Preperiodicity and systematic extraction of periodic orbits of the quadratic map

Jason A. C. Gallas
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Jason A. C. Gallas: Instituto de Altos Estudos da Paraíba, Rua Silvino Lopes 419-2502, 58039-190 João Pessoa, Brazil2Complexity Sciences Center, 9225 Collins Ave. Suite 1208, Surfside FL 33154, USA3Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany

International Journal of Modern Physics C (IJMPC), 2020, vol. 31, issue 12, 1-13

Abstract: Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree 4020, while for the 52 377 period-20 orbits the degree rises already to 1 047 540. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map.

Keywords: Arithmetic dynamics; preperiodic points; quadratic map; exact orbital equations (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1142/S0129183120501740

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