In this paper, a trajectory-based methodology is proposed for characterizing and visualizing the feasible region of the coupling multi-energy system. A nonlinear dynamic system called quotient gradient system (QGS) play a fundamental role, which is constructed by the set of equality and inequality constraints. The theoretical relationship between the feasible region of the coupling multi-energy system and the union of the stable equilibrium manifolds of QGS is established. In other words, the feasible region can be characterized by the stable equilibrium manifolds of QGS. Moreover, the fact that each trajectory of QGS converges to an equilibrium manifold is proven, which is useful in visualizing the feasible region. Finally, the results of coupling multi-energy feasibility region problems developed have been numerically verified in a simple coupling heat and electricity network.