Local and global stability indices for a riddled basin attractor of a piecewise linear map

ABSTRACT

We consider a piecewise expanding linear map with a Milnor attractor whose basin is riddled with the basin of a second attractor. To characterize the local geometry of this riddled basin, we calculate a stability index for points within the attractor as well as introducing a global stability index for the attractor as a set. Our results show that for Lebesgue almost all points in attractor, the index is positive and we characterize a parameter region, where some points have negative index. We show there exists a dense set of points for which the index is not converge. Comparing to recent results of Keller, we show that the stability index for points in the attractor can be expressed in terms of a global stability index for the attractor and Lyapunov exponents for this point.

KEYWORDS:

• Skew product
• Milnor attractor
• stability index
• riddled basins of attraction

AMS Classification:

• 28A75
• 37A25
• 37B25

Acknowledgments

The work in this paper was developed from ideas in the first author’s doctoral thesis [16] at the University of Exeter, and the first author thanks the Ministry of Education Malaysia and Universiti Malaysia Terengganu for their financial support. We would like to thank Mark Holland, Gerhard Keller, Alex Lohse, Ana Rodrigues and Charles Walkden for their helpful advice and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author would like to thanks the Ministry of Education Malaysia and Universiti Malaysia Terengganu for their financial support.