A Window into the World of KAM Theory. Issue 4 (7th August 2020)

• Record Type: Journal Article
• Authors: Walsh, James
• Is Part Of: Mathematics magazine. Volume 93:Issue 4(2020)
• Page Start: 244
• Page End: 260
• Publication Date: 2020-08-07
• Subjects: MSC: 37J40; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/0025570X.2020.1792238
• Languages: English
• ISSNs: 0025-570X

Abstract

Summary: What happens when you periodically force a nonlinear oscillator in the absence of damping? For linear oscillators, such as the mass-on-a-spring model typically encountered in that first differential equations course, the behavior of the forced system is easily and well understood. In this article, a simple mechanical model is used to illuminate the power and beauty of the theory of Kolmogorov, Arnol'd, and Moser. Known as KAM theory, this profound 20th century mathematical achievement answers the question posed above. Model simulations illustrating the complex coexistence of regular and chaotic motions are presented. Additionally, KAM theory is placed within its historical context, namely, the quest to determine the stability of the solar system.