The parameterized map . For each fixed μ value, the attracting set in [0, 1] is shown. There are infinitely many cascades. For this map, each cascade has precisely one saddle-node bifurcation. The branch of unstable orbits ending at the saddle-node bifurcation is shown for period-1 and period-3. Each regular periodic orbit within the map's horseshoe at is uniquely connected to one of the (infinitely many) cascades. The regular period-one and period-three orbits are shown here, depicted respectively by one and three dots at . This is the bifurcation diagram most frequently displayed to illustrate the phenomenon of period-doubling cascades. However, cascades occur for much more complex dynamical systems that are completely unrelated to quadratic maps, as shown for example in Fig. 2 .
The attracting set for the double-well Duffing equation: This equation is periodically forced with period . Therefore, the F = time- map is a diffeomorphism on parameterized by ω. Depicted here is the attracting set of F, projected to the -plane. The constant 0.01 has been added to destroy symmetry in order to avoid non-generic symmetry-breaking bifurcations.
A path in a component. There is one component that contains fixed points (as well as orbits of period 2 n for . The path h(s) shown in red passes through each orbit in this component exactly once. As shown, at each period-doubling bifurcation, h(s) passes through the upper branch, but this choice is arbitrary. If the path were chosen so it sometimes went through the lower branch of periodic points, it would still pass through the same orbits. Both branches of periodic points describe the same orbits. Note that the lower branch extends infinitely far to the right and infinitely far down. The limiting end point is not a periodic point and is not a point of the cascade. The path can be thought of as the image of a function for s in an open interval J, and the path h(s) passes through each orbit exactly once. Furthermore, for an appropriate change of variables, we can choose J = (–1, +1).
Paths on families of orbits. Each point in the curves of this figure represents an entire orbit. This figure depicts three different paths of periodic orbits. It outlines the situation in Theorem 1 and Lemma 1. Path segments between bifurcations consist of regular saddles (denoted S), attracting orbits, or repelling orbits (both denoted AR). The bifurcation orbits for a generic family are all saddle-node (SN), period-doubling, or period-halving (both denoted PD) bifurcations. If while following a path, the path reverses direction ( is increasing on one branch and decreasing on the other), as with p 3 and p 4 at SN, then one branch is Type S while the other is type AR. The path starting at p 1 reverses direction twice, both times at a SN point. If a path connects orbits at two ends of the parameter interval, as with p 1 and p 5, then they are the same type. We show that there are infinitely many Type S segments starting at that do not return to , and only a finite number of these may connect to orbits at , and these must be Type S. Hence, there are infinitely many paths like the one stemming from p 2, remaining forever in a bounded region between and and containing cascades.
Paired cascades. This figure depicts two sets of period-seven cascade components for the Hénon map , each containing paired cascades. Only one point of each of the period-7 orbits of the Hénon map are shown, so that it is clearer how each pair is connected by its component. The leftmost and rightmost cascades are paired; that is, they lie in the same component of orbits. They are connected by a path of unstable regular periodic orbits (the orbit connecting the lefthand and righthand sides of the figure). Likewise, the two middle cascades are paired. They lie on (and are connected by) a path of attracting period-seven orbits. Paired cascades are not robust to macroscopic changes in the map in that both can be simultaneously destroyed by a large enough local perturbation.
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