Chaos Science

Complexity Science and Chaos Theory are fascinating areas of scientific research with wide-ranging applications. The interdisciplinary nature and ubiquity of complexity and chaos are features that provides scientists with a motivation to pursue general theoretical tools and frameworks. Complex systems give rise to emergent behaviors, which in turn produce novel and interesting phenomena in science, engineering, as well as in the socio-economic sciences.

In this book we summarize the essential results of our efforts over the years to calculate energies, wave functions, and electromagnetic transitions of atoms as functions of the magnetic field strength from laboratory fields up to neutron star magnetic fields. Motivated by the observational evidence of huge magnetic 5 fields with strengths up to 10 T in the vicinity of white dwarf stars and of up 9 to 10 T in the vicinity of neutron stars the authors, together with coworkers and candidates for doctor and diploma degrees, have investigated this ,fasci­ nating quantum mechanical problem more or less continuously since 1978. The extensive tables and figures in the appendices represent the most complete data set to date in this field of research. For practical use all numbers are available by "anonymous ftp" over Internet. The first direct measurement of a neutron star magnetic field by Trum­ per and his group, who observed a cyclotron feature at about 50 ke V in the spectrum of the X-ray pulsar Hercules X-I corresponding to a field strength of 8 several 10 T, stimulated investigations of atoms within the framework of the adiabatic approximation, which is well justified for such field strengths. This method and its results are discussed in Chaps. 3, 5, and 6.

Quantum and chaos, key concepts in contemporary science, are incompatible by nature. This volume presents an investigation into quantum transport in mesoscopic or nanoscale systems which are classically chaotic and shows the success and failure of quantal, semiclassical, and random matrix theories in dealing with questions emerging from the mesoscopic cosmos. These traditional theories are critically analysed, and this leads to a new direction. To reconcile quantum with chaos and to restore genuine temporal chaos in quantum systems, a time-discrete variant of quantum dynamics is proposed.

The development of the supersymmetry technique has led to significant advances in the study of disordered metals and semiconductors.

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin­ tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead­ ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator.

This book discusses continuous and discrete nonlinear systems in systematic and sequential approaches. The unique feature of the book is its mathematical theories on flow bifurcations, nonlinear oscillations, Lie symmetry analysis of nonlinear systems, chaos theory, routes to chaos and multistable coexisting attractors. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, featuring a multitude of detailed worked-out examples alongside comprehensive exercises. The book is useful for courses in dynamical systems and chaos and nonlinear dynamics for advanced undergraduate, graduate and research students in mathematics, physics and engineering.

In this book we summarize the essential results of our efforts over the years to calculate energies, wave functions, and electromagnetic transitions of atoms as functions of the magnetic field strength from laboratory fields up to neutron star magnetic fields. Motivated by the observational evidence of huge magnetic 5 fields with strengths up to 10 T in the vicinity of white dwarf stars and of up 9 to 10 T in the vicinity of neutron stars the authors, together with coworkers and candidates for doctor and diploma degrees, have investigated this ,fasci­ nating quantum mechanical problem more or less continuously since 1978. The extensive tables and figures in the appendices represent the most complete data set to date in this field of research. For practical use all numbers are available by "anonymous ftp" over Internet. The first direct measurement of a neutron star magnetic field by Trum­ per and his group, who observed a cyclotron feature at about 50 ke V in the spectrum of the X-ray pulsar Hercules X-I corresponding to a field strength of 8 several 10 T, stimulated investigations of atoms within the framework of the adiabatic approximation, which is well justified for such field strengths. This method and its results are discussed in Chaps. 3, 5, and 6.