I am self studying Control Theory from a book and in it the author asks a question --
If the linear system of ordinary differential equation is unstable ( at x=0 ) prove that non linear system is also unstable. ( Use linearization in non linear system ie if system is $X' = AX + g(X)$ , then you can assume that $||g(X) ||<k ||X||$ for suitable $||X||< k$ and also assume that $A$ is a constant matrix) .
Can somebody please tell how to prove it.
Not true as stated. Consider the nonlinear system
$$ \eqalign{x' = y - x^3\cr y' = - y^3\cr} $$ whose equilibrium at $(0,0)$ is asymptotically stable, although the linearized system $$ \eqalign{ x' = y\cr y' = 0\cr}$$ is unstable.