Quote1.1 Why ChaosBook?
It seems sometimes that through a preoccupation with science, we acquire a firmer hold over the vicissitudes of life and meet them with greater calm, but in reality we have done no more than to find a way to escape from our sorrows.
- Hermann Minkowski in a letter to David Hilbert
The problem has been with us since Newton's first frustrating (and unsuccessful) crack at the 3-body problem, lunar dynamics. Nature is rich in systems governed by simple deterministic laws whose asymptotic dynamics are complex beyond belief, systems which are locally unstable (almost) everywhere but globally recurrent. How do we describe their long term dynamics?
The answer turns out to be that we have to evaluate a determinant, take a logarithm. It would hardly merit a learned treatise, were it not for the fact that this determinant that we are to compute is fashioned out of infinitely many infinitely small pieces. The feel is of statistical mechanics, and that is how the problem was solved; in the 1960's the pieces were counted, and in the 1970's they were weighted and assembled in a fashion that in beauty and in depth ranks along with thermodynamics, partition functions and path integrals amongst the crown jewels of theoretical physics.
This book is not a book about periodic orbits. The red thread throughout the text is the duality between the local, topological, short-time dynamically invariant compact sets (equilibria, periodic orbits, partially hyperbolic invariant tori) and the global long-time evolution of densities of trajectories. Chaotic dynamics is generated by the interplay of locally unstable motions, and the interweaving of their global stable and unstable manifolds. These features are robust and accessible in systems as noisy as slices of rat brains. Poincaré, the first to understand deterministic chaos, already said as much (modulo rat brains). Once this topology is understood, a powerful theory yields the observable consequences of chaotic dynamics, such as atomic spectra, transport coefficients, turbulent shapes.
That is what we will focus on in ChaosBook. The book is a self-contained graduate textbook on deterministic and quantum chaos. Your professor does not know this material, so you are on your own. We will teach you how to evaluate a determinant, take a logarithm–stuff like that. Ideally, this should take 100 pages or so. Well, we fail–so far we have not found a way to traverse this material in less than a semester, or 200-300 page subset of this text.
Question 1.1. Professor K. Zweistein asks
Q Perhaps it is painfully obvious to the experts, but I have so far failed to find what I need by a hyperlink-assisted walk through ChaosBook. Shouldn't the textbook be clear about this? At present, the barrier to entry (having to read ChaosBook entirely, cover to cover) appears too steep for the working scientists to learn. Perhaps a simple illustrative example? paper? would help...
A OK, Karen.