24 research outputs found
On the relationship between instability and Lyapunov times for the 3-body problem
In this study we consider the relationship between the survival time and the
Lyapunov time for 3-body systems. It is shown that the Sitnikov problem
exhibits a two-part power law relationship as demonstrated previously for the
general 3-body problem. Using an approximate Poincare map on an appropriate
surface of section, we delineate escape regions in a domain of initial
conditions and use these regions to analytically obtain a new functional
relationship between the Lyapunov time and the survival time for the 3-body
problem. The marginal probability distributions of the Lyapunov and survival
times are discussed and we show that the probability density function of
Lyapunov times for the Sitnikov problem is similar to that for the general
3-body problem.Comment: 9 pages, 19 figures, accepted for publication in MNRA
Shadowing unstable orbits of the Sitnikov elliptic 3-body problem
Errors in numerical simulations of gravitating systems can be magnified
exponentially over short periods of time. Numerical shadowing provides a way of
demonstrating that the dynamics represented by numerical simulations are
representative of true dynamics. Using the Sitnikov Problem as an example, it
is demonstrated that unstable orbits of the 3-body problem can be shadowed for
long periods of time. In addition, it is shown that the stretching of phase
space near escape and capture regions is a cause for the failure of the
shadowing refinement procedure.Comment: 9 pages, 13 figures, accepted in MNRA