The Harmonic Oscillator, Complex-Dynamics Predictability, and the Beauty of Trigonometry: Subharmonic Cascades towards Resonance
The forced and undamped harmonic oscillator revisits with new and fundamental aspects. The study discloses complementary and -so far overlooked- intrinsic properties. Despite its simplicity, the model is shown to be characterized by countless -theoretically unlimited- sequences of intricate solutions. Such hierarchies, including the familiar period-doubling series -or equivalently subharmonic cascades- usually typify complex nonlinear dynamical systems. The remarkable similarity between the numerically simulated and the analytically predicted solutions confers the model unquestionable credit. It takes simple trigonometry -at the reach of the willing undergraduate student- to fully grab the essence of the new outcomes.
