Minimal $ P $-symmetric periodic solutions in semi-positive definite Hamiltonian systems

Peng Liu; Duanzhi Zhang; Zhihao Zhao

doi:10.3934/dcds.2024029

Article Contents

2024, Volume 44, Issue 8: 2327-2341. Doi: 10.3934/dcds.2024029

This issue Previous Article Optimal control of a stationary Navier-Stokes hemivariational inequality with numerical approximation Next Article Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Minimal $ P $-symmetric periodic solutions in semi-positive definite Hamiltonian systems

1.
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
2.
Great Bay Institute for Advanced Study, Dongguan 523000, China
3.
University of Science and Technology of China, Hefei 230026, China
4.
School of Sciences, Great Bay University, Dongguan 523000, China

^*Corresponding author: Duanzhi Zhang
^*Corresponding author: Duanzhi Zhang

Received: November 2023

Revised: March 2024

Early access: March 21, 2024

Published online: March 21, 2024

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

Let $ P $ be a real symplectic matrix satisfying $ P^k = I_{2n} $, where $ k: = \min\{m\in \mathbf{N}^{\ast}\mid P^m = I_{2n}\} $. In this paper, we study the minimal $ P $-symmetric period problem of autonomous and semi-positive definite Hamiltonian systems in $ \mathbf{R}^{2n} $. Applying the iterative inequality and Hörmander index estimate of $ P $-index, we prove that for every $ \tau>0 $, there exists a non-constant $ P $-solution with minimal $ P $-symmetric period not less than $ \frac{\tau} {2n+1} $.

Keywords:

Mathematics Subject Classification: Primary: 70H05, 34C25; Secondary: 58E05, 53D12, 58J30.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	B. Booss-Bavnbek and C. Zhu, General spectral flow formula for fixed maximal domain, Cent. Eur. J. Math., 3 (2005), 558-577. doi: 10.2478/BF02475923.
[2]	S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186. doi: 10.1002/cpa.3160470202.
[3]	D. Dong and Y. Long, The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems, Trans. Amer. Math. Soc., 349 (1997), 2619-2661. doi: 10.1090/S0002-9947-97-01718-2.
[4]	I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Invent. Math., 81 (1985), 155-188. doi: 10.1007/BF01388776.
[5]	Z. Fan, D. Zhang, Y. Zhou and C. Zhu, Some progress on minimal period problems in reversible semi-positive Hamiltonian systems, J. Differential Equations, 334 (2022), 490-519. doi: 10.1016/j.jde.2022.06.027.
[6]	G. Fei, S.-K. Kim and T. Wang, Minimal period estimates of periodic solutions for superquadratic Hamiltonian systems, J. Math. Anal. Appl., 238 (1999), 216-233. doi: 10.1006/jmaa.1999.6527.
[7]	G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems, Nonlinear Anal., 27 (1996), 821-839. doi: 10.1016/0362-546X(95)00077-9.
[8]	X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit, Comm. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y.
[9]	C. Liu, Maslov $P$-index theory for a symplectic path with applications, Chinese Ann. Math. Ser. B, 27 (2006), 441-458. doi: 10.1007/s11401-004-0365-0.
[10]	C. Liu, Relative index theories and applications, Topol. Methods Nonlinear Anal., 49 (2017), 587-614. doi: 10.12775/tmna.2016.093.
[11]	C. Liu and S. Tang, Iteration inequalities of the Maslov $P$-index theory with applications, Nonlinear Anal., 127 (2015), 215-234. doi: 10.1016/j.na.2015.06.029.
[12]	C. Liu and B. Zhou, Minimal $P$-symmetric period problem of first-order autonomous Hamiltonian systems, Front. Math. China, 12 (2017), 641-654. doi: 10.1007/s11464-017-0627-2.
[13]	Y. Long, Index Theory for Symplectic Paths with Applications, Progr. Math., 207. Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.
[14]	Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${\bf R}^{2n}$, Ann. of Math., 155 (2002), 317-368. doi: 10.2307/3062120.
[15]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.
[16]	D. Zhang, Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems, Sci. China Math., 57 (2014), 81-96. doi: 10.1007/s11425-013-4598-9.
[17]	D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems, Discrete Contin. Dyn. Syst., 35 (2015), 2227-2272. doi: 10.3934/dcds.2015.35.2227.
[18]	X. Zhang and C. Liu, Minimal period estimates on $P$-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems, Front. Math. China, 16 (2021), 239-253. doi: 10.1007/s11464-021-0903-z.
[19]	B. Zhou, C. Liu, Z. Zhou and X. Zhang, Minimal $\mathcal{M}$-symmetric periodic solutions of general Hamiltonian systems and delay differential equations, J. Differential Equations, 329 (2022), 1-30. doi: 10.1016/j.jde.2022.04.039.
[20]	Y. Zhou, L. Wu and C. Zhu, Hörmander index in finite-dimensional case, Front. Math. China, 13 (2018), 725-761. doi: 10.1007/s11464-018-0702-3.
[21]	C. Zhu, A generalized Morse index theorem, Analysis, Geometry and Topology of Elliptic Operators, World Sci. Publ., Hackensack, NJ, (2006), 493-540. doi: 10.1016/s0022-1236(06)00025-5.
[22]	C. Zhu and Y. Long, Maslov-type index theory for symplectic paths and spectral flow. I, Chinese Ann. Math. Ser. B, 20 (1999), 413-424. doi: 10.1142/S0252959999000485.