Revised: September 30, 2016

Published: May 2017

The authors were partially supported by INdAM-GNAMPA.

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 34C25.

Citation:

Full Text(HTML)

Figure 1. The regions where we estimate the angular velocity of the solutions, in the three cases (a), (b) and (c)

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.