Solutions of DGL-Systems at infinity

Question

I have a solution $H(t)$ for the dynamical system $$\dot x = v(x)$$ with $v:\mathbb{R}^n\rightarrow \mathbb{R}^n$ a vector field and $\lim\limits_{t\rightarrow \infty} H(t)=\lim\limits_{t\rightarrow -\infty} H(t)$. Why trends $\lim\limits_{|t|\rightarrow \infty} H(t)$ to a fixed point?

Thank you!

Answer 1

Consider that in $$ H(n+1)-H(n)=\int_0^1 v(H(n+s))\,ds $$ the left side converges to zero for $n\to+\infty$ while the right side converges to $v(H_*)$ where $H_*=\lim_{t\to+\infty}H(t)$.

Thus $v(H_*)=0$ which makes $H_*$ a fixed point.