January  2019, 39(1): 585-606. doi: 10.3934/dcds.2019024

Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems

1. 

Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy

2. 

Centro de Matemática, Aplicações Fundamentais e Investigação Operacional (CMAF – CIO), Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edificio C6, 1749–016 Lisboa, Portugal

3. 

Departamento de Matemática and Centro de Matemática, Aplicações Fundamentais, e Investigação Operacional (CMAF – CIO), Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edificio C6, 1749–016 Lisboa, Portugal

* Corresponding author

Received  May 2018 Published  October 2018

In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0, i_∞$ at the origin and at infinity, has at least $|i_∞-i_0|$ periodic solutions, and an additional one if $i_0$ is even. Our argument combines the Poincaré-Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.

Citation: Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024
References:
[1]

A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC research notes in mathematical series 425, 2001.  Google Scholar

[2]

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 539-603.   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

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J. CamposA. MargheriR. Martins and C. Rebelo, A note on a modified version of the Poincaré-Birkhoff theorem, J. Differential Equations, 203 (2004), 55-63.  doi: 10.1016/j.jde.2004.03.022.  Google Scholar

[5]

C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253.  doi: 10.1002/cpa.3160370204.  Google Scholar

[6]

F. Dalbono and C. Rebelo, Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, Rend. Semin. Mat. Univ. Politec. Torino, 60 (2002), 233-263.   Google Scholar

[7]

F. Dalbono and F. Zanolin, Multiplicity results for asymptotically linear equations using the rotation number approach, Mediterr. J. Math, 4 (2007), 127-149.  doi: 10.1007/s00009-007-0108-z.  Google Scholar

[8]

Y. Dong, Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems, J. Differential Equations, 214 (2005), 233-255.  doi: 10.1016/j.jde.2004.10.030.  Google Scholar

[9]

A. Fonda and P. Gidoni, An avoiding cones condition for the Poincaré–Birkhoff Theorem, J. Differential Equations, 262 (2017), 1064-1084.  doi: 10.1016/j.jde.2016.10.002.  Google Scholar

[10]

A. Fonda and J. Mawhin, Iterative and variational methods for the solvability of some semilinear equations in Hilbert spaces, J. Differential Equations, 98 (1992), 355-375.  doi: 10.1016/0022-0396(92)90097-7.  Google Scholar

[11]

A. Fonda and J. Mawhin, An iterative method for the solvability of semilinear equations in Hilbert spaces and applications, in: Partial Differential Equations and Other Topics (J. Wiener and J. K. Hale eds.), Longman, London, (1992), 126–132. Google Scholar

[12]

A. FondaM. Sabatini and F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff theorem, Topol. Methods Nonlinear Anal., 40 (2012), 29-52.   Google Scholar

[13]

A. Fonda and A.J. Ureña, A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Lin aire, 34 (2017), 679-698.  doi: 10.1016/j.anihpc.2016.04.002.  Google Scholar

[14]

M. Garrione, A. Margheri and C. Rebelo, Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers, preprint. Google Scholar

[15]

I.M. Gel'fand and V.B. Liskii, On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Am. Math. Soc. Transl. Ser. 2, 8 (1958), 143-181.  doi: 10.1090/trans2/008/06.  Google Scholar

[16]

C.-G. Liu, A note on the monotonicity of the Maslov-type index of linear Hamiltonian systems with applications, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1263-1277.  doi: 10.1017/S0308210500004364.  Google Scholar

[17]

Y. Long, A Maslov type index for symplectic paths, Topological Meth. Nonlinear Anal., 10 (1997), 47-78.  doi: 10.12775/TMNA.1997.021.  Google Scholar

[18]

A. MargheriC. Rebelo and P. Torres, On the use of Morse index and rotation numbers for multiplicity of resonant BVPs, J. Math. Anal. Appl., 413 (2014), 660-667.  doi: 10.1016/j.jmaa.2013.12.005.  Google Scholar

[19]

A. MargheriC. Rebelo and F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations, 183 (2002), 342-367.  doi: 10.1006/jdeq.2001.4122.  Google Scholar

[20]

C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal., 29 (1997), 291-311.  doi: 10.1016/S0362-546X(96)00065-X.  Google Scholar

[21]

C.P. Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics, Invent. Math., 26 (1974), 187-200.  doi: 10.1007/BF01418948.  Google Scholar

show all references

References:
[1]

A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC research notes in mathematical series 425, 2001.  Google Scholar

[2]

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 539-603.   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]

J. CamposA. MargheriR. Martins and C. Rebelo, A note on a modified version of the Poincaré-Birkhoff theorem, J. Differential Equations, 203 (2004), 55-63.  doi: 10.1016/j.jde.2004.03.022.  Google Scholar

[5]

C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 37 (1984), 207-253.  doi: 10.1002/cpa.3160370204.  Google Scholar

[6]

F. Dalbono and C. Rebelo, Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, Rend. Semin. Mat. Univ. Politec. Torino, 60 (2002), 233-263.   Google Scholar

[7]

F. Dalbono and F. Zanolin, Multiplicity results for asymptotically linear equations using the rotation number approach, Mediterr. J. Math, 4 (2007), 127-149.  doi: 10.1007/s00009-007-0108-z.  Google Scholar

[8]

Y. Dong, Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems, J. Differential Equations, 214 (2005), 233-255.  doi: 10.1016/j.jde.2004.10.030.  Google Scholar

[9]

A. Fonda and P. Gidoni, An avoiding cones condition for the Poincaré–Birkhoff Theorem, J. Differential Equations, 262 (2017), 1064-1084.  doi: 10.1016/j.jde.2016.10.002.  Google Scholar

[10]

A. Fonda and J. Mawhin, Iterative and variational methods for the solvability of some semilinear equations in Hilbert spaces, J. Differential Equations, 98 (1992), 355-375.  doi: 10.1016/0022-0396(92)90097-7.  Google Scholar

[11]

A. Fonda and J. Mawhin, An iterative method for the solvability of semilinear equations in Hilbert spaces and applications, in: Partial Differential Equations and Other Topics (J. Wiener and J. K. Hale eds.), Longman, London, (1992), 126–132. Google Scholar

[12]

A. FondaM. Sabatini and F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff theorem, Topol. Methods Nonlinear Anal., 40 (2012), 29-52.   Google Scholar

[13]

A. Fonda and A.J. Ureña, A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Lin aire, 34 (2017), 679-698.  doi: 10.1016/j.anihpc.2016.04.002.  Google Scholar

[14]

M. Garrione, A. Margheri and C. Rebelo, Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers, preprint. Google Scholar

[15]

I.M. Gel'fand and V.B. Liskii, On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Am. Math. Soc. Transl. Ser. 2, 8 (1958), 143-181.  doi: 10.1090/trans2/008/06.  Google Scholar

[16]

C.-G. Liu, A note on the monotonicity of the Maslov-type index of linear Hamiltonian systems with applications, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1263-1277.  doi: 10.1017/S0308210500004364.  Google Scholar

[17]

Y. Long, A Maslov type index for symplectic paths, Topological Meth. Nonlinear Anal., 10 (1997), 47-78.  doi: 10.12775/TMNA.1997.021.  Google Scholar

[18]

A. MargheriC. Rebelo and P. Torres, On the use of Morse index and rotation numbers for multiplicity of resonant BVPs, J. Math. Anal. Appl., 413 (2014), 660-667.  doi: 10.1016/j.jmaa.2013.12.005.  Google Scholar

[19]

A. MargheriC. Rebelo and F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations, 183 (2002), 342-367.  doi: 10.1006/jdeq.2001.4122.  Google Scholar

[20]

C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal., 29 (1997), 291-311.  doi: 10.1016/S0362-546X(96)00065-X.  Google Scholar

[21]

C.P. Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics, Invent. Math., 26 (1974), 187-200.  doi: 10.1007/BF01418948.  Google Scholar

Figure 1.  An Hamiltonian flow corresponding to case ⅰ) of Remark 14. For small periods $T$, the system has $i_0=0$, $i_\infty=-1$, and the only non-zero $T$-periodic orbits are the fixed points $A$ and $B$
Figure 2.  An Hamiltonian flow corresponding to case ⅱ) of Remark 14. For small periods $T$, the system has $i_0=1$, $i_\infty=-1$, and the only non-zero $T$-periodic orbits are the fixed points $M$ and $S$
Figure 3.  An Hamiltonian flow corresponding to case ⅲ) of Remark 14. For small periods $T$, the system has $i_0=1$, $i_\infty=0$, and the only non-zero $T$-periodic orbit is the fixed point $S$
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