I am just discussing the solution of the differential $x'=f(x)=x(x-1)(x+1)$ with initial value $x(0)=x_0$. Since $f(x)$ as a polynomial is continuously differentiable indeed $f$ is Lipschitz-continious and hence, it exists a unique solution $x:I \to \mathbb{R}$ on a compact $I$ (Picard-Lindelöf). Let $x$ be a maximal solution of the given IVT. Then let $x:(a,b) \to \mathbb{R}$ and $x_0>1$. I know have to show/calculate the following:
- $a= -\infty$ and $\lim_{t \to -\infty}x(t)=1$.
- $\lim_{t\to b}x(t)$ and
- $b<\infty$.
Now, all these are very easily shown by solve the IVT straight forward getting $x(t)=\frac{x_0}{\sqrt{e^{2t}(1-x_{0}^2)+x_{0}^2}}$ for any $x_0>1$. But there must be a more rudimental way to get these results by qualitative considerations. One idea I had was the $\omega-$/$\alpha-$limes-set of the Orbit of the solution. But I can't find the arguments.
Is there anybody having ideas for me, please? I am looking forward for your answers! Thanks a lot in advance!