Stability of the linear system, especially the diagonalizable system, can be verified by computing the signs of the real parts of its eigenvalues. In this paper, we generalize this result to the diagonalizable nonlinear system in terms of nonlinear eigenvalues. According to the main results, the unique equilibrium point of the diagonalizable system is globally asymptotically stable if a specific set of nonlinear eigenvalues is not positive at each point in the state space.
Kawano, Y., & Ohtsuka, T. (2015). Stability criteria with nonlinear eigenvalues for diagonalizable nonlinear systems. Systems and Control Letters, 86, 41–47. https://doi.org/10.1016/j.sysconle.2015.10.001