This chapter is subtitled “A summary of lectures given at the International Center for Mechanical Studies (CISM) at Udine, Italy, 1986.” It consists of four sections: “Introduction: New Tools to Diagnose Chaotic Vibrations;” “Experimental Techniques;” “Chaos Diagrams;” and “Fractal Dimension.”

The introduction lists some experimental and numerical tools for studying systems with chaotic dynamics, provides a checklist for the identification of nonperiodic or chaotic motions, and discusses some of the tools in more detail; the vibration of a buckled beam serves as an illustrative example. The “new tools” include fast Fourier transforms, Poincaré maps, Lyapunov exponents, and fractal dimensions; it is strange to find these ideas described as new in a 1986 review.

The experimental techniques discussed in Section 2 include plotting Poincaré maps from digitally sampled data and constructing one-dimenaional maps, position-triggered Poincaré maps, and double Poincaré maps. A number of terms introduced in this section are not defined, such as “Cantor set structure” and “Cantor set sheets.”

Section 3, “Chaos Diagrams,” lists two kinds of criteria for chaos in physical systems: predictive rules enable one to determine the set of input or control parameters that will lead to chaos, and diagnostic criteria enable one to tell if a particular system is in a state of chaotic dynamics, based on measurements or signal processing of data from the history of the system.

Section 4, “Fractal Dimension,” is divided into subsections 4.1, “Correlation dimension,” and 4.2, “Fractal dimension of strange attractors.” Subsection 4.2 lists three principal definitions of fractal dimension: averaged pointwise dimension, correlation dimension, and Lyapunov dimension. A discussion of methods for calculating the averaged pointwise dimension follows, but the Lyapunov dimension is not discussed.

The set of lecture notes suffers from a number of typographical errors, some fairly serious (both equations in the pair numbered (1) have the same left side), and some minor (“beren” for “been,” p. 43, and an extraneous right parenthesis in equation 31). The paper has 56 references, 12 of which are cited in the text; in 7 of these, the date in the citation does not agree with the date in the bibliography. A citation of “Landau, 1941” is not listed in the bibliography. Most of the figures are uncaptioned and use notation inconsistent with that used in the text. Caveat lector.