Theory of intermittency applied to classical pathological cases

Abstract

State-by-state assignment of the bending spectrum of acetylene at 15000 cm1: A case study of quantumclassical correspondence J. Chem. Phys. 111, 600 (1999); 10.1063/1.479341 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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. Intermittency is a particular route to the deterministic chaos characterized by spontaneous transitions between laminar and chaotic dynamics. It is observed in a variety of different dynamical systems in Physics, Neuroscience, and Economics. Frequently, there is no feasible mathematical model for the process under study. Then reliable quantification of main characteristics of the intermittent process (e.g., the length of laminar phase) from experimental data is a challenging problem. The classical theory of intermittency has significant pitfalls. Though it works fine in some model systems, there exist a number of so-called pathological cases that deviate significantly from the classical predictions. In this work, we address the problem of unification of anomalous and standard intermittencies under single framework. The unified model can be fitted to experimental or numerical data. We note that to accomplish this step no a priori knowledge is required. We propose a procedure that can cope with reduced data sets consisting of several dozens of points. This makes our methodology useful for real-life applications. Using the experimentally obtained measures, we can classify intermittent processes into different theoretical types. We thoroughly test our method on two particular but canonical case...

Discontinuous Reinjection Probability Density functions in Type V Intermittency

“…In this methodology, the reinjection probability density function is not directly calculated from the data series. An auxiliary function, M(x), is previously evaluated [4,[25][26][27][28][29][30]…”

“…where the data set (reinjection points) fx j g N j¼1 must be sorted from the lowest to the highest, i.e., x j x jþ1 [25][26][27][28][29][30][31][32].…”

“…The M function methodology, developed in the last years, is a broader methodology to evaluate RPD functions. It includes the uniform reinjection as a particular case, and it has shown to be very accurate for a broad class of maps showing type I, II, and III intermittencies [4,[25][26][27][28][29][30][31][32][33][34][35]. This methodology has been recently applied to type V intermittency obtaining continuous and nonuniform RPDs [36].…”

Calculation of the Statistical Properties in Intermittency Using the Natural Invariant Density

“…To calculate this general RPD, the M function methodology was developed, which has worked accurately for maps with types I, II, III and V intermittencies without and with noise. The M function methodology uses a less restrictive assumption of a constant density at points that govern the reinjection process, i.e., pre-reinjection points [3,[26][27][28][29][30][31][32][33][34][35].…”

Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation

“…The method based on the function M{x) has been tested in a wide class of maps exhibiting intermittency: in type-I [25], type-II [22], and type-Ill [23] intermittencies, in type-II and III with presence of noise [24], in classical pathological cases [26]. In all cases M(x) satisfy a linear approximation:…”