While reading some basic introductory texts in nonlinear dynamics, it was asserted that no chaotic behaviour for flows can occur in $\mathbb{R}^2$ because of the Poincare-Bendixson Theorem. Intuitively, this makes sense; essentially the long term behaviour of any bounded trajectory must either tend to a critical point or a periodic orbit and so there is nothing "unexpected" or "unexplainable" that can occur. However, I was wondering if there exists a rigorous proof of such a statement? I am aware that a universal mathematical definition of chaos does not exist, so for my question to be precise, let us adopt Devaney's defintion of chaos in $\mathbb{R}^2$:
Let $U \subseteq \mathbb{R}^2$ be a closed invariant set and let $\dot{x} = f(x)$ be a continuous dynamical system, where $f: U \to \mathbb{R}^2$ is $\mathcal{C}^1$. Then we say that $f$ is chaotic on $U$ if it is 1) topologically transitive on $U$, 2) if $f$ has dense periodic orbits in $U$, and 3) $f$ has sensitive dependence on initial conditions.
Then is it possible to rigorous prove that no chaos exists for continuous dynamical systems on $\mathbb{R}^2$? I am asking because I can't quite see how it can be done directly from the Poincare-Bendixson Theorem, so any suggestions will be much appreciated!