# Techniques to prove that solutions of ODE are defined for all time.

What are the main techniques for proving that the solutions of ODE are defined for all times? Obviously I refer to qualitative study of $$y'=f(t,y)$$ with $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$ at least $C^0$. The standard way is to use classical reults (e.g. prove that $f$ is sublinear or globally lipschitz) but these rusults are often not applicable. In this case I try to show that the solutions are bounded (it is known that if the orbits are bounded then the solutions are defined for all time). To do this I usually try one of the following techniques:

• I look for a constant of the motion and I study the level sets to understand the shape of the orbits (and see if they are bounded);

• I find a particular solution that is a closed and curve (e.g. a circle): in fact for the uniqueness the orbits can not intersect and therefore all orbits that start from inside this closed curve will have to remain there and will therefore be limited;

• I study the function $h(t): = x^2(t) + y^2(t)$: I make the derivative and I try to show that it is limited;

• I try a change of variables (e.g. polar coordinates) that simplifies the equations.

In conclusion:

1) Do you know other general techniques to show that the orbits are bounded?

2) However, boundedness is only a sufficient condition to prove that solutions are defined for all time! In case the solutions are unbounded how can I prove that they are defined for all time? For example consider $$\begin{cases}\dot x=-y^2\\ \dot y = x^2\end{cases}$$ or $$\begin{cases}\dot x=y^2\\ \dot y = x^2\end{cases}$$ The level sets of the constants of motion ($E:=x^3\pm y^3$) of these system are open and unbounded curve. In this case the only techniques that I know is to cumpute $t=\int^{+\infty}_{x_0}...dx$ and understand if this integral converges or diverges (for other details see the end of this page: Qualitative study of $\dot x = - y^2$, $\dot y= x^2$).

• The simplest approach and really the standard, if there is one: Gronwall. – John B Jul 3 '18 at 21:40
• @JohnB it's included in the case "$f(t,y)$ sublinear in $y$". – Ef_Ci Jul 3 '18 at 21:44
• The first and third approaches can be subsumed under Lyapunov function. – user539887 Jul 3 '18 at 21:44
• @user539887 good idea! Thanks. – Ef_Ci Jul 3 '18 at 21:46
• @user539887 By the way, I kind of found the reference: in Nemytskii-Stepanov book (p.19) there is Vinograd theorem which states that reparameterization that makes vector field complete exists. However, they do it differently than I remembered. At least some reference exists :) – Evgeny Jul 5 '18 at 20:52

In mathematical population biology, a totally competitive system of ODEs is defined as $$x'_i = x_i f_i(x_1, \dots, x_n), \quad i = 1, \dots, n,$$ where $$\frac{\partial f_i}{\partial x_j} < 0$$ (of course $f_i$ are assumed to be $C^1$, at least). The natural phase space is the nonnegative orthant $C = \{\, x = (x_1, \dots, x_n) : x_i \ge 0$ for all $i = 1, \dots, n \,\}$.
As $x_i$ represents the population density of the $i$-th species, only solutions that are bounded for positive time have biological significance. There is a very simple, but extremely useful, sufficient condition for such boundedness:
For each $i = 1, \dots, n\$ there is $x_i^* > 0$ such that $f_i(0, \dots, 0, x_i^*, 0, \dots, 0) = 0$.
Indeed, let $x \in C$ be such that for some $j$, $x_j > x_j^*$. Then $$f_j(x) = f_j(x_1, \dots, x_{j-1}, x_j, x_{j+1}, \dots, x_n) \le f_j(0, \dots, 0, x_j, 0, \dots, 0) < f_j(0, \dots, 0, x_j^*, 0, \dots, 0) = 0.$$ From this it follows that all solutions are attracted toward the ($n$-dimensional) parallelepiped $[0, x_1^*] \times \ldots \times [0, x_n^*]$. Consequently, each solution in $C$ is defined for all positive times $t$.