The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

Thai Son Doan; Martin Rasmussen; Peter E. Kloeden

doi:10.3934/dcdsb.2015.20.875

Article Contents

2015, Volume 20, Issue 3: 875-887. Doi: 10.3934/dcdsb.2015.20.875

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The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

1.
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ
2.
School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074

Received: November 2013

Revised: April 2014

Published: May 2015

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Abstract

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.

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Mathematics Subject Classification: Primary: 37H15, 37H20, 60H10; Secondary: 37H10, 60H30.

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