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

Abstract

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 F μ (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).