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I am reading a book by Alligood et al. on dynamical systems. The following picture is supposed to show the stable and unstable manifold of the map. According to the book the cross sign seen on the leftmost and bottom corner of the picture is the saddle point of this map. Henon map is defined as: $$f(x,y)=(a-x^2+by,x)$$ enter image description here

and in the right picture $a=1.4$ and nothing is mentioned about $b$. The point is how can we get that many intersections between stable and unstable manifolds when we only have two fixed points?; how can unstable and stable manifold intersect each other but don't generate a fixed point???

Cupitor
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    Stable and unstable manifolds have a property that if they intersect once (transversally), they will intersect infinitely many times. They are not all fixed points though. Infact, the fixed point is the infinite time limit of such points. – nonlinearism Feb 07 '14 at 14:37
  • any reference to "Stable and unstable manifolds have a property that if they intersect once (transversally), they will intersect infinitely many times" ?@nonlinearism – BAYMAX Jul 03 '18 at 07:17

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