Singular perturbations of zⁿ with a pole on the unit circle

We consider the family of complex maps given by fλ,a (z) = z n + λ/(z − a)d where n, d ≥ 1 are integers, and a and λ are complex parameters such that a = 1 and is sufficiently small. We focus on the topological characteristics of the Julia and Fatou sets of fλ,a

Living in Critters’ world

We study several aspects of Critters, an invertible computation universal cellular automaton introduced by Norman Margolus in the 1980’s. We review and extend analysis of some of its interesting, complex characteristics

Singular perturbations in the quadratic family with multiple poles

Agraïments: The first author is partially supported by the European Community through the project 035651-1-2-CODY.We consider the quadratic family of complex maps given by qc(z) = z2 + c where c is the center of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded superattracting cycle by adding one pole to each point in the cycle. When c = −1 the Julia set of q−1 is the well known basilica and the perturbed map is given by fλ(z) = z2 − 1 + λ/(z d0 (z + 1)d1) where d0, d1 ≥ 1 are integers, and λ is a complex parameter such that (...) is very small. We focus on the topological characteristics of the Julia and Fatou sets of fλ that arise when the parameter λ becomes nonzero. We give sufficient conditions on the order of the poles so that for small λ the Julia sets consist of the union of homeomorphic copies of the unperturbed Julia set, countably many Cantor sets of concentric closed curves, and Cantor sets of point components that accumulate on them

The McMullen Domain: Rings Around the Boundary

Our goal in this paper is to consider the dynamics of families of rational maps of the form F*(z) = zn + *zn where * 6 = 0 is a complex parameter and n is a positive integer. The Julia sets corresponding to maps in these families have been shown to possess a number of interesting dynamical and topological properties. In this paper we discuss some of the properties of the parameter plane for these maps. Each of these maps has 2n "free " critical points. However, like the well-studied quadratic family Qc(z) = z2 + c, each of these families has only one free critical orbit since all forward orbits of the critical points behave symmetrically. Hence the *-plane is a natural parameter plane for these families

Nonlinear science: an interactive Mathematica notebook

This interactive Mathematica(TM) notebook provides a ready-made tool by which users can undertake their own mathematical experiments and explore the behavior of non-linear systems, from chaos in low-dimensional maps and coupled ordinary differential equations to solitons and coherent structures in nonlinear partial differential equations and "intrisic localized modes" and "discrete breathers" in extended lattice systems

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\Lambda Please address all correspondence to Robert L. Devaney, Department of Mathematics