American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space
  • DCDS Home
  • This Issue
  • Next Article
    Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach
November  2008, 21(4): 1071-1094. doi: 10.3934/dcds.2008.21.1071

Rotation numbers and Lyapunov stability of elliptic periodic solutions

Jifeng Chu 1, and  Meirong Zhang 2,

1. 

College of Science, Hohai University, Nanjing 210098, China
2. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received  March 2007 Revised  March 2008 Published  May 2008

Using the relation between the Hill's equations and the Ermakov-Pinney equations established by Zhang [27], we will give some interesting lower bounds of rotation numbers of Hill's equations. Based on the Birkhoff normal forms and the Moser twist theorem, we will prove that two classes of nonlinear, scalar, time-periodic, Newtonian equations will have twist periodic solutions, one class being regular and another class being singular.
Keywords: Ermakov-Pinney equation, twist coefficient., periodic solution, Lyapunov stability, rotation number, Hill's equation.
Mathematics Subject Classification: Primary 37J25; Secondary 34D20, 37E40, 34C2.
Citation: Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071
[1]

Özlem Orhan, Teoman Özer. New conservation forms and Lie algebras of Ermakov-Pinney equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 735-746. doi: 10.3934/dcdss.2018046
[2]

Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303
[3]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631
[4]

Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255
[5]

Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793
[6]

Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305
[7]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[8]

Szandra Beretka, Gabriella Vas. Stable periodic solutions for Nazarenko's equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3257-3281. doi: 10.3934/cpaa.2020144
[9]

Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173
[10]

Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763
[11]

Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007
[12]

Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823
[13]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[14]

Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
[15]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343
[16]

Jiann-Sheng Jiang, Kung-Hwang Kuo, Chi-Kun Lin. Homogenization of second order equation with spatial dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 303-313. doi: 10.3934/dcds.2005.12.303
[17]

Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719
[18]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455
[19]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[20]

Jaime Angulo Pava, Borys Alvarez Samaniego. Existence and stability of periodic travelling-wavesolutions of the Benjamin equation. Communications on Pure & Applied Analysis, 2005, 4 (2) : 367-388. doi: 10.3934/cpaa.2005.4.367

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences