Intermittency and Jakobson's theorem near saddle-node bifurcations

Ale Jan Homburg; Todd Young

doi:10.3934/dcds.2007.17.21

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2007, Volume 17, Issue 1: 21-58. Doi: 10.3934/dcds.2007.17.21

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Intermittency and Jakobson's theorem near saddle-node bifurcations

Ale Jan Homburg^1, and
Todd Young^2,

1.
KdV Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam
2.
Department of Mathematics, Ohio University, Athens, OH 45701

Received: December 31, 2004

Revised: May 31, 2006

Published: December 2006

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Abstract

We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to an atomic measure supported on the orbit of the saddle-node point. Using these measures we analyze the intermittent time series that result from the destruction of the periodic attractor in the saddle-node bifurcation and prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the periodic attractor.

Keywords:

absolutely continuous invariant measure,
saddle node bifurcation,
nonuniform hyperbolicity.

Mathematics Subject Classification: Primary: 37E05; Secondary: 37G10, 37D25.

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