Statistics – Computation
Scientific paper
May 2000
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000dda....31.0401n&link_type=abstract
American Astronomical Society, DDA Meeting #31, #04.01; Bulletin of the American Astronomical Society, Vol. 32, p.859
Statistics
Computation
4
Scientific paper
We determined analytically the dependence of the Lyapunov exponent upon time step for the linear paradigms of the simple harmonic oscillator (center) and simple repeller (homoclinic point) for several popular symplectic integration schemes. For the oscillator, we showed that there is a Hopf bifurcation resulting in the appearance of a non-zero Lyapunov exponent for sufficiently large step size, while the repeller produced a positive Lyapunov exponent for any step size. We explored the standard map, corresponding to integration of the pendulum problem, and showed how the Lyapunov exponent manifests similar behavior except that nonlinearity further reduces the value of the maximum time step that can be safely used before non-physical behavior manifests ( 1/10 period). Recently, Grazier et al. (1999. Icarus 140, 341--352) published the results of outer solar system planetesimal simulations that used an error-optimized modified 13 th order Störmer integration scheme with a 4-day time step. Their computation showed no evidence for chaotic behavior among the four outer planets and the sun over an 8 x 108 year period for eight randomly selected initial conditions taken over a 300 year time span from the DE245 ephemeris. Using a second-order Wisdom-Holman integrator, optimized to minimize round-off effects, and the commonly used 400 day time step, we recovered the commonly cited exponential growth consistent with a Lyapunov time scale between 5 My and 12 My. However, when we reduced the time step to 200 days, the behavior is no longer demonstrably chaotic. A further reduction to 100 days in the time step shows no chaotic behavior over a 100 My period, and a 50 day time step based computation shows that the Wisdom-Holman calculation has effectively converged to non-chaotic behavior.
Grazier Kevin R.
Hyman James M.
Kaula William M.
Lee A. Y.
Newman William I.
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