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Preface
Author(s) -
Hori Yoshiaki
Publication year - 1994
Publication title -
the journal of dermatology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.9
H-Index - 65
eISSN - 1346-8138
pISSN - 0385-2407
DOI - 10.1111/j.1346-8138.1994.tb03291.x
Subject(s) - citation , library science , medicine , computer science
From the pioneer works of H. Poincaré to present days, the nonlinear science has developed tremendously. In particular, chaos theory presents nowadays a number of fields of investigation. One of such fields is the so-called routes to chaos, where an interesting one is the chaotic intermittency, because this phenomenon has been observed in many different fields. The intermittency theory coming from the early 1980s was based on a strong hypothesis on the reinjection probability density function (RPD). From early times, “pathology cases” were found, that is, systems showing chaotic intermittency with statistic properties not fully explained basing on the classical theory. These cases demand a broader intermittency theory looking for a new and more general RPD function. By means of the Poincaré map, many continuous systems can be investigated by one-dimensional maps. In this book, new methodologies to investigate chaotic intermittency in one-dimensional maps are presented. A new general methodology to evaluate the RPD is developed. The core of this formulation is a new function, called M.x/, which is very useful to calculate the RPD function, even for a small number of numerical or experimental data. The M.x/ function is defined by means of integrals; hence the influence on the statistical fluctuations in the data series is reduced. As a result, a more general form for the RPD is found. By including the new RPD in the classical mathematical formulation of chaotic intermittency, new results have been obtained. For instance, 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. In this new framework, the classical theory is recovered as a particular case. Even more, the pathology cases are included in a natural way in the new theory. Also, we present a new analytical approach to obtain the RPD from the mathematical expression of the map. In this new framework, the noise effect on the system is evaluated by means of the analytical derivation of the noisy RPD (NRPD). This is an important difference with respect to the classical approach based on the Fokker–Planck equation or