ESAIM: Proceedings and Surveys (Nov 2014)

Invariant Curves of Quadratic Maps of the Plane from the One-Parameter Family Containing the Trace Map*

  • Belmesova S.S.,
  • Efremova L.S.,
  • Fournier-Prunaret D.

DOI
https://doi.org/10.1051/proc/201446009
Journal volume & issue
Vol. 46
pp. 98 – 110

Abstract

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The rigorous proofs are given: (1) for the existence of the unbounded invariant curves, containing the fixed point – source (μ + 1; 1), of the maps from the one-parameter family Fμ(x,y) = (xy, (x − μ)2), μ ∈ [0, 2]; (2) for the birth of the closed invariant curve from the elliptic fixed point (μ − 1; 1) for μ = 3 / 2. Numerical results are presented for the main steps of the evolution of this invariant curve, when μ changes in the interval (3 / 2, 2).

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