Find necessary and sufficient conditions so that $(0,0)$ is stable. Suppose we have the system
$$
\left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) = \left(\begin{array}{c} f(x) + y \\ g(x) \end{array}\right).
$$
Here $f,g: \mathbb{R} \to \mathbb{R}$ are smooth analytic functions of $x$ such that
$$
\lim_{x\to 0} \frac{f(x)}{x^k}
$$
and
$$
\lim_{x\to 0} \frac{g(x)}{x^l}
$$
exists and are non-zero for some $k, l \geq 2$. What are necessary and sufficient conditions on $f$ and $g$ so that $(0, 0)$ is (asymptotically) stable?
I have no idea where to start. I thought it has to do something with the signs of $f$ and $g$ near $0$, but the $y$ term causes a disturbance, and I don't know how to handle that, so to say.
 A: The existence of a Lyapunov function is a necessary and sufficient condition for the stability of dynamical systems, in the sense of Lyapunov. 
You may start with a candidate Lyapunov functon $V(x,y)=(x^2+y^2)/2$. Its Lie derivative along the system dynamics yields
\begin{equation*}
\dot{V}(x,y)=x f(x)+xy+yg(x)=xf(x)+y(g(x)+1),
\end{equation*}
where $\dot{V}$ stands for the inner product between the gradient of $V$ and the vector field. To ensure that $\dot{V}$ is a negative definite definite function, you may assume that $f$ is unbounded, and satisfies $xf(x)<0$, for every $x\in\mathbb{R}$ and $f(x)\Leftrightarrow x=0$. Concerning $g$, you may assume that there exists a function $\alpha:\mathbb{R}\to\mathbb{R}$ such that, for every $y\in\mathbb{R}$, $y\alpha(y)<0$ and $\alpha(y)=0\Leftrightarrow y=0$, and for every $x\in\mathbb{R}$, $g(x)+1\leq \alpha(y)$.
Note that the previous assumptions are dependent on the candidate Lyapunov function that you chose.
If you want to consider only the differential equation itself, you may start with a necessary condition regarding the stabilizability of the $x$-subsystem: 


*

*if $y=0$, then 0 is asymptotically stable for the system
$\dot{x}=f(x)$. Otherwise, whenever $g(x)=0$ and $f(x)\neq0$, there's nothing you can do.


You might find more information on the book Khalil, Nonlinear Systems.
