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March  2007, 18(2&3): 597-611. doi: 10.3934/dcds.2007.18.597

Global attractor and rotation number of a class of nonlinear noisy oscillators

Wenxian Shen 1,

1. 

Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310, United States

Received  March 2006 Revised  August 2006 Published  March 2007

The current paper is concerned with the global dynamics of a class of nonlinear oscillators driven by real or white noises, of which a typical example is a shunted Josephson junction exposed to some random medium. Applying random dynamical systems theory, it is shown that a driven oscillator in the class under consideration with a tempered real noise has a one-dimensional global random attractor provided that the damping is not too small. Moreover, restricted to the global attractor, the oscillator induces a random dynamical system on $S^1$. It is then shown that there is a rotation number associated to the oscillator which characterizes the speed at which the solutions of the oscillator move around the global attractor. The results extend the existing ones for time periodic and quasi-periodic Josephson junctions and can be applied to Josephson junctions driven by white noises.
Keywords: tempered function, global random attractor, invariant measure, ergodic theorem, random fixed point, real or white noise, random dynamical system, rotation number., Josephson junction.
Mathematics Subject Classification: 34C15, 34F05, 34D45,37H10, 60H1.
Citation: Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597
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