Phys. Rev. E 63, 016216 (2000) [5 pages]Infinite hierarchies of nonlinearly dependent periodic orbitsReceived 2 May 2000; revised 14 September 2000; published 27 December 2000 Quadratic maps are used to show explicitly that the skeleton of unstable periodic orbits underlying classical and quantum dynamics is stratified into a doubly infinite hierarchy of orbits inherited from a set of basic “seeds” through certain nonlinear transformations Tα(x). The hierarchy contains nonunique substructurings which arise from the different possibilities of sequencing the transformations Tα(x). The structuring of the orbital skeleton is shown to be generic for Abelian equations, i.e., for all dynamical systems generated by iterating rational functions. © 2000 The American Physical Society URL:
http://link.aps.org/doi/10.1103/PhysRevE.63.016216
DOI:
10.1103/PhysRevE.63.016216
PACS:
05.45.-a, 03.65.Fd, 45.05.+x
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