Chaotic intermittency is characterized by a signal that alternates aleatory between long regular (pseudo-laminar) phases and irregular bursts (pseudo-turbulent or chaotic phases). This phenomenon has been found in physics, chemistry, engineering, medicine, neuroscience, economy, etc. As a control parameter increases, the number of chaotic phases also increases. Therefore, intermittency presents a continuous route from regular behavior to chaotic motion. In this paper, a review of different types of intermittency is carried out. In addition, the description of two recent formulations to evaluate the reinjection processes is developed. The new theoretical formulations have allowed us to explain several tests previously called pathological. The theoretical background also includes the noise effects in the reinjection mechanism.

The classical theory of intermittency developed for return maps assumes uniform density of points reinjected from the chaotic to laminar region. Though it works fine in some model systems, there exist a number of so-called pathological cases characterized by a significant deviation of main characteristics from the values predicted on the basis of the uniform distribution. Recently, we reported on how the reinjection probability density (RPD) can be generalized. Here, we extend this methodology and apply it to different dynamical systems exhibiting anomalous type-II and type-III intermittencies. Estimation of the universal RPD is based on fitting a linear function to experimental data and requires no a priori knowledge on the dynamical model behind. We provide special fitting procedure that enables robust estimation of the RPD from relatively short data sets (dozens of points). Thus, the method is applicable for a wide variety of data sets including numerical simulations and real-life experiments. Estimated RPD enables analytic evaluation of the length of the laminar phase of intermittent behaviors. We show that the method copes well with dynamical systems exhibiting significantly different statistics reported in the literature. We also derive and classify characteristic relations between the mean laminar length and main controlling parameter in perfect agreement with data provided by numerical simulations. (C) 2013 AIP Publishing LLC.

This paper reports theoretical and numerical results about the reinjection process in type V intermittency. The M function methodology is applied to a simple mathematical model to evaluate the reinjection process through the reinjection probability density function (RPD), the probability density of laminar lengths, and the characteristic relation. We have found that the RPD can be a discontinuous function and it is a sum of exponential functions. The RPD shows two reinjection behaviors. Also, the probability density of laminar lengths has two different behaviors following the RPD function. The dependence of the RPD function and the probability density of laminar lengths with the reinjection mechanisms and the lower boundary of return are considered. On the other hand, we have obtained, for the analyzed map, that the characteristic relation verifies l¯≈ε−0.5. Finally, we highlight that the M function methodology is a suitable tool to analyze type V intermittency and there is a very high accuracy between the new theoretical equations and the numerical data.