Reliability of Lyapunov characteristic exponents computed by the two-particle method

For highly complex problems, such as the post-Newtonian formulation of compact binaries, the two-particle method may be a better, or even the only, choice to compute the Lyapunov characteristic exponent (LCE). This method avoids the complex calculations of variational equations compared with the variational method. However, the two-particle method sometimes provides spurious estimates to LCEs. In this paper, we first analyze the equivalence in the definition of LCE between the variational and two-particle methods for Hamiltonian systems. Then, we develop a criterion to determine the reliability of LCEs computed by the two-particle method by considering the magnitude of the initial tangent (or separation) vector ξ0 (or δ0), renormalization time interval τ, machine precision ε, and global truncation error ϵT. The reliable Lyapunov characteristic indicators estimated by the two-particle method form a V-shaped region, which is restricted by d0, ε, and ϵT. Finally, the numerical experiments with the Hénon-Heiles system, the spinning compact binaries, and the post-Newtonian circular restricted three-body problem strongly support the theoretical results.