The existence of periodic solutions for nonconservative superlinear second order ODEs: a rotation number and spiral analysis approach

Shuang Wang; FanFan Chen; Chunlian Liu; Shuang Wang; FanFan Chen; Chunlian Liu

doi:10.3934/era.2025003

Electronic Research Archive

2025, Volume 33, Issue 1: 50-67. doi: 10.3934/era.2025003

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Research article

The existence of periodic solutions for nonconservative superlinear second order ODEs: a rotation number and spiral analysis approach

1.
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224051, China
2.
School of Science, Zhejiang Sci-Tech University, Hangzhou 310000, China
3.
School of Mathematics and Statistics, Nantong University, Nantong 226019, China

Received: 30 September 2024 Revised: 03 December 2024 Accepted: 05 December 2024 Published: 06 January 2025

We investigate the existence of periodic solutions for nonconservative superlinear second-order differential equations in the sense of rotation numbers. Specifically, we focus on equations whose solutions at infinity behave comparably to a suitable linear system. By employing a rotation number approach, spiral analysis, and fixed-point theorems, we establish the existence of periodic solutions for nonconservative superlinear second-order differential equations. Among the equations we consider, a notable subclass is partially superlinear second-order differential equations, which provide a concrete illustration of our results. Our results extend several recent results, thereby advancing to a more comprehensive understanding of periodic behavior in nonconservative systems.
- periodic solutions,
- nonconservative,
- superlinear,
- rotation numbers,
- spiral analysis
Citation: Shuang Wang, FanFan Chen, Chunlian Liu. The existence of periodic solutions for nonconservative superlinear second order ODEs: a rotation number and spiral analysis approach[J]. Electronic Research Archive, 2025, 33(1): 50-67. doi: 10.3934/era.2025003

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Abstract

We investigate the existence of periodic solutions for nonconservative superlinear second-order differential equations in the sense of rotation numbers. Specifically, we focus on equations whose solutions at infinity behave comparably to a suitable linear system. By employing a rotation number approach, spiral analysis, and fixed-point theorems, we establish the existence of periodic solutions for nonconservative superlinear second-order differential equations. Among the equations we consider, a notable subclass is partially superlinear second-order differential equations, which provide a concrete illustration of our results. Our results extend several recent results, thereby advancing to a more comprehensive understanding of periodic behavior in nonconservative systems.

References

[1]	P. Gidoni, Existence of a periodic solution for superlinear second order ODEs, J. Differ. Equations, 345 (2023), 401–417. https://doi.org/10.1016/j.jde.2022.11.054 doi: 10.1016/j.jde.2022.11.054
[2]	M. Struwe, Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order, J. Differ. Equations, 37 (1980), 285–295. https://doi.org/10.1016/0022-0396(80)90099-6 doi: 10.1016/0022-0396(80)90099-6
[3]	A. Capietto, J. Mawhin, F. Zanolin, A continuation approach to superlinear periodic boundary value problems, J. Differ. Equations, 88 (1990), 347–395. https://doi.org/10.1016/0022-0396(90)90102-U doi: 10.1016/0022-0396(90)90102-U
[4]	S. Wang, F. Chen, D. Qian, The existence of periodic solution for superlinear second order ODEs by a new fixed point approach, J. Differ. Equations, 410 (2024), 481–512. https://doi.org/10.1016/j.jde.2024.07.031 doi: 10.1016/j.jde.2024.07.031
[5]	D. Qian, P. J. Torres, P. Wang, Periodic solutions of second order equations via rotation numbers, J. Differ. Equations, 266 (2019), 4746–4768. https://doi.org/10.1016/j.jde.2018.10.010 doi: 10.1016/j.jde.2018.10.010
[6]	S. Wang, D. Qian, Subharmonic solutions of indefinite Hamiltonian systems via rotation numbers, Adv. Nonlinear Stud., 21 (2021), 557–578. https://doi.org/10.1515/ans-2021-2134 doi: 10.1515/ans-2021-2134
[7]	S. Wang, C. Liu, Periodic solutions of superlinear planar Hamiltonian systems with indefinite term, J. Appl. Anal. Comput., 13 (2023), 2542–2554. http://dx.doi.org/10.11948/20220426 doi: 10.11948/20220426
[8]	S. Wang, Periodic solutions of weakly coupled superlinear systems with indefinite terms, Nonlinear Differ. Equations Appl. NoDEA, 29 (2022), 1–22. https://doi.org/10.1007/s00030-022-00768-1 doi: 10.1007/s00030-022-00768-1
[9]	C. Liu, D. Qian, P. J. Torres, Non-resonance and double resonance for a planar system via rotation numbers, Results Math., 76 (2021), 1–23. https://doi.org/10.1007/s00025-021-01401-w doi: 10.1007/s00025-021-01401-w
[10]	M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc., 64 (2001), 125–143. https://doi.org/10.1017/S0024610701002277 doi: 10.1017/S0024610701002277
[11]	J. Chu, G. Meng, Z. Zhang, Continuous dependence and estimates of eigenvalues for periodic generalized Camassa-Holm equations, J. Differ. Equations, 269 (2020), 6343–6358. https://doi.org/10.1016/j.jde.2020.04.042 doi: 10.1016/j.jde.2020.04.042
[12]	P. Yan, M. Zhang, Rotation number, periodic Fu$\breve{\rm{c}}$ik spectrum and multiple periodic solutions, Commun. Contemp. Math., 12 (2010), 437–455. https://doi.org/10.1142/S0219199710003877 doi: 10.1142/S0219199710003877
[13]	J. Chu, M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21 (2008), 1071–1094. https://doi.org/10.3934/dcds.2008.21.1071 doi: 10.3934/dcds.2008.21.1071
[14]	J. Li, Y. Cheng, Barycentric rational interpolation method for solving time-dependent fractional convection-diffusion equation, Electron. Res. Arch., 31 (2023), 4034–4056. https://doi.org/10.3934/era.2023205 doi: 10.3934/era.2023205
[15]	J. Li, Barycentric rational collocation method for fractional reaction-difusion equation, AIMS Math., 8 (2023), 9009–9026. https://doi.org/10.3934/math.2023451 doi: 10.3934/math.2023451
[16]	J. Li, X. Su, K. Zhao, Barycentric interpolation collocation algorithm to solve fractional differential equations, Math. Comput. Simul., 205 (2023), 340–367. https://doi.org/10.1016/j.matcom.2022.10.005 doi: 10.1016/j.matcom.2022.10.005
[17]	J. K. Hale, Ordinary Differential equations, (eds. R. E. Krieger and P. Co. Huntington), New York, 1980.
[18]	V. Lakshmikantham, S. Leela, Differential and Integral Inequalities: Theory and Applications, Academic Press, New York and London, 1969.

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