The golden number seen in a mechanical oscillator

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Mathematical Physics

Background:

• The golden number (φ) is an irrational number observed in natural phenomena and human creations.
• Its presence in physical systems, particularly in oscillatory behaviors, remains an area of interest.

Purpose of the Study:

• To investigate the emergence of the golden number in the periodic solutions of an underactuated mass-spring oscillator.
• To analytically and experimentally validate the occurrence of the golden number in such systems.

Main Methods:

• Two-time scale perturbation method for analytical derivation.
• Poincaré method to assess solution stability.
• Numerical simulations for result illustration.
• Experimental validation of theoretical findings.

Main Results:

• The golden number appears in the ratio of amplitudes of the periodic solutions.
• The golden number dictates the oscillation frequency, defining a 'golden solution'.
• The 'golden solution' is demonstrated to be asymptotically stable.

Conclusions:

• The study confirms the unexpected appearance of the golden number in a nonlinear driven mass-spring oscillator.
• Analytical, numerical, and experimental evidence supports the existence of a stable 'golden solution'.