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In a dynamical system, the stable and unstable manifold of a fixed point can intersect outside this point. I can understand the existence of homoclinic connections (orbits), which, as far as I'm concerned, just means that the stable and unstable manifold overlap exactly. But for homoclinic tangles, the picture here shows that the two manifolds can intersect for infinite times.

Do these intersections violate the deterministic feature of this dynamical system, since at each intersection point there are two different velocity directions ?

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  • $\begingroup$ If I remember correctly, the actual velocity at the "connection" point is zero, which corresponds to the fact that you can only reach the connection in infinite time. But I may misremember. $\endgroup$ – Ian Aug 25 '15 at 13:56
  • $\begingroup$ @lan Do you remember from which book you get the detailed description ? $\endgroup$ – kevin Aug 25 '15 at 14:01
  • $\begingroup$ Actually, the picture from link is drawn for diffeomorphism, not for system of ODEs. In case of diffeomorphism this tangle violates nothing. $\endgroup$ – Evgeny Aug 25 '15 at 15:33
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    $\begingroup$ @Evgeny I am not a math major, I assume you are referring to maps such that the intersection point cannot be reached since the trajectory is discrete. But from the setup equation (2.1) and (2.2) in this short essay, it seems that homoclinic tangle exists for continous dynamical system. I am not sure. $\endgroup$ – kevin Aug 25 '15 at 16:17
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    $\begingroup$ Strictly speaking, the homoclinic tangle (in the sense that it's the consequence of transverse homoclinic intersection) exists in Poincare return map, not in an initial system of ODEs. But the presence of such complicated structure in the return map implies that life is far from easy in the original system of ODEs. $\endgroup$ – Evgeny Aug 25 '15 at 17:01
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Of course not. As already mentioned in the comments, homoclinic tangle is considered for maps, usually derived by taking a Poincare section in phase space of continuous systems. For example, consider a continuous systems with hyperbolic periodic orbit with period T in $R^3$ (or equivalently, consider a non-autonomous continuous system in $R^2$). Then the time-T map of this system, which will be map $P:R^2\rightarrow R^2$, you may see a homoclinic tangle. This is exactly what is done in the notes that you mention (since the second term in eq. 2.2 is time-dependent). Once you have time-dependence, you are really working in one higher dimension that the dimension of $x$. Hence, the tangles are NOT intersecting in the full 2 (for x)+1 (for time) dimensional space, but rather in a 'collapsed' 2D space, in which time dimension has been collapsed. It is like looking from above on to the x-y plane if time is Z axis.

"Deterministic law" as mentioned in original question, is nothing but uniqueness of solution of the system of ODEs under consideration. If the r.h.s is Lipschitz (or even better differentiable/smooth), it is guaranteed to have uniqueness. And we surely get tangles for many smooth systems.

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  • $\begingroup$ Great :) that's almost the same that I wanted to write in full answer but was too tired to do it yesterday :) $\endgroup$ – Evgeny Aug 26 '15 at 5:07

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