Reviewed by
Hans-Walter Lorenz
Department of Economics, Friedrich-Schiller-University Jena
It is relatively uncommon that advances made in sciences – and in particular in mathematics – attract any attention in a basically open-minded public audience. Exceptions to this rule can be seen in the popularity of catastrophe theory in the 1980s and (to some minor extend) of the mysteries of relativity theory in the post-war era. The last two decades saw an interest in complexity issues because the term often has been used as a synonym for non-simplicity with the consequence that addressed problems have appeared as being interesting per se; “chaos” as a particular form of complex phenomena has attracted most attention. It did not come as a big surprise that “chaos” was supposed to be of interest in fields like economics, management science, sociology and other so-called “soft sciences” as well.Actually, the term “chaos” describes a rather unambiguous phenomenon although several formal definitions exist in the literature (depending, among others, on the basic type of dynamical system). For more practical scientific purposes it might be fair to restrict the chaos property of deterministic (i.e., completely non-stochastic, full-specified) dynamical systems (i.e., sets of equations that are designed as unambiguously describing the motion of an assumed reality) to the following properties: i) time series generated by a chaotic system look random although they are generated by a completely deterministic system, ii) the development of two time series starting at very neighboring starting points might differ completely (the “butterfly effect”), and iii) different initial points can belong to different “basins of attraction” with different sets of points which are eventually reached.
It is now well-established that chaos can prevail in weather-forecasting systems (not necessarily in the actually observed weather!) and other natural-science fields dealing with fluids in general. Whether chaos is present in economic time series is still a somewhat open question, probably because external stochastic disturbances play a major role in the realization of economic variables. However, attempts have been undertaken during the, say, last twenty years to establish “chaos” even in fields for which the mathematical concept was not developed. The present book contains a collection of papers dealing with “chaos” (in a very broad sense) in political environments with topics ranging from general peace/war considerations, the Arab Spring, and counter-intelligence problems to the butterfly effect in politics, to gender problems in politics or to economic decision making processes.
Unfortunately, an introduction to the collection with an overview of the collection by the editors is missing. Since most of the topics covered by the papers in the present collection are not (and cannot be) expressed in a mathematical way, the “chaos” phenomenon can only be understood as a metaphor, emphasizing the basically uncertain, stochastic nature of future events even in case of seemingly well-understood scenarios.
Quite typical for many papers dealing with topics genuinely uncommon to their authors, the present book abounds in negative examples of dwelling into unfamiliar grounds. For example, the author of the paper on economic decision making (Chapter 4) would have been well-advised to consult a beginners’ textbook on microeconomics. Most economists will probably consider the simple model of a single market at least as unknown (and perhaps as economic nonsense). The 5th-order dynamical system is not only strictly linear (not non-linear, as claimed by the author) but the projected trajectory simply exhibits a regular although explosive behavior. The paper in Chapter 5 is an example of ignoring an important property of chaotic systems, namely that it is often relevant to concentrate on the boundary of a basin of attraction. In other papers the authors talk about rather miraculous phenomena – on page 92, the authors claim that “two or more different basins of attraction are consecutively visited by the trajectory” (!), on the next pages they describe the renowned Henon map – without any hint on its relation to the present paper and eventually describe the solution behavior of a linear and therefore non-chaotic particular case!).
My summarizing assessment in this review of the present book therefore is twofold (and basically reflects experiences made after having been confronted with similar projects): the formal, mathe-matical apparatus available for the study of (specific) nonlinear dynamical systems is not used entirely flawlessly; mathematical concepts are often used inadequately and examples taken from other fields such as economics may exhibit a sort of innocence (or ignorance) in the light of rather known results. Whatsoever, this critique does not diminish the relevance of challenging the still common linear world-view which advocates the dominance of simple dynamic, regular phenomena such as stable fixed points that are perhaps disturbed by unexplained stochastic influences.
The collection of examples from diverse scenarios in this book emphasizes the prevalence of – in a very broad sense – non-linear phenomena in circumstances which perhaps might not have been expected by a layman in these fields. However, the book would have benefited a lot when no attempts were made to employ actually unambiguous formal instruments to general problems which have difficulties to be formalized at all. The use of the somewhat mysterious and still somewhat unfamiliar terms like chaos, butterfly effects, sensitivity to initial condition and motion towards and on strange attractors pretends a sort of scientific seriousness of the basic arguments discussed in most contributions in the volume. However, the careless use of the mathematics and the employed terms can indeed be taken as an argument against the volume. The editors would have been well-advised to keep an eye on the incorrect use of borrowed concepts from other fields. A reduction of the instrumental arguments to weaker heuristics and general metaphors would not have diminished the possible relevance of the investigated problems.
A last remark: unfortunately, a final copy-editing of the volume by the publisher obviously did not happen!
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