I am reading the following textbook:
Introduction to Applied Nonlinear Dynamical Systems and Chaos by Wiggins
p.92 (top)
Consider $$\dot{x} = f(x)$$ where $f(x)$ is $C^r$, $r\geq 1$, on some open set $U\in \mathbb{R}^n$. Suppose that the solutions exist for all time (Leave it as an exercise to make the necessary modifications when solutions exist only on finite time intervals).
My problem is what are the conditions to make solutions exist for all time.
Consider and example: $$\dot{x} = x^2, \ \ \ \ x\in \mathbb{R}$$
the solution through $x_0$ at $t = 0$ is $$x(t,0,x_0) = \frac{-x_0}{x_0t – 1}$$
this solution (trajectory) does not exist for all time, since it becomes infinite at $t = 1/x_0$.
So what are the conditions to make solution exist for all time
Best Answer
Usual conditions for the infinite existence of solutions of $\dot x=f(x)$ are restrictions to linear growth like $$ |\dot x|=|f(x)|\le C(1+|x|) $$ or $$ \frac12\frac{d}{dt}|x|^2=|x^T\dot x|=|x^Tf(x)|\le C(1+|x|^2) $$ which impose an exponential bound on the growth of solutions. The solution of the corresponding differential inequalities is with the Grönwall lemma.
These are not the only conditions, only the relatively easy ones. If the vector field $f(x)$ points inwards on the boundary of a bounded set $U$, all solutions are restricted to $U$ and thus bounded and exist for all positive times. This looks geometrically simple but can be hard to establish