On the unfolding of a blowout bifurcation

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

Subexponential instability in one-dimensional maps implies infinite invariant measure

“…This conjecture is valid in the drift-diffusion model. 29 Moreover, the transformation Tx =2x with probability 1/2 and Tx = x / 2 with probability 1/2 has an infinite invariant measure absolutely continuous with respect to the Lebesgue measure. We conjecture that a nontrivial zero Lyapunov exponent would imply infinite invariant measure.…”

On statistical attractors and the convergence of time averages

“…In this paper, we show, in Theorem 2·3, that the statistical attractor can be defined using the notion of essential ω-limit set of trajectories previously defined in [2]. We also examine the convergence (or otherwise) of (Birkhoff) time averages of observables along trajectories.…”

Milnor attractors and topological attractors of a piecewise linear map

“…These exotic objects arise naturally as part of a blowout bifurcation: as a parameter is varied we may imagine that some orbits of the synchronized attractor lose stability in transverse directions although typical synchronized orbits remain transversally stable. At this stage the synchronized state is no longer Liapunov stable, although it is possible that almost all orbits are eventually attracted to the synchronized state (the precise details depend on global features of the system [2,3,23]) which is called a Milnor attractor. As a parameter is varied further the typical orbits in the synchronized state may lose transverse stability in a blowout bifurcation, leading to dynamics with intermittent characteristics { orbits spend a long time close to the synchronized state interspersed with larger uctuations away from the synchronized state.…”

“…decreases from 1 2 it is possible to determine the parameter at which the rst synchronized orbit loses transverse stability (the point at which the synchronized state as a whole loses asymptotic stability), the point at which`typical' trajectories lose transverse stability (the blowout bifurcation point in the language of [3,8]) and nally the point beyond which all synchronized states are transversely unstable. For the range of values of a considered below, there is on-o intermittency immediately after the blowout bifurcation [1,2,3,19,24,25,26], and between the loss of asymptotic stability and the blowout bifurcation the synchronized state is a Milnor attractor (i.e. it attracts almost all initial conditions locally in a measure theoretic sense [1,3,4,8,17,21,25]).…”