Iteration of a quaternionic quadratic family

1996 ◽  
Vol 52 (1) ◽  
pp. 180-194 ◽  
Author(s):  
Meta Škapin-Rugelj
Keyword(s):  
Quadratic Family

Attractors in the quasi-periodically perturbed quadratic family

Nonlinearity ◽  
2012 ◽  
Vol 25 (5) ◽  
pp. 1537-1545 ◽  
Author(s):  
Kristian Bjerklöv
Keyword(s):  
Quadratic Family

One-Level Density of Low-lying Zeros of Quadratic and Quartic Hecke -functions

2019 ◽  
Vol 72 (2) ◽  
pp. 427-454 ◽  
Author(s):  
Peng Gao ◽  
Liangyi Zhao
Keyword(s):  
Level Density ◽  
Central Point ◽  
Gaussian Field ◽  
Quadratic Family ◽  
Density Results

AbstractIn this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.

scholarly journals Real Perturbation of Complex Analytic Families: Points to Regions

1998 ◽  
Vol 08 (01) ◽  
pp. 73-93 ◽  
Author(s):  
Bruce B. Peckham
Keyword(s):  
Numerical Study ◽  
Sufficient Conditions ◽  
Real Parameter ◽  
Complex Parameter ◽  
Mandelbrot Set ◽  
Complex Conjugate ◽  
Real Plane ◽  
Phase Variable ◽  
Bifurcation Points ◽  
Quadratic Family

This study provides some connections between bifurcations of one-complex-parameter complex analytic families of maps of the complex plane C and bifurcations of more general two-real-parameter families of real analytic (or Ck or C∞) maps of the real plane ℛ2. We perform a numerical study of local bifurcations in the families of maps of the plane given by [Formula: see text] where z is a complex dynamic (phase) variable, [Formula: see text] its complex conjugate, C is a complex parameter, and α is a real parameter. For α=0, the resulting family is the familiar complex quadratic family. For α≠ 0, the map fails to be complex analytic, but is still analytic (quadratic) when viewed as a map of ℛ2. We treat α in this family as a perturbing parameter and ask how the two-parameter bifurcation diagrams in the C parameter plane change as the perturbing parameter α is varied. The most striking phenomenon that appears as α is varied is that bifurcation points in the C plane for the quadratic family (α=0) evolve into fascinating bifurcation regions in the C plane for nonzero α. Such points are the cusp of the main cardioid of the Mandelbrot set and contact points between "bulbs" of the Mandelbrot set. Arnold resonance tongues are part of the evolved scenario. We also provide sufficient conditions for more general perturbations of complex analytic maps of the plane of the form: [Formula: see text] to have bifurcation points for α=0 which evolve into nontrivial bifurcation regions as α grows from zero.

scholarly journals Rigorous numerics for critical orbits in the quadratic family

2020 ◽  
Vol 30 (7) ◽  
pp. 073143
Author(s):  
A. Golmakani ◽  
C. E. Koudjinan ◽  
S. Luzzatto ◽  
P. Pilarczyk
Keyword(s):  
Rigorous Numerics ◽  
Quadratic Family

scholarly journals ALTERNATED JULIA SETS AND CONNECTIVITY PROPERTIES

2009 ◽  
Vol 19 (06) ◽  
pp. 2123-2129 ◽  
Author(s):  
MARIUS-F. DANCA ◽  
M. ROMERA ◽  
G. PASTOR
Keyword(s):  
Julia Sets ◽  
Quadratic Family ◽  
Totally Disconnected

In this work we present the alternated Julia sets, obtained by alternated iteration of two maps of the quadratic family [Formula: see text] and prove analytically and computationally that these sets can be connected, disconnected or totally disconnected verifying the known Fatou–Julia theorem in the case of polynomials of degree greater than two. Some examples are presented.

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