Does Homoclinic tangle violate deterministic law? 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 ?
 A: 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.
