The invariance test of the previous problem is continued at : the fourth-order…

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

The invariance test of the previous problem is continued at

A_d=1.2\,\mathrm{s^{-2}}

: the fourth-order Runge-Kutta run at

\Delta t=0.01\,\mathrm{s}

, which reached the saturation separation

\delta_{\rm sat}=1\,\mathrm{rad}

(from

\delta_0=1.0\times10^{-15}\,\mathrm{rad}

) at

t=261.7\,\mathrm{s}

, is now compared with a run using a different integrator, the Euler-Cromer method, at

\Delta t=0.0005\,\mathrm{s}

, starting from the same two states. The Euler-Cromer run reaches the same saturation separation at

t=259.7\,\mathrm{s}

. (a) Find the Lyapunov exponent implied by the Euler-Cromer run, to three significant figures, using

\delta(t)=\delta_0e^{\lambda_{\rm L}t}

. (b) Compare it with the Runge-Kutta value (

\lambda_{\rm L}=0.1320\,\mathrm{s^{-1}}

\Delta t=0.01\,\mathrm{s}

) and state whether replacing the integrator, in addition to changing the step, has altered the exponent by more than the quoted uncertainty of

\pm0.004\,\mathrm{s^{-1}}