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In this paper, we have drawn the periodic orbits in the restricted four-body problem. The fourth body of infinitesimal mass moves under the Newtonian gravitational law due to three mass points (known as primaries) in the collinear Euler's configuration. We have studied the dynamics of the fourth body in the spatial collinear restricted four-body problem (SCR4BP) with non-spherical primaries. We have determined the periodic orbits by using the Fourier series method only around collinear libration points in SCR4BP with non-spherical primaries. The effects of the orbital parameter ε ( 0 < ε < 1 ), which controls the size of the orbit, on the periodic orbits and their period are investigated. In order to study the effect of this parameter, we have retained the terms up to third order with respect to the orbital dimensions. We have also drawn the variational graphs to determine the variation in the time period T of the periodic orbits. Moreover, we have unveiled that how oblateness and mass parameters influence the shape, size, and period of the periodic orbits. [ABSTRACT FROM AUTHOR]

In the current manuscript, we have explored the spatial collinear restricted four-body problem(SCR4BP) in which the three bodies, known as primaries, are situated in the collinear Euler's configuration, where the primary P 0 ( m 0 = β m ) (taken as central primary) is placed at the origin, and other two primaries P 1 ( m 1 ) and P 2 (m 2) , which are non-spherical bodies, are placed at the same distance from the origin having masses m 1 = m 2 = m . The massless fourth body known as test particle (or infinitesimal body) is moving under the Newtonian gravitational attraction of the primaries. This fourth body does not influence the motion of the primaries but its motion is effected by them as in the restricted three-body problem (R3BP). We have determined the long- and short-period orbits around those libration points which lie on the y - axis only in the SCR4BP for the case of two equal masses by using Fourier series method. We have studied the evolution of families of these orbits as the oblateness parameter A and mass parameter β evolve. The time period T of the periodic orbit is studied by using the variational graphs. By retaining the terms up to third order in the Fourier expansions, we have explored how the oblateness of the peripheral primaries as well as the mass parameter affect not only the shape, size and period but also the networks of long- and short-period orbits. [ABSTRACT FROM AUTHOR]

We consider a system in which the charged particle orbits under the influence of the electromagnetic field of three dipoles located on a system of three celestial bodies. Using well-known bivariate iterative scheme, known as Newton–Raphson (NR) iterative scheme, we numerically evaluated the positions of the stationary points (SPs) or equilibrium points (EPs) or libration points (LPs) and the linked basins of convergence (BoCs), and we also evaluated their linear stability. Moreover, we unveiled how the parameters, entering the effective potential function, affect the convergence dynamics of the system. Moreover, we also unveiled how the involved parameters affect the geometry of the zero velocity curves (ZVCs). Further, the correlation with the required number of iterations and the regions of convergence as well as the probability distributions associated to the BoCs is illustrated. In order to quantify the degree of final-state uncertainty of the BoCs, the basin entropy (BE) and for the fractality of boundaries of BoCs, the boundary basin entropy (BBE) are computed. [ABSTRACT FROM AUTHOR]