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January  2022, 21(1): 337-354. doi: 10.3934/cpaa.2021180

Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

Fanfan Chen 1, , Dingbian Qian 1,, , Xiying Sun 1, and  Yinyin Wu 2,

1. 

School of Mathematical Sciences, Soochow University, Suzhou 215006, China
2. 

Department of Fundamental Courses, Wuxi Institute of Technology, Wuxi 214121, China

* Corresponding author

Received  April 2021 Revised  September 2021 Published  January 2022 Early access  October 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (No. 12071327, No. 11671287)

We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.

Keywords: Hamiltonian systems, Coupled system, Subharmonic solutions, sub-linear growth, Poincaré-Birkhoff theorem.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34C15, 46N20.
Citation: Fanfan Chen, Dingbian Qian, Xiying Sun, Yinyin Wu. Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth. Communications on Pure & Applied Analysis, 2022, 21 (1) : 337-354. doi: 10.3934/cpaa.2021180
References:
[1]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular-Laplacian, J. Differ. Equ., 243 (2007), 536-557.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[2]

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

[3]

A. Calamai and A. Sfecci, Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations, Nonlinear Differ. Equ. Appl., 24 (2017), 1-17.  doi: 10.1007/s00030-016-0427-5.  Google Scholar

[4]

T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differ. Equ., 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

[5]

T. Ding and F. Zanolin, Subharmonic solutions of second order nonlinear equations: A time-map approach, Nonlinear Anal., 20 (1993), 509-532.  doi: 10.1016/0362-546X(93)90036-R.  Google Scholar

[6]

A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differ. Equ., 200 (2004), 162-184.  doi: 10.1016/j.jde.2004.02.001.  Google Scholar

[7]

A. Fonda, M. Garrione and P. Gidoni, Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal., 5 (2016), 367-382.   Google Scholar

[8]

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

[9]

A. Fonda and M. Ramos, Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities, J. Differ. Equ., 109 (1994), 354-372.  doi: 10.1006/jdeq.1994.1055.  Google Scholar

[10]

A. Fonda, Z. Schneider and F. Zanolin, Periodic oscillations for a nonlinear suspension bridge model, J. Comput. Appl. Math., 52 (1994), 113-140.   Google Scholar

[11]

A. Fonda and A. Sfecci, Multiple periodic solutions of Hamiltonian systems confined in a box, Discrete Contin. Dyn. Syst., 37 (2017), 297-301.  doi: 10.3934/dcds.2017059.  Google Scholar

[12]

A. Fonda and A. Sfecci, Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differ. Integral Equ., 25 (2012), 993-1010.   Google Scholar

[13]

A. Fonda and A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260 (2016), 2150-2162.  doi: 10.1016/j.jde.2015.09.056.  Google Scholar

[14]

A. Fonda and R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc, 140 (2012), 1331-1341.   Google Scholar

[15]

A. Fonda and R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583-602.  doi: 10.1515/anona-2017-0040.  Google Scholar

[16]

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

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M. Garcia-Huidobro, R. Manásevich and F. Zanolin, Strongly nonlinear second-order ODEs with unilateral conditions, Differ. Integral Equ., 8 (1993), 1057-1078.   Google Scholar

[18]

P. Hartman, On boundary value problems for second order differential equations, J. Differ. Equ., 26 (1977), 37-53.  doi: 10.1016/0022-0396(77)90097-3.  Google Scholar

[19]

H. Jacobowitz, Periodic solutions of $x''+f(x, t) = 0$ via the Poincaré-Birkhoff theorem, J. Differ. Equ., 20 (1976), 37-52.  doi: 10.1016/0022-0396(76)90094-2.  Google Scholar

[20]

J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations, in Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Mathematics, 2065, Springer Verlag, 2013. doi: 10.1007/978-3-642-32906-7_3.  Google Scholar

[21]

J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Institute of Mathematical Sciences, New York University, 2005. doi: 10.1090/cln/012.  Google Scholar

[22]

D. Qian, L. Chen and X. Sun, Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differ. Equ., 258 (2015), 3088-3106.  doi: 10.1016/j.jde.2015.01.003.  Google Scholar

[23]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.  doi: 10.1137/S003614100343771X.  Google Scholar

[24]

D. Qian, P. J. Torres and P. Wang, Periodic solutions of second order equations via rotation numbers, J. Differ. Equ., 266 (2019), 4746-4768.  doi: 10.1016/j.jde.2018.10.010.  Google Scholar

[25]

C. Rebelo and F. Zanolin, Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities, Trans. Amer. Math. Soc., 348 (1996), 2349-2389.  doi: 10.1090/S0002-9947-96-01580-2.  Google Scholar

[26]

X. Sun, Q. Liu, D. Qian and N. Zhao, Infinitely many subharmonic solutions for nonlinear equations with singular $\phi$-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 279-292.  doi: 10.3934/cpaa.20200015.  Google Scholar

show all references

References:
[1]

C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular-Laplacian, J. Differ. Equ., 243 (2007), 536-557.  doi: 10.1016/j.jde.2007.05.014.  Google Scholar

[2]

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

[3]

A. Calamai and A. Sfecci, Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations, Nonlinear Differ. Equ. Appl., 24 (2017), 1-17.  doi: 10.1007/s00030-016-0427-5.  Google Scholar

[4]

T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differ. Equ., 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

[5]

T. Ding and F. Zanolin, Subharmonic solutions of second order nonlinear equations: A time-map approach, Nonlinear Anal., 20 (1993), 509-532.  doi: 10.1016/0362-546X(93)90036-R.  Google Scholar

[6]

A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differ. Equ., 200 (2004), 162-184.  doi: 10.1016/j.jde.2004.02.001.  Google Scholar

[7]

A. Fonda, M. Garrione and P. Gidoni, Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal., 5 (2016), 367-382.   Google Scholar

[8]

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

[9]

A. Fonda and M. Ramos, Large-amplitude subharmonic oscillations for scalar second-order differential equations with asymmetric nonlinearities, J. Differ. Equ., 109 (1994), 354-372.  doi: 10.1006/jdeq.1994.1055.  Google Scholar

[10]

A. Fonda, Z. Schneider and F. Zanolin, Periodic oscillations for a nonlinear suspension bridge model, J. Comput. Appl. Math., 52 (1994), 113-140.   Google Scholar

[11]

A. Fonda and A. Sfecci, Multiple periodic solutions of Hamiltonian systems confined in a box, Discrete Contin. Dyn. Syst., 37 (2017), 297-301.  doi: 10.3934/dcds.2017059.  Google Scholar

[12]

A. Fonda and A. Sfecci, Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differ. Integral Equ., 25 (2012), 993-1010.   Google Scholar

[13]

A. Fonda and A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260 (2016), 2150-2162.  doi: 10.1016/j.jde.2015.09.056.  Google Scholar

[14]

A. Fonda and R. Toader, Periodic solutions of radially symmetric perturbations of Newtonian systems, Proc. Amer. Math. Soc, 140 (2012), 1331-1341.   Google Scholar

[15]

A. Fonda and R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583-602.  doi: 10.1515/anona-2017-0040.  Google Scholar

[16]

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

[17]

M. Garcia-Huidobro, R. Manásevich and F. Zanolin, Strongly nonlinear second-order ODEs with unilateral conditions, Differ. Integral Equ., 8 (1993), 1057-1078.   Google Scholar

[18]

P. Hartman, On boundary value problems for second order differential equations, J. Differ. Equ., 26 (1977), 37-53.  doi: 10.1016/0022-0396(77)90097-3.  Google Scholar

[19]

H. Jacobowitz, Periodic solutions of $x''+f(x, t) = 0$ via the Poincaré-Birkhoff theorem, J. Differ. Equ., 20 (1976), 37-52.  doi: 10.1016/0022-0396(76)90094-2.  Google Scholar

[20]

J. Mawhin, Resonance problems for some non-autonomous ordinary differential equations, in Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Mathematics, 2065, Springer Verlag, 2013. doi: 10.1007/978-3-642-32906-7_3.  Google Scholar

[21]

J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Institute of Mathematical Sciences, New York University, 2005. doi: 10.1090/cln/012.  Google Scholar

[22]

D. Qian, L. Chen and X. Sun, Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differ. Equ., 258 (2015), 3088-3106.  doi: 10.1016/j.jde.2015.01.003.  Google Scholar

[23]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.  doi: 10.1137/S003614100343771X.  Google Scholar

[24]

D. Qian, P. J. Torres and P. Wang, Periodic solutions of second order equations via rotation numbers, J. Differ. Equ., 266 (2019), 4746-4768.  doi: 10.1016/j.jde.2018.10.010.  Google Scholar

[25]

C. Rebelo and F. Zanolin, Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities, Trans. Amer. Math. Soc., 348 (1996), 2349-2389.  doi: 10.1090/S0002-9947-96-01580-2.  Google Scholar

[26]

X. Sun, Q. Liu, D. Qian and N. Zhao, Infinitely many subharmonic solutions for nonlinear equations with singular $\phi$-Laplacian, Commun. Pure Appl. Anal., 19 (2020), 279-292.  doi: 10.3934/cpaa.20200015.  Google Scholar

Figure 1.  $ z_k(t)\in \mathcal{D}_{1} $ and $ z_k(t)\in\mathcal{D}_{1}\cup \mathcal{E}_{k, y} $
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Figure 2.  $ z_k(t)\in \mathcal{D}_{2} $
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