The golden number seen in a mechanical oscillator

A seemingly ubiquitous irrational number often appearing in nature and in man-made things like structures, paintings, and physical systems, is the golden number. Here, we show that this astonishing number appears in the periodic solutions of an underactuated mass-spring oscillator driven by a nonlinear self-excitation. Specifically, by using the two-time scale perturbation method, it is analytically demonstrated that the golden number appears in the ratio of amplitudes, as well as in the oscillation frequency of the periodic solution, which is referred to as golden solution and, by applying the Poincaré method, it is demonstrated that this solution is asymptotically stable. Additionally, the analytic results are illustrated by means of numerical simulations and also, an experimental study is conducted.