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Similarities with Other Attractors

المؤلف:  Garnett P. Williams

المصدر:  Chaos Theory Tamed

الجزء والصفحة:  187

11-3-2021

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Similarities with Other Attractors

Before examining the characteristics of a chaotic attractor, let's see why it still qualifies as an attractor. Here are several features it has in common with all attractors.
• It's still the set of points (but in this case an infinite number of points) that the system settles down to in phase space.
• It occupies only certain zones (and is therefore still a shape) within the bounded phase space. All data points are confined to that shape. That is, all possible trajectories still arrive at and stay "on" the attractor. (As with nonchaotic attractors, a trajectory technically never gets completely onto a chaotic attractor but only approaches it asymptotically.) In that sense, a chaotic attractor is a unit made up of all chaotic trajectories. Figure 1 shows a two-dimensional view of the well known Rössler (1976) attractor, a three-dimensional strange attractor designed as a simplification of the Lorenz attractor.
• A chaotic attractor shows zones of recurrent behavior in the form of orderly periodicity, as explained below.
• It's quite reproducible.
• It has an invariant probability distribution, as explained in the following section.

Figure 1: Two-dimensional projection of the Rössler strange attractor. Computer-generated graphics by Sebastian Kuzminsky.