Problem statement
I have been solving the sample variant of the control work on differential equations, which I will be having soon, when stumbled upon the following problem, which I find quite interesting, but challenging as well. $$ \left\{\begin{matrix} \; \dot{x} &=&-2y-x^3, \\ \; \dot{y} &=&x-y^3. \\ \end{matrix}\right. \tag{1}$$ Given the system of nonlinear ODEs $(1)$, I have to explore the stability of the equilibrium node $(0, 0)$, by using Lyapunov's function of the following form: $\mathbb{V}(x,y) = ax^2 + by^2,$ for some constants $a, b$.
My attempt
As we know, in order to verify the node's stability, we have to find the derivative of $\mathbb{V}$ over $t$, if the derivative is greater than zero, then the equilibrium is unstable, else stable.
$$\dot{\mathbb{V}} = \frac{\partial \mathbb{V}}{\partial x} \frac{\partial \mathbb{x}}{\partial t} + \frac{\partial \mathbb{V}}{\partial y} \frac{\partial \mathbb{y}}{\partial t} = 2ax(-2y-x^3) + 2by(x-y^3) = -2a(2xy +x^4) + 2b(xy-y^4).$$
And there is the point, where I don't actually know, what exactly to do. First, I have tried to consider several cases for $a$ and $b$ ($a=b=0$; $a \gt b \gt 0$, $a=b\gt 0$, etc.), but then I realized, that if I am to do this separately, I will have to consider quite a lot of cases, even for $x$ and $y$, as it is not that clear overall, whether $\dot{\mathbb{V}}$ is less or greater than zero.
I want to know, by that point, whether there is a more simple method to understand, whether the node is stable or not. Also, I want to clarify, maybe I get the task incorrectly, and I just have to chose some $a, b$ by myself in order to verify the stability of the node?
Anyways, I would appreciate any help of yours!
Thank you in advance.