Questions: Poincare-Bendixson Theorem

Question 1

You have a 2D system where you can prove that trajectories enter a bounded annular region (a ring) and never leave, and that the region contains no fixed points. What can you conclude?

• A. Nothing — you need to solve the equations to determine the long-term behavior
• B. The system must have at least one stable limit cycle inside the annular region
• C. The system is chaotic because trajectories are trapped and can't reach a fixed point
• D. The system must have a strange attractor inside the region

Answer: (B)

This is exactly the setup for the Poincare-Bendixson theorem. Trajectories are confined to a bounded, closed region (the annulus) with no fixed points. The theorem guarantees the omega-limit set must be a periodic orbit. In practice, this means there is at least one limit cycle in the annulus. This is one of the most powerful techniques for proving the existence of limit cycles without explicitly finding them.

Question 2

A researcher claims to have found chaos in a two-dimensional autonomous continuous-time system. Is this possible?

• T. True
• F. False

Answer: False

The Poincare-Bendixson theorem rules out chaos in 2D continuous autonomous systems. In two dimensions, the only possible omega-limit sets are fixed points, periodic orbits, and heteroclinic/homoclinic cycles (connections between fixed points). There is no room for the stretching-and-folding that produces sensitive dependence on initial conditions. Chaos requires at least three continuous dimensions (like the Lorenz system), or a discrete map in lower dimensions. The researcher must have an error, a non-autonomous system, or a discrete-time map.

Question 3

The Poincare-Bendixson theorem applies to all dynamical systems, including discrete maps and systems in three or more dimensions.

• T. True
• F. False

Answer: False

The theorem is specific to continuous flows in two dimensions (or on two-dimensional surfaces). It fails completely in three or more dimensions — the Lorenz system is 3D and chaotic. It also doesn't apply to discrete maps — the logistic map is one-dimensional and chaotic. The theorem exploits the topology of the plane: trajectories can't cross in 2D (by uniqueness), which severely constrains what limit sets look like. In 3D, trajectories can pass over and under each other, allowing the stretching and folding that produces chaos.

Question 4

Explain the practical strategy for using the Poincare-Bendixson theorem to prove a limit cycle exists in a specific system.

Model answer: Construct a trapping region — a bounded, closed subset of the phase plane that trajectories enter but cannot leave (the vector field points inward on the boundary). Then show that the trapping region contains no stable fixed points (either no fixed points at all, or only unstable ones). By Poincare-Bendixson, the omega-limit set of any trajectory in the region must be a periodic orbit. Often the trapping region is an annulus: the outer boundary is established using a Lyapunov-like argument showing trajectories can't escape to infinity, and the inner boundary comes from the instability of a fixed point (trajectories spiral away from it).

For the van der Pol oscillator, the strategy works beautifully. Near the origin, the fixed point is unstable (eigenvalues have positive real part for μ > 0), so trajectories spiral outward — the inner boundary of the annulus. Far from the origin, the strong damping (x² ≫ 1) drives trajectories inward, establishing the outer boundary. The annulus contains no fixed points (the only one is at the origin, excluded by the inner boundary). Poincare-Bendixson guarantees a limit cycle exists between the boundaries.