Kellert begins by defining chaos theory as "the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems". He goes on to explain this in non-technical language, but he doesn't provide a general introduction to chaos theory: readers are expected to have read at least popular accounts of the subject.
Chapter two addresses the topics of predictability and sensitive dependence on initial conditions. Kellert argues that the problems of prediction raised by chaos theory require a new form of impossibility, between theoretical impossibility (violation of accepted laws of nature) and practical impossibility (ruled out by human limitations). He suggests the term transcendental impossibility for describing tasks which are not theoretically impossible but whose achievement is inconsistent with fundamental, non-contingent facts about enquiry.
Chapter three tackles some of the gnarly issues surrounding the concept of determinism. Kellert examines four of the many different meanings of the term (differential dynamics, unique evolution, value determinateness, and total predictability). He argues that the combination of chaos theory and quantum mechanics leaves anything except weak, local forms of determinism untenable. (He does seem a little unsure of himself when it comes to quantum mechanics, but his arguments are still convincing.)
When it comes to our notions of understanding, Kellert argues that chaos theory has consequences both methodological — discouraging microreductionism and encouraging experimental approaches and diachronic perspectives — and epistemological — favouring qualitative over quantitative prediction, geometric structures over causal mechanisms, and the search for order over the search for laws. All of these are changes in emphasis within science, however, and Kellert warns:
"to see chaos theory as a revolutionary new science that is radically discontinuous with the Western tradition of objectifying and controlling nature falsifies both the character of chaos theory and the history of science. ... any expectation that chaos theory will re-enchant the world will meet with disappointment."
The final chapter explores possible explanations for the long delay in the development of chaos theory. Kellert argues that social and ideological factors played an central role, along with more technical ones such as the lack of computers and mathematical prejudice against the intractably non-linear.
In the Wake of Chaos does cover a lot of material rather rapidly: those without any background in the philosophy of science may find it a little too concise; those with only a shallow grasp of the necessary mathematics may find some parts hard to follow. But a better presented or more clearly argued alternative is hard to imagine. In my opinion it behooves anyone who wants to make philosophical claims about chaos to acquire at least enough understanding of it to be able to follow Kellert's book.