Liapunov function for a system of quadratic ODEs I have been trying to find a Liapunov function which would give me information about the stability of the following system of differential equations, however, I am not able to come up with any. The problem is an exercise from the book 'Differential Equations and Dynamical Systems' by Lawrence Perko (Third Edition), Chapter Two, Problem Set $9$, Question $5(c)$, so I guess, there does exist a Liapunov function for the same:
\begin{align}
\frac{\mathrm{d}x_1}{\mathrm{d}t} &= -4x_2 + x_1^2 \\
\frac{\mathrm{d}x_2}{\mathrm{d}t} &= 4x_1 + x_2^2
\end{align}
 A: I do not know which Lyapunov function will work for this example, but here is a hint how to prove that the origin is Lyapunov stable (a center). The system 
$$
\dot x_1=f_1(x_1,x_2),\\
\dot x_2=f_2(x_1,x_2),
$$
is said to be symmetric with respect to $x_1$-axis if it is invariant with the change of the variables $(t,x_2)\to(-t,-x_2)$ (similarly, for $x_2$-axis).
Theorem: If a system symmetric with respect to $x_1$-axis, and the linear approximation is a center, then for the full nonlinear system the origin is a center. 
An idea of proof is given in Perko, Section 2.10 (the very end of it).
A: I have partially solved this question using the following approach. Transform the problem into polar co-ordinates, that is 
\begin{align}
r &= \sqrt{x_1^2 + x_2^2} \\
\theta &= \arctan \frac{x_1}{x_2}
\end{align}
we get the equations
\begin{align}
\frac{\mathrm{d}r}{\mathrm{d}t} &= r^2 \left(\cos^3\theta + \sin^3\theta \right) \\
\frac{\mathrm{d}\theta}{\mathrm{d}t} &= -4 
\end{align}
Now, we try to determine the stability of the system using this co-ordinate system, which should be somewhat easier, since $r$ is always positive and $\mathrm{d}\theta/\mathrm{d}t$ is always negative. I will work over it tomorrow morning though. 
