Tangled Separatrices on Two Degrees of Freedom Swing Equation System

Abstract

One of the applications of two degrees of freedom swing equation system is the transient stability problem of an electrical power system. Using the Liapunov function method, there are many reports on the sufficient conditions of normal operations. However the basin structure of the stable equilibrium point, which corresponds to the normal operating state, is not entirely known. In this paper we will approach this problem by investigating the structure of the separatrix of the corresponding conservative system. This separatrix decomposes the phase space into regions of bound motions and divergent motions. This boundary concerns the invariant manifolds of the closed orbits on the center-manifolds of the saddle-center equilibrium points, which appear under the conservative condition. By numerical simulations, we will confirm in this report that there are transverse homoclinic intersections on these manifolds, whose cross-sectional view is different from a familiar homoclinic structure due to the asymmetric feature of the potential. Such circumstances are not observed, for example, on the well-known Hénon-Heiles system whose saddle-center equilibrium point has a trivial homoclinic loop. *