In this paper, the spectral decompositions of Lyapunov functions are applied for the first time to the analysis of the behavior of a bilinear model of a two-area electric power system. In contrast to the technique of normal forms and modal series methods, we consider the spectral decomposition not for the dynamics of state variables, but for Lyapunov functions, which characterize the L2-norms of variables or signals in the time domain. The solution of the generalized Lyapunov equation for a bilinear system is represented as the sum of Hermitian matrices corresponding to individual eigenvalues of the system or their pairwise combinations. An iterative algorithm for calculating spectral terms is developed for stable bilinear systems. In a test experiment for the purpose of transient stability analysis, we evaluate the value of individual eigenmodes and their pairwise combinations depending on the magnitude of bilinear terms. The obtained results are consistent with an intuitive interpretation derived from the model equations and eigenvalue analysis. The spectral decompositions of Lyapunov functions allowed us to indicate the range of applicability of linear model and to reveal dominant eigenmodes in the analysis of transient stability of electric power system.