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I am trying to formalize the following proof on Perko's Differential Equations and Dynamical Systems, which says that a periodic orbit has index +1.

enter image description here

My only problem is trying to prove that the map $g$ is continuous on $T$. Geometrically it seems obvious, and I am trying to prove it using limits, but I do not get anything. The following figure (also from Perso's book) is a nice illustration:

enter image description here

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All follows from the definition of the vector field $\bf u$. For example, on the line $s=t$ you have $$ \lim_{t\to s^+}\frac{x(t)-x(s)}{\|x(t)-x(s)\|}=\lim_{t\to s^+}\frac{\frac{x(t)-x(s)}{t-s}}{\left\|\frac{x(t)-x(s)}{t-s}\right\|}=\frac{x'(s)}{\|x'(s)\|}={\bf u}(x(s)). $$

The proof and figures are copied from Coddington and Levinson's book, really the best book there is on this particular topic.

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  • $\begingroup$ If I am not mistaken, you are only approaching the line $s=t$ from "above", but I think we need to compute that limit aproaching from everywhere (for example on a diagonal) $\endgroup$ – user203327 Mar 23 '16 at 20:53
  • $\begingroup$ Of course, but on the diagonal it is trivial. $\endgroup$ – John B Mar 23 '16 at 20:57
  • $\begingroup$ I mean on something like, on the point $(1/2,1/2)$, the line $y=-1/4+3/2*x$ aproaching $(1/2,1/2)$ from the right $\endgroup$ – user203327 Mar 23 '16 at 21:09
  • $\begingroup$ That is a very particular case of my answer. $\endgroup$ – John B Mar 23 '16 at 21:12

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