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It is shown that the topology of periodic orbits belonging to different series of period-doubling bifurcations can be distinguished by the structure of a template which represents a topological nature of orbits. In order to do this, a new quantity will be introduced, the local crossing number , which counts the number of half-twists of a period-doubled orbit along the tubular neighborhood of the orbit just before the bifurcation. The local crossing number can be extracted by inspecting the power spectrum of period-doubling orbits. The topological nature of the orbits is further investigated by symbolic dynamics with the help of the template to find that any series of period-doubling orbits can be characterized by a unique series of periodic blocks: W = { y , xy , xy 3 , xy 3 ( xy ) 2 ,…}. A new concept, irreducible and reducible templates, is introduced. With the irreducible template, various models treated by others are reexamined. |
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