New Formulation for Chaotic Intermittency with Noise

Abstract

The proper description of turbulent flows presents difficulty for researchers in fluid mechanics.
One feature of some of these flows is intermittency. The intermittency phenomenon in Chaotic Dynamics theory is understood as a specific route to the deterministic chaos when spontaneous transitions between laminar and chaotic dynamics occur. A correct characterization of the intermittency is very important, principally, to study those problems having partially unknown governing equations or there are experimental or numerical data series. In this paper a new methodology to investigate systems showing chaotic intermittency phenomenon with noise is presented. The methodology used for system without noise is extended to evaluate the noisy reinjection probability density (NRPD), the noisy probability of the laminar length and the noisy characteristic relation. The approach also provides information to accurately describe the noiseless system. It also was found that, for type-II and type-III intermittencies, for large values of the instability parameter the characteristic relations approach the associated ones to the noiseless intermittency. However, for low values of the instability parameter, the characteristic relations reach a saturation level that depends on the NRPD. Also, this new methodology does not need to satisfy noise strength lower that the control parameter, it allows to analyze the noise effect on the intermittency statistical properties for large noise strengths. However, in few cases for type-I intermittency, the description of the noiseless system using the noisy data can be inaccurate. In addition, it is shown that occasionally the presence of noise could be not detected due to the results behave as they would be corresponding to a noiseless system. This aspect may have important consequences especially when working with experimental data. To validate the new theoretical formulation, the analytical results are tested by several numerical computations, showing an excellent agreement between the analytical models and the numerical results.