A model for the nonautonomous Hopf bifurcation. (26th June 2015)

• Record Type: Journal Article
• Authors: Anagnostopoulou, V; Jäger, T; Keller, G
• Is Part Of: Nonlinearity. Volume 28:Number 7(2015:Jul.)
• Page Start: 2587
• Page End: 2616
• Publication Date: 2015-06-26
• Subjects: skew products -- random dynamical systems -- nonautonomous Hopf bifurcation; 39A28 -- 37H20 -- 37A30; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/28/7/2587
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Inspired by an example of Grebogi et al (1984 Physica D13 261–8 ), we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold (1998 Random Dynamical Systems (Berlin: Springer)). The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant 'generalised torus' splitting off a previously stable central manifold after the second bifurcation point. The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measure-preserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.