The birth of the global stability theory and the theory of hidden oscillations

TL;DR

This paper reviews the historical development of global stability theory, highlighting the emergence of hidden oscillations and their significance in dynamical systems analysis.

Contribution

It provides a historical overview of the origins of global stability theory and discusses the concept and challenges of hidden oscillations.

Findings

Global stability boundaries are linked to the emergence of oscillations.

Hidden oscillations are difficult to detect and analyze.

Historical development of stability criteria is summarized.

Abstract

The first mathematical problems of the global analysis of dynamical models can be traced back to the engineering problem of the Watt governor design. Engineering requirements and corresponding mathematical problems led to the fundamental discoveries in the global stability theory. Boundaries of global stability in the space of parameters are limited by the birth of oscillations. The excitation of oscillations from unstable equilibria can be easily analysed, while the revealing of oscillations not connected with equilibria is a chalfilenging task being studied in the theory of hidden oscillations. In this survey, a brief history of the first global stability criteria development and corresponding counterexamples with hidden oscillations are discussed.