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 $B$
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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 |
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.
Keywords:
Poincaré–Birkhoff theorem,
Maslov index,
topological degree,
asymptotically linear Hamiltonian systems,
periodic solutions.
Mathematics Subject Classification:
37J45, 34C25, 70H12.
Citation:
Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete and Continuous Dynamical Systems,
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. |
| [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.
|
| [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. |
| [4] |
J. Campos, A. Margheri, R. 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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
A. Fonda, M. 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.
|
| [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. |
| [14] |
M. Garrione, A. Margheri and C. Rebelo, Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers, preprint. |
| [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. |
| [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. |
| [17] |
Y. Long,
A Maslov type index for symplectic paths, Topological Meth. Nonlinear Anal., 10 (1997), 47-78.
doi: 10.12775/TMNA.1997.021. |
| [18] |
A. Margheri, C. 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. |
| [19] |
A. Margheri, C. 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. |
| [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. |
| [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. |
show all references
References:
| [1] |
A. Abbondandolo,
Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC research notes in mathematical series 425, 2001. |
| [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.
|
| [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. |
| [4] |
J. Campos, A. Margheri, R. 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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
A. Fonda, M. 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.
|
| [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. |
| [14] |
M. Garrione, A. Margheri and C. Rebelo, Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers, preprint. |
| [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. |
| [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. |
| [17] |
Y. Long,
A Maslov type index for symplectic paths, Topological Meth. Nonlinear Anal., 10 (1997), 47-78.
doi: 10.12775/TMNA.1997.021. |
| [18] |
A. Margheri, C. 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. |
| [19] |
A. Margheri, C. 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. |
| [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. |
| [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. |
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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