Periodic orbits of the perturbed relative motion.

• New perturbed relative motion equations are derived. • Periodic orbits of deputy satellite are obtained numerically. • One, two, and three loops periodic orbits are analyzed. • The velocity and position of deputy satellite are also analyzed. • Single variable regression model developed to find the true anomaly initial value. This paper presents a novel dynamical system for the perturbed relative motion of deputy satellite. The governing equations are derived in the LVLH frame under the second zonal harmonic coefficient effect of the Earth on both chief and deputy satellites. An extended numerical analysis on the deputy satellite motion is performed, where the initial state vector of deputy satellite and eccentricity of chief satellite are used as initial conditions for finding its own periodic orbits. The interior loops of periodic orbits are obtained using numerical integrator RKEOM. Such orbits are analysed for different parameters and their stability are also discussed, where these orbits have considerable importance in the resonance analysis of motion The second zonal harmonic coefficient effect on the periodicity, initial position, initial true anomaly is studied, alongside exploring these orbits with one, two and three loops. Furthermore, a single variable nonlinear regression model is developed to predict initial true anomaly using eccentricity of chief satellite's orbit. [ABSTRACT FROM AUTHOR]