I think you are correct that the Poincaré-Bendixson theorem applies on a torus, and so you would not expect chaos with a positive LE. Since the orbit cannot intersect itself, the most complicated trajectory is quasiperiodic and eventually fills the entire torus. Such an orbit is ergodic. Of course periodic orbits are also possible as well as orbits that approach an equilibrium point.

Professor's Sprott answer is clear and to my understanding correct. I would only add that in the case of quasiperiodic behavior -- resulting from two frequencies whose ratio is irrational, from @Andrew's remark -- the surface of the 2D torus (T2) is covered gradually in a *predictable* way -- because the trajectory cannot intersect itself, from @Sprott's answer. For a T2 solution of a 3D dissipative system, I think you will find two null and one negative LEs. In a torus breakdown route to chaos one of the null LEs becomes positive and the asymptotic solution will no longer be on a T2.

Professors Julien Clinton Sprott, Luis Antonio Aguirre, I had some conversation with a mathematician about this question.

He have said me few confusing thins. For example, in the Poincaré-Bendixon theorem, you deal with the \omega limit sets. But in the Schwartz theorem (which is the generalization for the 2d torus), you deal with the minimal sets (see Article The Poincaré-Bendixson Theorem: From Poincaré to the XXIst century

The trick is in fact that the minimal set and the \omega limit set mean different things. Minimal set means such a set that is invariant under the system's flow, not the points lying on the system's limit trajectories.

He have also said me that 0 largest Lyapunov exponent (LLE) does not mean that the system is insensitive to its initial conditions. For instance, if the trajectories diverge not in exponential way, but in some power law, then the system has the sensitive to the initial condition, but 0 LLE. (see https://iopscience.iop.org/article/10.1088/0951-7715/19/9/001/meta)

So, to this moment, I understand that the statement that chaos is absent in 2d smooth dynamical systems holds on a physical level of severity, not the mathematical one.

The usual definition of chaos requires that errors in initial conditions grow EXPONENTIALLY in time on average, and hence a positive Lyapunov exponent. There are many systems for which the error grows as a linear or even a polynomial function of time and are thus somewhat sensitive to initial conditions but are not considered chaotic.

A simple example is two race cars traveling at the same speed on a circular track. If one starts and remains at a slightly larger radius, their separation will grow linearly in time, hence not chaotic.

It's also possible for the error to grow exponentially in a system that is not chaotic. Imagine two people investing slightly different amounts of money in an account with the same compound interest rates. The difference in their wealth will increase exponentially, but the system is not chaotic because the solution is unbounded. Hence the need for a nonlinearity to bring things back close to where they started (Poincare recurrence).