WIAS Preprint No. 668, (2001)

The effect of additive noise on dynamical hysteresis


  • Berglund, Nils
  • Gentz, Barbara

2010 Mathematics Subject Classification

  • 37H20 60H10 34C55 34E15 82C31


  • dynamical systems, singular perturbations, hysteresis cycles, scaling laws, non-autonomous stochastic differential equations, double-well potential, pathwise description, concentration of measure


We investigate the properties of hysteresis cycles produced by a one-dimensional, periodically forced Langevin equation. We show that depending on amplitude and frequency of the forcing and on noise intensity, there are three qualitatively different types of hysteresis cycles. Below a critical noise intensity, the random area enclosed by hysteresis cycles is concentrated near the deterministic area, which is different for small and large driving amplitude. Above this threshold, the area of typical hysteresis cycles depends, to leading order, only on the noise intensity. In all three regimes, we derive mathematically rigorous estimates for expectation, variance, and the probability of deviations of the hysteresis area from its typical value.

Appeared in

  • Nonlinearity 15 (2002), 605-632

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