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OOK REVIEWS 157

Averaging Methods in Nonlinear Dynamical Systems. By J A. SANDERSandF. VERHULST.Springer-Verlag, New York, 1985. x + 247 pp. $28.00, paper.ISBN 0-387-96229-8. Applied Mathematical Sciences, Vol. 59.Averaging and homogenization are approaches that the applied analyst should

be ready to employ in appropriate contexts.Thus,

we can begrateful

that these twoDutch authors have written a short monograph which presents the analytical aspectsof the method of averaging, generally at a level of readability that graduate studentsand other potential users can learn and benefit from. They largely omit the moregeometric approaches to nonlinear dynamical systems, referring readers instead toother recent books such as Arnold [1] and Guckenheimer and Holmes [3]. Theirpresentation is no t totally elementary, however. It includes a considerable number ofclever personal observations and insightful perspectives, generated in their ownresearch and teaching, which has emphasized mathematical and asymptotic analysisrelated to celestial mechanics.

The introductory material is actually important in many contexts. It explainshow to estimate solutions of ordinary differential equations through appropriate useof the Gronwall inequality. n unusual aspect is the authors concern for obtainingestimates on expanding intervals of validity e.g. on 0 _ _ 1/e as e 0 . This is oneexample of an ordinary topic in stability theory that is treated in a somewhatnonstandard fashion (which thereby will help maintain the interest of the more well-read . Elementary concepts of asymptotic expansions and regular perturbation theoryare presented, with interesting and worthwhile side comments added.

To most simply define averaging, suppose

where fis T-periodic in t and let y satisfy the averaged equation

fo(y)= Under mild smoothness assumptions, the text shows that

]x(t)-y(t)l

8/13/2019 Book Review Averaging Methods Nonlinear Dynamical Systems

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Systems with slowly-varying frequencies, like

(with ft nonvanishing) continue to challenge analysts. A linear example, the Einstein

pendulum, ledto the

asymptotic theory of adiabatic invariance. Whenft

has zeros,we need to worry about passagethrough resonance. Such problems may be attackedby matching local expansions, as is common in singular perturbations theory. Thesurvey by Kevorkian [4] discusses this topic more extensively than this monographdoes. Related work, briefly presented, includes multitime expansions and the use ofMelnikov functions.

Sanders and Verhulst end their monograph by considering normal forms for avector field and for Hamiltonian systems (as well as related resonances).These topicsrequire a more sophisticated reader, as well as more notation and machinery. Tosummarize, the authors have given us a valuable text for learning about averaging.With the basis they provide, the student will be able to read a vast new literature and,hopefully, will be able to solve many significant applied problems.

REFERENCES

[1] V. ARNOLD, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.

[2] N. N. BOGOLIUBOV N D YU . A. MITROPOLSKn,Asymptotic Methods in the Theory of NonlinearOscillations, Gordon and Breach, New York, 1961.

[3] J. GtCrNHEIMER ND P. HOLMES,Nonlinear Oscillations, Dynamical Systems, and BifurcationsofVector Fields , Springer-Verlag, New York, 1983.

[4] J KEVORKAN, Perturbation techniques for oscillatory systems with slowly varying coefficients, t sReview, to appear.

R. E. OMALLEY,JR.Rensselaer Polytechnic Institute

Game Theory in the Social SciencesConcepts and Solutions. By MARTIN SHUBIK.MIT Press, Cambridge, MA, 1984. x 514 pp. 12.95, paper. ISBN 0-262-19195-4.The Theory of Games is now, by most reckonings, about a half century old. It

arrived on the public scene in remarkably mature form with a birth announcement

from von Neumann and Morgenstern entitled Theory of Games and EconomicBehavior (1944). The reaction generated by this landmark book, both public andacademic, was swift and in many ways unprecedented.

The idea of game theory seemed to intrigue a broad cross-section of the literatepopulace; and its formal aspects were eagerly embraced by many quantitative socialscientists as the long-awaited conceptual framework needed to do real (deductive)science. t was hoped that game theory would do fo r the social sciences (primarilyeconomics, political science, and psychology) what calculus had done fo r the physicalsciences. Study and research in game theory were encouraged, embraced, and gener-ously funded. Inevitably, after such high initial enthusiasm and expectations, disillu-

sionment se t in . This was reinforced by the inherent difficulty in formulating game-theoretic models for most real life situations, the resultant overquantification anddehumanization that this difficulty fostered, and an overidentification of game theorywith military problems. Since the early 1970s a happy equilibrium has been reached,and game theory has been enjoying a revival in mathematics and the social sciences.

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