Periodic Orbit Identification in the CaledonianFour Body Problem

We present a numerical method for identifying periodic orbits in the Caledonian Symmetric Four Body Problem (CSFBP), a reflection-symmetric gravitational system with rich dynamical structure. Using a Poincaré return map defined on a reduced phase space, we construct a scalar function \(( K )\) whose vanishing corresponds to periodic motion. Applying a grid-based root-finding algorithm, we identify twelve first-order periodic orbits in the equal-mass case (\(( \mu = 1 )\)), classified into two hierarchical types: ``12'' orbits (binary pairs orbiting each other) and ``23'' orbits (two single bodies orbiting a central binary). A linear stability analysis based on the eigenvalues of the Jacobian matrix confirms that all detected orbits are linearly stable. The method demonstrates strong potential for generalization to higher-order orbits and symmetric \(( 2n )\)-body Caledonian systems. The periodic solutions also offer dynamical templates relevant to observed quadruple stellar systems.