PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos · Nonlinear dynamics and chaos
Modes, Packets and Media, Ch Driven and pumped nonlinear oscillators and the route to chaos
Problem
The same damped, driven pendulum is run at two drive settings, A and B, and each setting is run twice, the second run starting from an initial angle differing from the first by only δ0=1.0×10−8rad. In both cases the long-time spectrum is broadband: a continuous background with no sharp lines.
((a)) Drive A: the angular separation between the two runs grows as δ(t)=δ0eλt with λ=0.45s−1 until it saturates at order one radian; when the tiny initial difference is removed (both runs given exactly the same starting angle and angular velocity), the two strobe sets and the two long-run averaged spectra are identical to the precision of the integrator.
((b)) Drive B: the angular separation between the two runs grows roughly as t, not exponentially, and even with exactly the same nominal starting angle and angular velocity the two strobe sets share no point in common. Identify which drive shows broadband chaos and which shows a noise-corrupted signal, and give the criterion used for each case.