New Advances on Chaotic Intermittency and its Applications

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Sergio Elaskar
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Contributes to the state-of-the-art knowledge about chaotic intermittency
Chapter 1. Introduction
to chaotic intermittency

1.1. Basic concepts

1.2. Types of chaotic intermittency

1.3. Classics theory of chaotic intermittency

1.3.1. Type-I Intermittency

1.3.2. Type-II Intermittency

1.3.3. Type-III Intermittency

Chapter 2. Other
types of intermittency

2.1. Type V intermittency

2.2. Type X intermittency

2.3. Types on-off and in-out intermittency

2.4. Type eyelet intermittency

2.5. Type ring intermittency

2.6. Spatio-temporal intermittency

2.7. Crisis-induced intermittency

Chapter 3. Some
recent advances in the study of chaotic Intermittency

3.1. Introduction

3.2. Fine structure in intermittency

3.3. Renormalization and scaling properties of

3.4. Intermittency with LBR

3.5. Two dimensional intermittency

3.6. Horizontal visibility graphs method

3.7. Intermittency cascade

3.8. Multichannel of reinjection

3.9. Experimental evidence in electric circuits

Chapter 4. Some
applications of the chaotic intermittency

4.1. Chaotic intermittency in Engineering

4.2. Chaotic intermittency in Physics

4.3. Chaotic intermittency in Medicine and Economy

4.4. Chaotic intermittency in Neuroscience

Chapter 5.
Classical theory about noise effects in chaotic intermittency

5.1 Type-I intermittency

5.2 Type-II and III intermittency

Chapter 6. New
formulation of the chaotic intermittency

6.1 The
function M

6.2 The
reinjection probability density function
(RPD) evaluation

6.3 Type-II intermittency

6.4 Type-III intermittency

6.5 Type-I intermittency

Chapter 7. New
formulation of the noise effects in chaotic intermittency

7.1 Noisy reinjection probability density
function (NRPD)

7.2 Application to type-II intermittency

7.3 Application to type-III Intermittency

7.4 Application to type-I Intermittency

7.5 Influence of the boundary of reinjection

Chapter 8.
Application of the new formulation to pathological cases

8. 1 Introduction to pathological cases

Laugessen map

Pikovsky map

Chapter 9.
Application to dynamical systems. An example
9.1 Three wave truncation for
the Derivative non-linear Schrödinger equation (DNLS)

9.2 Type-I intermittency in the DNLS

9.3 Influence of the lower bound of reinjection

Chapter 10. Mathematical approximation of the RPD

10.1 The g-expansion

10.2 Critical slopes

10.3 Reinjection probability density function

Chapter 11. Evaluation of the intermittency statistical
properties using the Perron-Frobenius operator

11.1 Measures

11.1 The Perron-Frobenius operator

11.2 Piecewise monotonic maps

11.3 Application to intermittency

Appendix A. Basic
foundations of the probability density function.

A1. Definition of continuous random variables.

A2. Probability
distribution for a continuous random variable.

A3. Probability density function for a
continuous random variable.
One of the most important routes to chaos is the chaotic intermittency. However, there are many cases that do not agree with the classical theoretical predictions. In this book, an extended theory for intermittency in one-dimensional maps is presented. A new general methodology to evaluate the reinjection probability density function (RPD) is developed in Chapters 5 to 8. The key of this formulation is the introduction of a new function, called M(x), which is used to calculate the RPD function. The function M(x) depends on two integrals. This characteristic reduces the influence on the statistical fluctuations in the data series. Also, the function M(x) is easy to evaluate from the data series, even for a small number of numerical or experimental data.
As a result, a more general form for the RPD is found; where the classical theory based on uniform reinjection is recovered as a particular case. The characteristic exponent traditionally used to characterize the intermittency type, is now a function depending on the whole map, not just on the local map. Also, a new analytical approach to obtain the RPD from the mathematical expression of the map is presented. In this way all cases of non standard intermittencies are included in the same frame work.

This methodology is extended to evaluate the noisy reinjection probability density function (NRPD), the noisy probability of the laminar length and the noisy characteristic relation. This is an important difference with respect to the classical approach based on the Fokker-Plank equation or Renormalization Group theory, where the noise effect was usually considered just on the local Poincaré map.

Finally, in Chapter 9, a new scheme to evaluate the RPD function using the Perron-Frobenius operator is developed. Along the book examples of applications are described, which have shown very good agreement with numerical computations.