Effect of noise on the reinjection probability density in intermittency

This paper analyzes the effect of noise in type-II and type-III intermittency, with respective local Poincaré maps of x n + 1 = ( 1 + ε ) x n + ( 1 - ε ) x n 3 and x n + 1 = - ( 1 + ε ) x n - ax n 3 ( a > 0 ). We develop a method to obtain the noisy reinjection probability density (NRPD), which basically consists in extending a recent procedure used to derive the noiseless reinjection probability density (RPD). Our approach also provides information to accurately describe the noiseless system. We also derive the probability density of the laminar length. Our analytical results show a good agreement with numerical simulations. Finally, we have also found that, for large values of the instability parameter ε , the characteristic relations approach the associated ones to the noiseless intermittency. However, for low values of the instability parameter ε, the characteristic relations reach a saturation level that depends on the NRPD.