September  2003, 9(5): 1133-1148. doi: 10.3934/dcds.2003.9.1133

Homoclinic bifurcations, fat attractors and invariant curves

1. 

Departamento de Matemática, Facultad de Ciencias, La Hechicera, Mérida, 5101, Venezuela

Received  October 2000 Revised  November 2002 Published  June 2003

Here we show examples of homoclinic bifurcations which can be perturbed to produce invariant curves and attractors with high Hausdorff dimension.
Citation: Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133
[1]

G. A. Leonov. Generalized Lorenz Equations for Acoustic-Gravity Waves in the Atmosphere. Attractors Dimension, Convergence and Homoclinic Trajectories. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2253-2267. doi: 10.3934/cpaa.2017111

[2]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139

[3]

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

[4]

Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693

[5]

Jordi-Lluís Figueras, Àlex Haro. A note on the fractalization of saddle invariant curves in quasiperiodic systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1095-1107. doi: 10.3934/dcdss.2016043

[6]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006

[7]

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

[8]

Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281

[9]

Sergey Gonchenko, Ivan Ovsyannikov. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 273-288. doi: 10.3934/dcdss.2017013

[10]

Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179

[11]

Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335

[12]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021

[13]

Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069

[14]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[15]

Katsutoshi Shinohara. On the index problem of $C^1$-generic wild homoclinic classes in dimension three. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 913-940. doi: 10.3934/dcds.2011.31.913

[16]

Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699

[17]

Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755

[18]

Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287

[19]

V. Afraimovich, T.R. Young. Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 691-704. doi: 10.3934/dcds.2000.6.691

[20]

Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]