A natural question is whether we can classify all possible types of long-term behaviors of solutions of two-dimensional systems. We saw two types of behaviors: as $t\to+\infty$, solutions that tend to a fixed points, or solutions that wrap around a closed trajectory (limit cycle). Are there other types of behaviors?
We consider a two-dimensional system
$$ \begin{cases} \dot{x}=f(x,y)\\ \dot{y}=g(x,y) \end{cases} $$
where $f$ and $g$ are continuously differentiable functions.There is a first dichotomy: a solution can escape to infinity, or remain in a bounded region of the plane. In applications, the first case is obviously uninteresting. We now state a theorem which can be viewed as a generalization of Poincaré-Bendixson theorem.
Theorem. Suppose that a solution of the system lies in a bounded region of the plane in which there are finitely many fixed points. Then, as $t\to+\infty$, it approaches either- a single fixed point;- a single closed trajectory;- a graphic, that is, a finite set of trajectories, each of them connecting one of the fixed points as $|t|\to +\infty$.
Let us give an example of the third possibility. Consider the system
$$ \begin{cases} \dot{x}= \sin(x) \big(-0.1 \cos(x)-\cos(y)\big)\\ \dot{y}= \sin(y) \big(\cos(x)-0.1 \cos(y)\big). \end{cases} $$
There are fixed points which are saddles at the corners of the square $(0,0)$, $(0,\pi)$, $(\pi,0)$, and $(\pi,\pi)$, as well as many other points (due to the periodicity of the vector field). There is also a spiral source at $(\pi/2,\pi/2)$. All solutions emanating from a point inside the square (except, of course, $(\pi/2,\pi/2)$) accumulate on the boundary of the square. Each side of the square is a trajectory which connects two fixed points (it takes an infinite amount of time to make the connection).
The conclusion that can be drawn from the previous theorem is that the behavior of solutions in the long term is pretty simple. In particular, no ‘chaos’ is possible in phase plane! We shall see in the next part that this drastically changes for higher dimensional systems: solutions can accumulate on strange complicated sets.