How a swing behaves
This paper is an advertisement for \textit{R. MacKay}'s thesis [Princeton University, Univ. Microfilm Intern., Ann Arbor, MI (order {\#}8301411)]. The present state of art of the theory of asymptotic behaviour of two conservative, coupled, nonlinear oscillators is dealt with. The author gives first a historical introduction, then ''the general nature'' of the problem is described. The role of area preserving maps in this theory is emphasized, for this purpose the swing with oscillating length is introduced as a model for explanation; the notion of KAM (Kolmogorov- Arnold-Moser) curves appears here too. The text shows further possibility in physical terms how the parametric instability (\(\equiv\) period doubling) the stability of periodic orbits destroys, and island structures disintegrate into chaos with increasing perturbation. Surface of the critical maps (which represents the transition from isolated islands to chaotic motion) is practically characterized. Notion of winding number and using of continued fractions is recapitulated. Importance of renormalization is described. Paper is a large collection of very interesting physical interpretations for some mathematical facts and methods and also a stimulus for mathematicians.
conservative, coupled, nonlinear oscillators
transition from isolated islands to chaotic motion
scientific article; zbMATH DE number 3976441