Nonlinear regression multivariate model for first order resonant periodic orbits and error analysis

In this work, nonlinear multivariate regression models are developed under the structure of the circular restricted three-body problem (CRTBP) with and without perturbations for exterior and interior resonant periodic orbits of order one. The nonlinear multivariate regression can be used for simply symmetric retrograde periodic orbits, which are first-order resonant orbits. Three real systems are taken into consideration for the study namely, Saturn-Hyperion, Saturn-Titan and Earth-Moon with small, moderate and high mass ratio respectively. In the given three systems, the first primary is taken as an oblate spheroid. Also, coefficient of oblateness, Jacobi constant and mass ratio are taken as parameters. The nonlinear multivariate regression model is used to find the initial position of periodic orbit without constructing PSS for time saving and avoid complicated procedure. In this context, fourth-fifth degree polynomial obtained from regression gives the best possible predicted value for the initial condition of periodic orbit by assigning parameter value such as mass ratio, coefficient of the oblateness of first primary and Jacobi constant. Also, error analysis of these models for three real systems Saturn-Hyperion, Saturn-Titan and Earth-Moon is discussed in detail. From error analysis, it can be seen that these models give the best approximation for the initial position of the periodic orbits.