I would like to know if there is a way to represent the solution of a nonlinear system of ode in a general way.
For example, in the case of a linear system, the solution $\varphi(t)$ can be represented as a function of the eigenvalues $\lambda_{k}$ and eigenvectors $v_{k}$ for the jacobian matrix in the equilibrium point, i.e,
$$\varphi(t) = \sum_{k=1}^{n}C_{k}e^{\lambda_{k} t}v_{k}$$
where $C_{k}$ are de constants. I have found very little literature about it; here I mean that we talk about nonlinear systems but I have not found a way to represent the solution.