American Institute of Mathematical Sciences

Advanced
April  2006, 14(2): 343-354. doi: 10.3934/dcds.2006.14.343

Heteroclinic orbits and rotation sets for twist maps

Héctor E. Lomelí 1,

1. 

Department of Mathematics, Instituto Tecnológico Autónomo de México, Río Hondo # 1., Tizapán San Angel, México DF 01000, Mexico

Received  November 2004 Revised  May 2005 Published  November 2005

We show how the shadowing property can be used in connection to rotation sets. We review the concept of periodic chains and the shadowing rotation property, and study a class of diffeomorphisms with invariant sets that have such property. In particular, we consider invariant sets that arise from homoclinic and heteroclinic connections for twist maps in higher dimensions. As a consequence, we can show the existence of a family of twist maps, each one with an open set of rotation vectors that are realized by points that are close to a fixed point.
Keywords: heteroclinic orbits, Rotation set, Melnikov method, twist maps, shadowing., Farey.
Mathematics Subject Classification: Primary: 37E45, 37E40 Secondary: 37C50, 37J4.
Citation: Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343
[1]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
[2]

Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255
[3]

Sergey V. Bolotin. Shadowing chains of collision orbits. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 235-260. doi: 10.3934/dcds.2006.14.235
[4]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631
[5]

John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047
[6]

Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629
[7]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617
[8]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[9]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097
[10]

Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79
[11]

Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 2017, 4 (1&2) : 119-141. doi: 10.3934/jcd.2017004
[12]

Kazuyuki Yagasaki. Application of the subharmonic Melnikov method to piecewise-smooth systems. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2189-2209. doi: 10.3934/dcds.2013.33.2189
[13]

Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039
[14]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
[15]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[16]

Miguel Mendes. A note on the coding of orbits in certain discontinuous maps. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 369-382. doi: 10.3934/dcds.2010.27.369
[17]

Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399
[18]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[19]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[20]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences