Effects of manifold correction methods on chaos indicators

Abstract

The manifold approach of Nacozy et al. (Astrophys Space Sci 14:40–51, 1971), the approximate velocity correction method of Wu et al. (Astron J 133:2643–2653, 2007), and the velocity scaling method of Ma et al. (New Astron 13:216–223, 2008a) are some of the available manifold correction methods. They have been highly successful at maintaining invariant integrals in two-body problems and the Sun–Jupiter–Saturn system. This paper discusses their efficiency on chaos indicators. Because the planar circular restricted three-body problem involves the Jacobi constant $$C_\mathrm{j}$$ and chaotic phenomena, it is preferable to check the numerical performances of manifold corrections. First, we find that a low-order algorithm combined with manifold corrections can greatly improve the precision of the Jacobi constant $$C_\mathrm{j}$$ . Then, numerical experiments show that these manifold correction methods have the same performance in Poincare sections, Lyapunov exponents, fast Lyapunov indicators, smaller alignment indices, and relative finite time Lyapunov indicators. Moreover, manifold corrections not only allow for the use of larger step sizes compared to low-order algorithms without correction but also save substantial computation time compared to the high-order algorithm RKF7(8). In particular, the velocity scaling method of Ma et al. (2008a) lends itself to practical application in long-term integration.