Le the following non linear system:
$$\begin{cases} \dot x_1=x_2\\ \dot x_2=-x_1^3-x_2 \end{cases}$$ with $$V(x)=\frac{1}{4}x_1^3+\frac{1}{2}x_2^2$$
by Lyapunov Theorem I have proven that the origin is an asymptotically stable point.
Since the jacobian of the function $f$ (let's rewrite the system in vectorial form as $\dot x=f(x)$) has an eigenvalue equal to $0$, I would say that surely the origin cannot be exponentially stable. Do you think it is right my intuition?
If we understand exponential stability as the existence of a lyapunow function $V(x)$ such that $V(x) = -\lambda \dot V(x)$ then from
$$ \begin{cases} \dot x_1=x_2\\ \dot x_2=-x_1^3-x_2 \end{cases}\Rightarrow\begin{cases} x_1^3\dot x_1=x_1^3x_2\\ x_2\dot x_2=-x_1^3x_2-x_2^2 \end{cases} $$ after addition
$$ \frac 12\frac{d}{dt}\left(\frac 12x_1^4+ x_2^2\right)= -x_2^2\ge-\left(\frac 12x_1^4+ x_2^2\right) $$
so
$$ V(x)\ge -\lambda \dot V(x) $$