Within the past decade scientists, mathematicians and engineers have realized that a large variety of systems exhibit complicated evolution with time. This complicated behavior, called chaos, occurs so frequently that it has become important for the workers in many disciplines to have a good grasp of the fundamentals and basic tools of the emerging science of chaotic dynamics.
Topics in the book include: attractors; basins of attraction; one-dimensional maps; fractals; natural measure; strange attractors; delay coordinate embedding; fat fractals; Hausdorff dimension; symbolic dynamics; stable and unstable manifolds; Lyapunov exponents; metric and topological entropy; controlling chaos; chaotic transients; fractal basin boundaries; chaotic scattering; quasiperiodicity; Hamiltonian systems; KAM tori; period doubling cascades; the intermittency transition to chaos; crises; bifurcations to chaos in scattering problems and in fractal basin boundaries; the characterization of dynamics by unstable periodic orbits; and quantum chaos in time-independent bounded systems, as well as in temporally kicked and scattering problems. Homework problems are also included throughout the book.
Mathematicians have been aware of chaotic dynamics since Poincaré’s work at the turn of the century. But, as the turn of yet another century approaches, physical scientists and engineers have begun to use their understanding of chaos theory to analyze chaotic experimental time series data. Some researchers have even used the presence of chaos to achieve practical goals. To do this, they have had to work with dynamical processes for which the equations were either not known or were too complex to be useful. In other words, they have been coping with chaos.
Coping With Chaos is the first book to bring together recent advances in the interpretive and practical applications of chaos, which hold great promise for broad applicability throughout the physical sciences and engineering. Together with an introduction to chaos theory, this book provides detailed reports on methods of analyzing experimental time series data from chaotic systems and studies in which the unique attributes of chaos are put to practical use. Topics discussed in this book include:
• Theory of chaotic dynamics
• Embedding techniques for the analysis of experimental data
• Calculation of dimension and Lyapunov exponents
• Determination of periodic orbits and symbolic dynamics
• Prediction of chaotic time series
• Noise filtering of chaotic data
• Control of chaotic systems
• The use of Chaotic signals for communication
• And more