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**Homework Statement::**Can someone explain the finite number of equilibria outcome of the Poincaré-Bendixson Theorem?

**Relevant Equations::**Poincaré-Bendixson Theorem

**[Mentor Note -- General question moved from the schoolwork forums to the technical math forums]**

Hi,

I was reading notes in dynamical systems and have the following question about the Poincaré-Bendixson theorem.

**Context:**The Poincaré-Bendixson Theorem states:

"Let M be a positively invariant region of a vector field, containing only a finite number of equilibria. Let ## x \in M ## and consider ##\omega(\mathbf{x})##. Then one of the following possibilities holds:

(i) ##\omega(\mathbf{x})## is an equilibria

(ii) ##\omega(\mathbf{x})## is a closed orbit

(iii) ##\omega(\mathbf{x})## consists of a finite number of equilibria ## x_1 ^{*}, x_2 ^{*}, ..., x_n ^{*} ## and orbits ##\gamma## with ##\alpha (\gamma) = \mathbf{x}_i ^{*} ## and ##\omega (\gamma) = \mathbf{x}_j ^{*} ##

"

'Positively invariant' basically means that once a trajectory enters a region, it won't escape (a non-mathematical explanation, but it helps me to visualize what is going on)

**Question:**What is meant by possibility (iii) about the finite number of equilibria and the ##\omega## and ##\alpha## limits (/cycles)? I cannot visualize what is going on and would appreciate any help (or sketch if possible).

I understand what is meant by the first two possibilities, but not the third.

**Attempt:**

Breaking down the 'sentence':

- I don't understand how the ##\omega## set can contain multiple equilibria, without it being a cycle.

- I don't understand what is meant by the latter half at all

Many thanks in advance.

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