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On the unfolding of a blowout bifurcation

Ashwin, Peter, Aston, Philip J. and Nicol, Matthew (1998) On the unfolding of a blowout bifurcation Physica D: Nonlinear Phenomena, 111 (1-4). pp. 81-95. ISSN 01672789

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Abstract

<p>Suppose a chaotic attractor <i>A</i> in an invariant subspace loses stability on varying a parameter. At the point of loss of stability, the most positive Lyapunov exponent of the natural measure on <i>A</i> crosses zero at what has been called a ‘blowout’ bifurcation.</p> <p>We introduce the notion of an <i>essential</i> basin of an attractor <i>A</i>. This is the set of points <i>x</i> such that accumulation points of the sequence of measures <img src="http://www.sciencedirect.com/cache/MiamiImageURL/B6TVK-3SYS86V-6-4/0?wchp=dGLbVlz-zSkzV" alt="Image" align="absbottom" border="0" height=22 width=123> are supported on <i>A</i>. We characterise supercritical and subcritical scenarios according to whether the Lebesgue measure of the essential basin of <i>A</i> is positive or zero.</p> <p>We study a drift-diffusion model and a model class of piecewise linear mappings of the plane. In the supercritical case, we find examples where a Lyapunov exponent of the branch of attractors may be positive (‘hyperchaos’) or negative, depending purely on the dynamics far from the invariant subspace. For the mappings we find asymptotically linear scaling of Lyapunov exponents, average distance from the subspace and basin size on varying a parameter. We conjecture that these are general characteristics of blowout bifurcations.</p>

Item Type: Article
Additional Information: This is a pre-copy-editing, author-prepared, peer-reviewed PDF of an article published in Physica D, 111, 81-95. Click here to access the publisher's version. © 1998 Elsevier Science B. V.
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:40
Last Modified: 23 Sep 2013 18:32
URI: http://epubs.surrey.ac.uk/id/eprint/1382

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