I need help to determine the general solution (or the flow) of a 2x2 nonlinear system for a particular case and in general how the general solution in nonlinear systems can be understood
We considerer the next non linear system:
$$\left\{\begin{array}{l} x^{\prime}(t)=y \\ y^{\prime}(t)=-x+\left(1-x^{2}\right) y \end{array}\right.$$
and i want to find general solution. For my aproach i understand that you have to considerer the linearization of the system in the equilibrium point, i.e,
$$J(0,0)=\left[\begin{array}{cc} 0 & 1 \\ -1 & 1 \end{array}\right]$$
and therefore the linear system is
$$\left\{\begin{array}{l} x^{\prime}(t)=y, \\ y^{\prime}(t)=-x+y . \end{array} \quad \text { Or } \quad\left(\begin{array}{l} x \\ y \end{array}\right)^{\prime}=\left[\begin{array}{cc} 0 & 1 \\ -1 & 1 \end{array}\right]\left(\begin{array}{l} x \\ y \end{array}\right) .\right.$$
with $\lambda_{1,2} = \frac{1\pm \sqrt{3}i}{2}$ eigenvalues and $v_1,v_2$ (determinated) eigenvectors. Then solution for the linear system is
$$(x(t),y(t)) = (C_1e^{\lambda_1t}v_1, C_2e^{\lambda_2t}v_2)$$
without going to the transformation in sin() or cos(). But my question is What about the nonlinear part of the original system? How does it influence the linear solution?