February  2010, 27(1): 285-300. doi: 10.3934/dcds.2010.27.285

Heterodimensional tangencies on cycles leading to strange attractors

1. 

Department of Mathematics, Kyoto University of Education, Fushimi, Kyoto 612-8522, Japan

2. 

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan, Japan

Received  December 2008 Revised  October 2009 Published  February 2010

In this paper, we study a two-parameter family $\{\varphi_{\mu,\nu}\}$ of three-dimensional diffeomorphisms which have a bifurcation induced by simultaneous generation of a heterodimensional cycle and a heterodimensional tangency associated to two saddle points. We show that such a codimension-$2$ bifurcation generates a quadratic homoclinic tangency associated to one of the saddle continuations which unfolds generically with respect to some one-parameter subfamily of $\{\varphi_{\mu,\nu}\}$. Moreover, from this result together with some well-known facts, we detect some nonhyperbolic phenomena (i.e., the existence of nonhyperbolic strange attractors and the $C^{2}$ robust tangencies) arbitrarily close to the codimension-$2$ bifurcation.
Citation: Shin Kiriki, Yusuke Nishizawa, Teruhiko Soma. Heterodimensional tangencies on cycles leading to strange attractors. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 285-300. doi: 10.3934/dcds.2010.27.285
[1]

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

[2]

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

[3]

Antonio Pumariño, Joan Carles Tatjer. Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 971-1005. doi: 10.3934/dcdsb.2007.8.971

[4]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1

[5]

Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765

[6]

Lorenzo J. Díaz, Jorge Rocha. How do hyperbolic homoclinic classes collide at heterodimensional cycles?. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 589-627. doi: 10.3934/dcds.2007.17.589

[7]

Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko, J. Tomás Lázaro. Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4483-4507. doi: 10.3934/dcds.2018196

[8]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295

[9]

Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934/dcdss.2011.4.1511

[10]

Àlex Haro. On strange attractors in a class of pinched skew products. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 605-617. doi: 10.3934/dcds.2012.32.605

[11]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalizable Expanding Baker Maps: Coexistence of strange attractors. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1651-1678. doi: 10.3934/dcds.2017068

[12]

Yongluo Cao, Stefano Luzzatto, Isabel Rios. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61

[13]

Dongchen Li, Dmitry V. Turaev. Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4399-4437. doi: 10.3934/dcds.2017189

[14]

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

[15]

Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939

[16]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 941-966. doi: 10.3934/dcds.2018040

[17]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 657-670. doi: 10.3934/dcdsb.2018201

[18]

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

[19]

Maurizio Grasselli, Hao Wu. Robust exponential attractors for the modified phase-field crystal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2539-2564. doi: 10.3934/dcds.2015.35.2539

[20]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]