Existence, uniqueness and lifetime of solutions

Theorem. Consider the equation $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ and a point $\boldsymbol{x}_0$. Suppose that $\boldsymbol{f}$ is differentiable and that all its partial derivatives, namely the functions $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, $\frac{\partial g}{\partial x}$ and $\frac{\partial g}{\partial x}$, are continuous functions of $\boldsymbol{x}$. Then there exists a time interval $(-\tau,\tau)$ about $t=0$, with $\tau>0$, such that the equation has a unique solution $\boldsymbol{x}(t)$ such that $\boldsymbol{x}(0)=\boldsymbol{x}_0$.

In short, existence and uniqueness of solutions are guaranteed if $\boldsymbol{f}$ is continuously differentiable. The proof of this theorem is the same as in dimension one and it is also the same in higher dimension! From now on, all our vector fields are smooth enough, so that the above theorem applies. In most examples, indeed, vector fields will be most of the time polynomial functions in the variables $x,y$, or sometimes rational functions with a non vanishing denominator.

This theorem has an important corollary: different trajectories never cross. If two trajectories did intersect, then there would be two solutions starting from the same point (the crossing point), and this would violate the uniqueness part of the theorem. Therefore, trajectories partition the $xy$-plane. In particular, if there is a fixed point $\bar{\boldsymbol{x}}$ (meaning that $\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{0}$), and if it can be reached, the time it takes is infinite (if one does not start at this point, of course).

For exactly the same reason as the one explained in dimension one, there is no loss of generality in considering $\boldsymbol{x}(0)=\boldsymbol{x}_0$ at time $t=0$ because the vector field does not depend on time. In dimension one, we saw a simple example showing that, in general, solutions do not exist for all times. This is not a simple issue but, fortunately, for all applications we shall see, solutions exist for all times.

For two-dimensional systems (as opposed to higher-dimensional ones), the uniqueness part of the above theorem have strong topological consequences that we explore later on. For example, suppose that there is a closed trajectory $C$ in the $xy$-plane, which corresponds to a periodic solution. (We already met examples of such solutions in the opening part.) Then any trajectory traced out by a solution starting at a point inside $C$ is trapped in there forever. What is the fate of such a bounded solution? We shall see that we can answer this question thanks to the Poincaré-Bendixson theorem.