Transition to a fractal attractor via on-off intermittency
in a model with periodic dichotomous noise
P.V. Kuptsov and S.P. Kuznetsov
Saratov State University, Saratov, Russia
Noise induced phase transitions are a subject of great interest in many fields. Recently, a paradigm for on-off intermittency was declared in this connection [1]. It relates to a situation when dynamics of y variable is governed by multiplicative random influence from chaotic variable x, such as
where f(x) may accept both values of 
and 
. Many difficult
problems in nonlinear dynamics when one can not find a definite
bifurcation in dependence on a control parameter, but rather a finite
bifurcation interval, fit to this paradigm.
We consider a model demonstrating transition via on-off intermittency to Cantor-like fractal attractor. This is the system with discrete time:
with probability 
and 
with probability 
.
For 
we distinguish following regions in the (a, b)-plane:
1) For 
the fractal Cantor-like attractor is realized. At
the curve 
the fractal dimension becomes equal 1.
2) In the domain between the former curve and lines a=1 and b=1 we observe the probability distributions with peaks at "resonances" noted by Kapral et al.[2,3]. The attractor may be also considered as a fractal, but with latent fractal structure that can be revealed using embedding into a higher dimension space.
3) A domain between the lines a=1, b=1 and the curve ab=1. This is a domain for on-off intermittency. The boundary ab=1 corresponds to critical situation when we have equal probability to observe existing oscillations or their full damping over an arbitrary time interval.
The work was supported by Russian Fund of Fundamental Research (grant 96-02-00717).