Duffing Oscillator

At the end of Post 5, we left the Duffing oscillator on the threshold of chaos. We saw the period-doubling cascade in the bifurcation diagram, watched the single stable orbit split into two, then four, then dissolve into a dense fog of points. But we did not look closely at what the system is actually doing in that regime. Where does it go? What shape does its motion trace out over time? And how does that shape connect to the design of a metamaterial that suppresses vibration?

Everything in the last four posts has rested on a single assumption, stated quietly but used constantly: the oscillations are small. Small enough that restoring forces are proportional to displacement. Small enough that the wave equation stays linear, that superposition holds, that the dispersion relation describes behaviour across all amplitudes equally. The band gaps we derived in Post 4 are real, they work in experiments, and they are beautiful. But they belong to a world where amplitude doesn’t matter.