Consider the equations: $$z_1'=z_2 - t z_2^3,~~~~z_2'=-\frac{1}{2}z_2^3.$$ Question. Comment on the stability of the zero solution.
Here the matrix associated with the linear part of the equation is $$ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, $$ which has two zero eigen values. Hence the zero solution is unstable. Since the stability of the linear solution is same as the stability of the nonlinear solution, if suffices to comment about the stability of the linear solution. Do I have to justify more ? Thank you for your time.