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

Question

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.

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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 $).

3) Advice on books and readings about this problem are welcome.

Answer 1

An answer to 1):

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 $t$.

For (much) more on such systems, see, e.g., Zhao's book Dynamical Systems in Population Biology. See also the classical paper by M. W. Hirsch Systems of differential equations which are competitive or cooperative: III. Competing species.