American Institute of Mathematical Sciences

Advanced
January  2010, 26(1): 1-42. doi: 10.3934/dcds.2010.26.1

Generation of homoclinic tangencies by $C^1$-perturbations

Nikolaz Gourmelon 1,

1. 

Université Bordeaux 1, 351, cours de la Libération, F 33405 TALENCE cedex, France

Received  January 2008 Revised  September 2009 Published  October 2009

Given a $C^1$-diffeomorphism $f$ of a compact manifold, we show that if the stable/unstable dominated splitting along a saddle is weak enough, then there is a small $C^1$-perturbation that preserves the orbit of the saddle and that generates a homoclinic tangency related to it. Moreover, we show that the perturbation can be performed preserving a homoclinic relation to another saddle. We derive some consequences on homoclinic classes. In particular, if the homoclinic class of a saddle $P$ has no dominated splitting of same index as $P$, then a $C^1$-perturbation generates a homoclinic tangency related to $P$.
Keywords: homoclinic tangency, homoclinic class., bifurcation, Dominated splitting.
Mathematics Subject Classification: Primary: 37C20, 37C29; Secondary: 37G25, 37D3.
Citation: Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1
[1]

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
[2]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[3]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1
[4]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[5]

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
[6]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579
[7]

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
[8]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[9]

Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020
[10]

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
[11]

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
[12]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure and Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269
[13]

Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022
[14]

Karsten Matthies. Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 585-602. doi: 10.3934/dcds.2003.9.585
[15]

Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757
[16]

Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555
[17]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[18]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
[19]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, 2021, 29 (5) : 2987-3015. doi: 10.3934/era.2021023
[20]

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

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (125)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy