The same invariance test is run once more at

PHYSICS · Multi-part · 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

The same invariance test is run once more at Ad=1.2 s−2A_d=1.2\,\mathrm{s^{-2}}, this time halving the step of the Euler-Cromer integrator itself. Starting the same pair of states with δ0=1.0×10−15 rad\delta_0=1.0\times10^{-15}\,\mathrm{rad}, a run at Δt=0.001 s\Delta t=0.001\,\mathrm{s} reaches δsat=1 rad\delta_{\rm sat}=1\,\mathrm{rad} at t=260.7 st=260.7\,\mathrm{s}; halving the step to Δt=0.0005 s\Delta t=0.0005\,\mathrm{s} gives saturation at t=262.7 st=262.7\,\mathrm{s}. ((a)) Compute the Lyapunov exponent implied by each of the two Euler-Cromer steps, to three significant figures, using δ(t)=δ0eλLt\delta(t)=\delta_0e^{\lambda_{\rm L}t}. ((b)) The individual angle θ(t)\theta(t) at t=300 st=300\,\mathrm{s} differs by of order 1 rad1\,\mathrm{rad} between these two runs. Explain why this large disagreement about the trajectory is consistent with the close agreement you found for λL\lambda_{\rm L}, rather than contradicting it.

Browse the problem library