Perturbations of the quadratic family of order two

Abstract

Define the quadratic family of order two as F µ (x, y) = (y, −x 2 + µx), where µ is a real parameter. The boundary of the basin of attraction of the fixed point at ∞ is an invariant curve for µ < 4, and is a Cantor set for µ > 4. Perturbations of F µ with µ = 4 were studied in Romero et al (2001 Discrete Continuous Dynam. Syst. 7 35) (also in higher dimension), where it was proved that these situations persist. Now we study perturbations of the bifurcation point µ = 4, where the explosion of the basin, B ∞ , occurs. We prove that either there exists a connected invariant curve J contained in the boundary of the basin, or the set of critical points is a subset of B ∞ and the boundary has uncountably many components accumulated by the pre-images of the analytic continuation of the fixed point at the origin. The curve J undergoes a fractalization process until it ceases to exist.

Dynamics of vertical delay endomorphisms

“…It is easy to see that for all µ < µ 0 = − 1 12 the vertical delay endomorphism F µ has no fixed points. On the other hand, although the criterion for the attracting property is not available to calculate explicitly the number µ 1 , numerical experiments show that for µ 0.4 the nonwandering set of F µ has uncountably many components because the set of critical points of F µ is contained in B ∞ (F µ ) (see theorem 1 of [6]). These conditions do not imply that the nonwandering set of F µ is expanding or a Cantor set.…”

“…In [5] and [6] some of these changes were studied for perturbations of the two dimensional quadratic family F µ (x, y) = (y, −x 2 + µx), where µ is a real parameter bigger than 1. It was proven there for this family, and it is conjectured for any vertical delay endomorphism, that for every µ such that the set of critical points of F µ is not contained in the basin of attraction of ∞, there exists a connected set J containing one of the fixed points of F µ and some critical points; moreover, J is forward invariant and it is contained in the boundary of B ∞ (F µ ).…”

“…Before µ 0 , the basin of attraction of ∞ is the whole space, and after µ 1 , it is the complement of a Cantor set. In between, a lot of changes occur in the dynamics of the mappings F µ , some of which were explained in [5] and [6] for a particular class of vertical delay endomorphisms. See section 5 for numerical examples.…”

Classification of critical sets and their images for quadratic maps of the plane

“…Most such studies, with good reason, have restricted their studies to families with only one or two parameters. For representative studies, see [Abraham et al(1997), Aronson et al(1982), Frouzakis et al(2003), Gumowski & Mira(1980a), Gumowski & Mira(1980b), Lorenz(1989), Mira et al(1996b), Romero et al(2001), Romero et al(2007), Romero et al (2014)]. Research more in the spirit of our classification approach (described below), but still for a restricted set of quadratic maps, includes [Bofill et al(2004)], where the authors study quadratic maps with no fixed points, and [Nien(1998)] where maps with bounded critical sets (points or ellipses) are studied.…”

Perturbations of the quadratic family of order two: the boundary of hyperbolicity

“…In an attempt to better understand the dynamics of critical maps, we have considered the family of perturbations of the quadratic family. This paper is a continuation of [12,13], where some dynamical properties of the perturbations of the family F µ in any dimension were analysed; it was shown that for any C 2 perturbation F , the point at ∞ is an attractor; moreover, if B ∞ (F ) is the basin of attraction of ∞, then its boundary is an invariant manifold containing the analytic continuation of the fixed point that the mapping F µ has at the origin. This in spite of the fact that this submanifold is not necessarily smooth nor normally hyperbolic.…”