• Previous Article
    A class of doubly degenerate parabolic equations with periodic sources
  • DCDS-B Home
  • This Issue
  • Next Article
    An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches
October  2010, 14(3): 1181-1197. doi: 10.3934/dcdsb.2010.14.1181

Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems

1. 

Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan

2. 

Department of Applied Science, Naval Academy, Zuoying District, Kaohsiung City, 813, Taiwan

Received  August 2009 Revised  April 2010 Published  July 2010

A rigorous numerical proof for establishing existence of a transversal homoclinic orbit for a saddle fixed point with higher dimensional unstable eigenspaces is presented. As the first component of this method, a shadowing theorem that guarantees the existence of such a homoclinic orbit near a suitable pseudo orbit given the invertibility of a certain Jacobian is proved. The second component consists of a refinement procedure for numerically computing a pseudo homoclinic orbit with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component verifies that the homoclinic orbit is transversal. In [6], they proved the existence of transversal homoclinic orbits near anti-integrable limits and near singularities for the Arneodo-Coullet-Tresser maps. In this paper, the existence of transversal homoclinic orbits were proved far away from anti-integrable limits and singularities for these maps.
Citation: Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181
[1]

Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367

[2]

Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289

[3]

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

[4]

Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205

[5]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210

[6]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128

[7]

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

[8]

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

[9]

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

[10]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778

[11]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203

[12]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

[13]

Jianquan Li, Yanni Xiao, Yali Yang. Global analysis of a simple parasite-host model with homoclinic orbits. Mathematical Biosciences & Engineering, 2012, 9 (4) : 767-784. doi: 10.3934/mbe.2012.9.767

[14]

Yingxiang Xu, Yongkui Zou. Preservation of homoclinic orbits under discretization of delay differential equations. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 275-299. doi: 10.3934/dcds.2011.31.275

[15]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851

[16]

Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589

[17]

W.R. Derrick, P. van den Driessche. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 299-309. doi: 10.3934/dcdsb.2003.3.299

[18]

Flaviano Battelli, Ken Palmer. A remark about Sil'nikov saddle-focus homoclinic orbits. Communications on Pure and Applied Analysis, 2011, 10 (3) : 817-830. doi: 10.3934/cpaa.2011.10.817

[19]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[20]

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

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]