At drive amplitude (with , , ) the driven damped pendulum's motion is chaotic

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Mechanics & Dynamical Systems · Simulating classical dynamical systems

Computational Physics, Ch When trajectories separate: the driven pendulum and the logistic map

Problem

At drive amplitude Ad=1.2 s−2A_d=1.2\,\mathrm{s^{-2}} (with ω0=1 s−1\omega_0=1\,\mathrm{s^{-1}}, γ=0.5 s−1\gamma=0.5\,\mathrm{s^{-1}}, ωd=2/3 s−1\omega_d=2/3\,\mathrm{s^{-1}}) the driven damped pendulum's motion is chaotic. Two trajectories are started with initial angular separation δ0=1.0×10−15 rad\delta_0=1.0\times10^{-15}\,\mathrm{rad} and integrated with the fourth-order Runge-Kutta method. The separation grows as δ(t)=δ0eλLt\delta(t)=\delta_0e^{\lambda_{\rm L}t} until it saturates at the width of the region the motion visits, about δsat=1 rad\delta_{\rm sat}=1\,\mathrm{rad}. With a step of Δt=0.02 s\Delta t=0.02\,\mathrm{s} this saturation is reached at t=259.7 st=259.7\,\mathrm{s}; repeating the same experiment from the same starting states with the step halved to Δt=0.01 s\Delta t=0.01\,\mathrm{s} gives saturation at t=261.7 st=261.7\,\mathrm{s}. (a) Use δ(t)=δ0eλLt\delta(t)=\delta_0e^{\lambda_{\rm L}t} to find the Lyapunov exponent λL\lambda_{\rm L} implied by each run, to three significant figures. (b) A set of 6464 such measurements at a fixed step gives λL=0.132±0.004 s−1\lambda_{\rm L}=0.132\pm0.004\,\mathrm{s^{-1}}. State whether halving the step has changed the exponent by more than this uncertainty, and what that tells about whether the exponential divergence at Ad=1.2 s−2A_d=1.2\,\mathrm{s^{-2}} is caused by the integration method.

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