Question: Consider the following autonomous vector field on $\mathbb{R}$:
$\dot x = x-x^3, x \in \mathbb{R}$
Compute all equilibria and determine their stability, i.e., are they Lyapunov stable, asymptotically stable, or unstable?
Compute the flow generated by the vector field and verify the stability results for the equilibria directly from the flow.
My problem here is that we have gone very little into the idea of the flow of the solutions of ODEs. My instinct to find the stability of the equilibria directly from the flow of the solutions is to analyse it as we let $t \rightarrow \pm \infty$, however sometimes this is not always clear to do.
Attempt:
$\dot x = x-x^3 = x(1-x^2) = x(1-x)(1+x) \Rightarrow \bar x = 0, \pm 1$ being the equilibria of the system
Letting$\ \dot x = f(x)$
$\Rightarrow f'(x) = 1-3x^2 $
$f'(0) = 1$
$f'(1) = -2$
$f'(-1) = -2$
We can see here that $\bar x = 0$ is unstable, and $\bar x = \pm 1$ is stable (Lyapunov)
Calculating the flow:
$$\dot x = x-x^3 \Rightarrow \frac{\dot x}{x-x^3} = 1$$
Integrating: $$\int \frac{\dot x}{x-x^3} dt = \int \frac{dx}{x-x^3} = \int dt$$
$$\frac{1}{x-x^3} = \frac{1}{x(1-x)(1+x)} = \frac{1}{x} + \frac{1}{2(1-x)} -\frac{1}{2(1+x)}$$ via partial fractions.
Thus: $$\int \frac{dx}{x-x^3} = \int \frac{1}{x} + \frac{1}{2(1-x)} -\frac{1}{2(1+x)} dx = \log x - \frac{\log (1-x)}{2} - \frac{\log(1+x)}{2} = t+c $$ where c is a constant
Multiplying both sides by 2 and taking expontentials:
$$\frac{x^2}{(1-x)(1+x)}= e^{2(t+c)}$$
$$\Rightarrow x^2 = (1-x^2)e^{2(t+c)}$$
$$\Rightarrow x = \pm \sqrt\frac{e^{2(t+c)}}{1+e^{2(t+c)}} $$
Taking an arbitrary initial condition $ x_0 = x(0)$:
$$x_0 = \pm \sqrt\frac{e^{2c}}{1+e^{2c}} $$
Rearranging to get:
$$\frac{x_0 ^2}{1-x_0 ^ 2} = e^{2c}$$
Thus the flow is:
$$x = \pm \sqrt\frac{e^{2t}(\frac{x_0^2}{1-x_0^2})}{1+e^{2t}(\frac{x_0^2}{1-x_0^2})} = \pm \frac{e^t x_0}{\sqrt{ 1-x_0^2 +x_0^2 e^{2t}}}$$
The problem that I now have is getting an intuition and feeling of how this expression behaves as $t \rightarrow \pm \infty$ and how one would go about analyzing the stability directly, if I am wrong? I have searched texts however many use the notions from topology and such which I have not yet studied.
I must apologise for the formatting awkwardness in some parts.