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I am interested in a proof of Theorem 1.116 in Application of ODEs (p. 82).

i) a rest point, ii) a periodic orbit follows if we assume $\omega (p)$ i) is an union of rest points or ii) doesnt include rest points, with theorem 1.111 and Poincare-Bendrixon mentioned in that book.

My problem is I cannot see how to proof iii) a union of finitely many rest points and perhaps a countably infinite set of connecting orbits.

The only case which is left open is when in $\omega (p)$ both rest points and normal points with $f(y) \ne0$ are included. How can I argument form here?

Greetings.

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    The omega-limit set is connected. – 2017-01-29
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    How can you derivate "...perhaps a countably infinite set of connecting orbits." from that fact? Can someone give me an example of a system which has infinitely many connected orbits? – 2017-01-31
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    When there are infinitely many connecting orbits, only finitely many of them are heteroclinic. Again: The omega-limit set is connected. But there are so many books with the proof. – 2017-01-31
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    Can you recommend me one then? I cannot see why a heteroclinic orbit should be in the limit set. Thank you for your help, btw. – 2017-01-31

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