I'm reading the Poincaré-Bendixson Theorem's proof in the book I've mentioned. I feel kind of curious about the concept shown there: the transversal, and how they used it. I would like to know if there are other uses of this concept in other proofs, specially if they are oriented to study properties in ODE's. Thank you.
There are more uses of the concept: Transversal, given in Coddington-Levinson book of Theory of ODE?
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ordinary-differential-equations
dynamical-systems
bifurcation
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0I am not quite sure, but maybe start [from here](https://en.wikipedia.org/wiki/Transversality_(mathematics)) – 2017-01-01
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0@polfosol Actually what OP is asking about is closer to [this concept](https://en.wikipedia.org/wiki/Poincar%C3%A9_map) :) Transversals (or cross-sections) are key instruments in studying such low-dimensional objects as flows without singularities on tori (could be reduced to circle mappings), limit cycles on plane and even for occurence of chaotic behaviour in case of Shilnikov saddle-focus loop. The behaviour of trajectories in the neighbourhood of periodic orbit also could be studied by means of studying properties of fixed point of corresponding Poincar\'e mapping. – 2017-01-02