Fully Chaotic Maps and Broken Time Symmetry

Abstract

phenomena and complex systems, which is one of the most fascinating fields of science today, to publish books that cover the essential concepts in this area, as well as the latest developments. As the number of scientists involved in the subject increases continually, so does the number of new questions and results. Nonlinear effects are essential to understand the behaviour of nature, and the methods and ideas introduced to treat them are increasingly used in new applications to a variety of problems ranging from physics to human sciences. Most of the books in this series will be about physical and mathematical aspects of nonlinear science, since A C.I.P. CataIogue record for this book is available from the Library of Congress.

Looking at the Dimension of Time among Science, Psychology and Everyday Reality

“…This is a well-known situation in sociology: the behaviour of a population cannot be attributed to the behaviour of individuals. The same appears to be true for a lot of important classes of systems such as the thermodynamic systems and the interacting fields [18][19][20].…”

Non-Poisson processes: regression to equilibrium versus equilibrium correlation functions

“…The adoption of the more conventional reduced map of Ref. [17] would have made it possible for us to adopt the elegant prescriptions of Driebe [19], as done in Ref. [22].…”

“…In the ordinary Fokker-Planck approach the role of the stochastic force is played by a second-order derivative, which is not as unusual as the integral operator of Eq. (19). Here the role of randomness is played by the back-injection process, which, from within the density perspective is described by an operator that selects from all possible values pðy; tÞ; the specific value of pðy; tÞ at y ¼ 1: The idealization that we have adopted, of reducing the size of the chaotic region to zero, with the choice of the process of back injection located at y ¼ 1; makes it possible for us to use the continuous time representation and the equation of motion Eq.…”

Ensemble dynamics of chaos

“…It has been recognized since the first constructions of generalized spectral decompositions for simple maps that “fractal properties … invade the treatment” 7. The properties were derived in a number of ways 2, 7, 8. As a rule, the generalized eigenstates, which are functionals, involve a Riemann–Stieltjes integration over a function with fractal properties that turns out to be associated with families of singular invariant measures of the system.…”