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The Topology of Chaos

Gilmore, Robert — Lefranc, Marc

A new approach to understanding nonlinear dynamics and strange attractors

The behavior of a physical system may appear irregular or chaotic even when it is completely deterministic and predictable for short periods of time into the future. How does one model the dynamics of a system operating in a chaotic regime ? Older tools such as estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions do not sufficiently answer this question. In a significant evolution of the field of Nonlinear Dynamics, The Topology of Chaos responds to the fundamental challenge of chaotic systems by introducing a new analysis method-Topological Analysis-which can be used to extract, from chaotic data, the topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and are responsible for generating chaotic data. Beginning with an example of a laser that has been operated under conditions in which it behaved chaotically, the authors convey the methodology of Topological Analysis through detailed chapters on :
* Discrete Dynamical Systems : Maps
* Continuous Dynamical Systems : Flows
* Topological Invariants
* Branched Manifolds
* The Topological Analysis Program
* Fold Mechanisms
* Tearing Mechanisms
* Unfoldings
* Symmetry
* Flows in Higher Dimensions
* A Program for Dynamical Systems Theory

Laser Dynamics

- Thomas Erneux, Université Libre de Bruxelles
- Pierre Glorieux, Laboratoire de Physique des Lasers, Atomes et Molécules

Bridging the gap between laser physics and applied mathematics, this book offers a new perspective on laser dynamics. Combining fresh treatments of classic problems with up-to-date research, asymptotic techniques appropriate for nonlinear dynamical systems are shown to offer a powerful alternative to numerical simulations. The combined analytical and experimental description of dynamical instabilities provides a clear derivation of physical formulae and an evaluation of their significance. Starting with the observation of different time scales of an operating laser, the book develops approximation techniques to systematically explore their effects. Laser dynamical regimes are introduced at different levels of complexity, from standard turn-on experiments to stiff, chaotic, spontaneous or driven pulsations. Particular attention is given to quantitative comparisons between experiments and theory. The book broadens the range of analytical tools available to laser physicists and provides applied mathematicians with problems of practical interest, making it invaluable for graduate students and researchers.

Equilibrium Molecular Structures : From Spectroscopy to Quantum Chemistry

Molecular structure is the most basic information about a substance, determining most of its properties. Determination of accurate structures is hampered in that every method applies its own definition of "structure" and thus results from different sources can yield significantly different results. Sophisticated protocols exist to account for these differences, but until now, no textbook has been written to discuss such topics in a widely accessible manner.

Balancing quantum theory with practical experiments, Equilibrium Molecular Structures focuses on the theory involved in determining and converting measured and computed data sets into accurate and well-defined equilibrium structures.

This textbook begins with a discussion of quantum chemistry and the concept of potential energy surfaces, quantum chemical computation of structures and anharmonic force fields. The reader is next introduced to the method of least squares and the problem of ill-conditioning, leverage points, perturbation theory, computational aspects of determining semi-experimental equilibrium structures, the determination of moments of inertia from spectra, and the treatment of resonances. The textbook also examines the determination of diatomic molecular potentials using semiclassical and quantum mechanical methods as well as position and distance averages.

From basic elements to the latest advances and current best practices, Equilibrium Molecular Structures contains abundant references, examples, and exercises and includes a CD with additional examples. These features make the book ideal for class instruction but also user-friendly for self-instruction. It is recommended for newcomers to the field and also for experienced spectroscopists who want to expand their area of knowledge.