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Extra info for Can Technical Analysis Still Beat Random Systems?
12a, where the attracting set A is a fixed point. The stable set of the saddle separates the basins of attraction of A and P*. The branch a;i of W^ (5*) turns around P*. The branch ai of the unstable set W^ (5*) tends to P* whereas the cj-limit set of the points of the branch 0^2 is the attracting set A. After the homoclinic loop, or homoclinic tangle, of the two branches 33 1 Some Methods for the Global Analysis Figure 11: Qualitative representation of a mechanism leading to the appearance of an attracting closed curve.
1 Some Methods for the Global Analysis 31 All these loops correspond to structurally unstable situations and cause a qualitative change in the dynamic behavior of the dynamical system. , a study of the linear approximation of the map, we classify them as global bifurcations. Indeed, we study this kind of bifurcation looking at the asymptotic behavior of the stable and unstable sets of the saddle: If a bifurcation associated with a loop has occurred, before and after the bifurcation the involved branch of the unstable set converges to different attracting sets, and the points of the involved stable branch have a different a-limit set, as well.
Summarizing, we have seen that the coexistence of two closed invariant curves, one attracting and one repelling, in discrete maps can be achieved by a double mechanism: Starting from a repelling cycle and a saddle cycle, a first saddle connection (or tangle) causes the appearance of the attracting one associated with an (unstable) heteroclinic connection saddle - repelling cycle that plays the role of separatrix of basins, which is then replaced by the second closed curve, repelling, whose appearance is associated with a second saddle connection (or tangle).