Significant difference in the long-term evolution of a three-body system using the exact N-body integration (black) and the approximated DA integration (red). The system consists of a test particle orbiting a mass m with semimajor axis a and a perturber with the same mass m. The outer orbit has semimajor axis a_per = 10a and eccentricity e_per = 0.2. The rest of the parameters are described in the text. The top, middle and bottom panels show the evolution of 1 − e, inclination and j_z = (1 − e²)^1/2cos i, respectively. The time is normalized to the secular time-scale t_sec defined in equation (2).