Persistent singular attractors arising from singular cycle undersymmetric conditions

Mario E. Chávez-Gordillo; Bernardo San Martín; Jaime Vera

doi:10.3934/dcds.2011.30.671

Article Contents

2011, Volume 30, Issue 3: 671-685. Doi: 10.3934/dcds.2011.30.671

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Persistent singular attractors arising from singular cycle under symmetric conditions

1.
Departamento de Matemáticas, Universidad Católica del Norte, Av. Angamos 0610, Casilla 1280, Antofagasta
2.
The Boeing Company, P.O.Box 3707 MC OX-CC Seattle, WA 98124-2207

Received: April 30, 2010

Revised: October 31, 2010

Published: August 2011

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Abstract

We prove that generic symmetric $C^{r}$-vector field families on $\mathbb{R}^{3}$ unfolding a contracting singular cycle, exhibits singular attractors for a positive lebesgue measure set of parameter values. Essentially the cycle is formed by a real contracting singularity, like those in the geometric contracting Lorenz attractor, whose unstable branches go to periodic orbits in the cycle. We obtain a lower estimate for the density of this set at the first bifurcation value. Furthermore, for parameter values in this set the corresponding vector field admits a unique SRB measure, whose support coincides with the attractor.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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