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Counting the number of periods in one-dimensional maps with multiple critical points

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  • Xie, Fa-geng
  • Hao, Bai-lin

Abstract

The problem of counting the number of different periodic orbits in continuous maps of an interval is solved. A map with m monotone pieces (laps) and m - 1 critical points may have at most m - 1 independent kneading sequences, which provide the most convenient parameters for the map. When one or more kneading sequences are kept constant or bound to vary simultaneously, various degenerated cases of the map arise. The number of period n orbits of the general m-lap map as various degenerated cases is given by combinations of Nink(n) with k⩽m, where Nm(n) is the number of period n orbits of a particular kind of degenerated m-lap map with only one kneading sequence. The quantity Nm(n) may be calculated or enumerated by many different methods, which are all discussed in this paper. We also give tabulated values for Nm(n) for m=2 to 7.

Suggested Citation

  • Xie, Fa-geng & Hao, Bai-lin, 1994. "Counting the number of periods in one-dimensional maps with multiple critical points," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 202(1), pages 237-263.
    • Handle: RePEc:eee:phsmap:v:202:y:1994:i:1:p:237-263
    DOI: 10.1016/0378-4371(94)90176-7
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