American Institute of Mathematical Sciences

Advanced
January  2000, 6(1): 243-253. doi: 10.3934/dcds.2000.6.243

Homoclinic points and intersections of Lagrangian submanifold

Zhihong Xia 1,

1. 

Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Received  November 1999 Published  December 1999

In this paper, we prove certain persistence properties of the homoclinic points in Hamiltonian systems and symplectic diffeomorphisms. We show that, under some general conditions, stable and unstable manifolds of hyperbolic periodic points intersect in a very persistent way and we also give some simple criteria for positive topological entropy. The method used is the intersection theory of Lagrangian submanifolds of symplectic manifolds.
Keywords: Lagrangian submanifold., Homoclinic points.
Mathematics Subject Classification: 37G2.
Citation: Zhihong Xia. Homoclinic points and intersections of Lagrangian submanifold. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 243-253. doi: 10.3934/dcds.2000.6.243
[1]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
[2]

Qian Liu, Xinmin Yang, Heung Wing Joseph Lee. On saddle points of a class of augmented lagrangian functions. Journal of Industrial and Management Optimization, 2007, 3 (4) : 693-700. doi: 10.3934/jimo.2007.3.693
[3]

Boris Buffoni, Laurent Landry. Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 75-116. doi: 10.3934/dcds.2010.27.75
[4]

Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429
[5]

Frederic Gabern, Àngel Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 143-182. doi: 10.3934/dcdsb.2001.1.143
[6]

Xiao Wen. Structurally stable homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1693-1707. doi: 10.3934/dcds.2016.36.1693
[7]

Victoria Rayskin. Homoclinic tangencies in $R^n$. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 465-480. doi: 10.3934/dcds.2005.12.465
[8]

Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143
[9]

Wolf-Jürgen Beyn, Thorsten Hüls. Continuation and collapse of homoclinic tangles. Journal of Computational Dynamics, 2014, 1 (1) : 71-109. doi: 10.3934/jcd.2014.1.71
[10]

Patrick Henning, Anders M. N. Niklasson. Shadow Lagrangian dynamics for superfluidity. Kinetic and Related Models, 2021, 14 (2) : 303-321. doi: 10.3934/krm.2021006
[11]

Adrian Constantin. Solitons from the Lagrangian perspective. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 469-481. doi: 10.3934/dcds.2007.19.469
[12]

Andrew James Bruce, Katarzyna Grabowska, Giovanni Moreno. On a geometric framework for Lagrangian supermechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 411-437. doi: 10.3934/jgm.2017016
[13]

Xingbo Liu, Deming Zhu. On the stability of homoclinic loops with higher dimension. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 915-932. doi: 10.3934/dcdsb.2012.17.915
[14]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Unfolding globally resonant homoclinic tangencies. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022043
[15]

Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615
[16]

Keith Promislow, Hang Zhang. Critical points of functionalized Lagrangians. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1231-1246. doi: 10.3934/dcds.2013.33.1231
[17]

Paula Kemp. Fixed points and complete lattices. Conference Publications, 2007, 2007 (Special) : 568-572. doi: 10.3934/proc.2007.2007.568
[18]

K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 55-62.

[19]

Jorge Rebaza. Uniformly distributed points on the sphere. Communications on Pure and Applied Analysis, 2005, 4 (2) : 389-403. doi: 10.3934/cpaa.2005.4.389
[20]

Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (203)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy