The two following theorems appear to be contradictory to me. I'm sure I must have overlooked something significant here.
The Poincaré-Bendixson II(a) says that if $R$ is a Type I invariant region, and has a finite number of nodes or spiral points, then given any initial position $X_0$ in $R$, $\lim_{t\to \infty}=X_1$ for some critical point $X_1$.
But how can this be possible? That the region $R$ has a finite number of nodes or spiral points, yet has no periodic solutions in $R$. Because Poincaré-Bendixson I(a) says that if $R$ is a Type I region that has a single unstable note or an unstable spiral point in its interior (and thus has a finite number of nodes or spiral points), then there must be at least one periodic solution in $R$.
Any help is greatly appreciated.
Best Answer
These two versions of the Poincaré-Bendixson Theorem are not contradicting each other. Instead, I think you overlooked some of the hypotheses.
The first theorem tells you that a positively invariant, compact subset of the phase plane always contains at least one closed orbit, provided there are no fixed points in it (or provided it has just one unstable node or spiral point in its interior). While the proof of this fact relies on the topology of $\mathbb{R}^2$ a lot, the idea behind its formulation is easy to understand: if a trajectory is trapped in a compact subset of the phase plane which does not contain any stable fixed points, then it would have nothing to do but to approach a limit cycle.
The second theorem relies on different hypotheses: $R$ is still a positively invariant, compact subset of the phase plane, but it does not contain any periodic solutions now. So you should ask yourself: if a solution is forever trapped in $R$, and there are no limit cycles to be approached, what can that solution possibly do? Flowing toward a stable equilibrium point, of course. Indeed, this answers your question: it is possible to have a region $R$ that is a positively invariant, compact subset of the phase plane, with no periodic solutions in it, that still contains a finite number of nodes or spiral points: you'd just need some of those fixed points to be stable.
Also, pay attention to the fact your book says nothing about a positively invariant, compact subset of the phase plane that is both free from equilibrium points and limit cycles. Such a region cannot exist, and this is the very essence of the Poincaré-Bendixson Theorem: once you manage to trap a solution inside a region like $R$, then it can only either approach a (stable) fixed point, or spiral toward a limit cycle.
There's actually a third possibility, that of approaching a more complicated object called a cycle graph (which consists of a finite number of fixed points together with the orbits connecting them). Yet the spirit of the theorem does not change: a bounded solution may not wander in a chaotic fashion, because nothing different from approaching a stable fixed point, a limit cycle or a cycle graph is allowed. As a matter of fact, the Poincaré-Bendixson Theorem is what tells you that chaos can never occur in a two-dimensional dynamical system.