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Noisy one-dimensional maps near a crisis. I. Weak Gaussian white and colored noise

Abstract

We study one-dimensional single-humped maps near the boundary crisis at fully developed chaos in the presence of additive weak Gaussian white noise. By means of a new perturbation-like method the quasi-invariant density is calculated from the invariant density at the crisis in the absence of noise. In the precritical regime, where the deterministic map may show periodic windows, a necessary and sufficient condition for the validity of this method is derived. From the quasi-invariant density we determine the escape rate, which has the form of a scaling law and compares excellently with results from numerical simulations. We find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz>1. Finally, we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak Gaussian noise. Within this extended class of noises and for single-humped maps with any fixed orderz>0 of the maximum, in the scaling law for the escape rate both the critical exponents and the scaling function are universal.

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Reimann, P. Noisy one-dimensional maps near a crisis. I. Weak Gaussian white and colored noise. J Stat Phys 82, 1467–1501 (1996). https://doi.org/10.1007/BF02183392

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Key Words

  • Noisy map
  • crisis
  • escape rate
  • scaling and universality
  • invariant density
  • transient chaos
  • colored noise