In this paper, a novel dynamical trajectory-based methodology is developed for systematically computing multiple local optimal solutions of general nonlinear programming problems with disconnected feasible components satisfying nonlinear equality/inequality constraints. The proposed methodology, deterministic in nature, exploits trajectories of two different non-linear dynamical systems to find multiple local optimal solutions. The methodology consists of two phases: Phase I starts from an arbitrary (infeasible) initial point and rinds systematically multiple or all the disconnected feasible components; Phase II finds an adjacent local optimal solution from a local optimum via a decomposition point, thereby systematically locating multiple local optimal solutions which lie within each feasible component found in Phase I. By alternating between these two phases, the methodology locates multiple or all the local optimal solutions which lie in all the disconnected feasible components. A theoretical foundation for the proposed methodology is also developed. The methodology is illustrated with a numerical example with promising results.