Publications
Dynamics
-Singular
Perturbations in the Quadratic Family. Journal
of Difference Equations and Applications, 4, No. 6 (2008), pp. 581-595. In this paper, we consider families
of maps of the form z2 + c + λ/z2 where c
is in a hyperbolic component of the Mandelbrot set but not at its center and λ
is a small complex parameter. We show that, if | λ | is
sufficiently small, then the Julia set of these maps consists of countably many
curves that are homeomorphic to inverted copies of the Julia set of z2
+ c and uncountably many point components that accumulate on each one of
these curves. Submitted to the Journal of
Difference Equations and Applications. Preprint [pdf]
-Rabbits, Basilicas,
and Other Julia Sets Wrapped in Sierpinski Carpets. With Paul Blanchard,
Robert L. Devaney, Antonio Garijo and Elizabeth Russell. In Complex Dynamics:
Families and Friends, A. K. Peters (2009), 277-296. In this paper, we consider
families of maps of the form z2 + c + λ/z2
where c is the center of a hyperbolic component of the Mandelbrot set with
period greater than 1 and λ is a small complex parameter. We show
that, if | λ | is sufficiently small, then the Julia set of these
maps are bounded on the outside by a homeomorphic copy of the Julia set of z2
+ c. Also, assuming all of the critical orbits of the map eventually enter
the basin of infinity, then each internal component of filled Julia set of this
map is "wrapped" in a set homeomorphic to the Sierpinski carpet.
Preprint [pdf]
-Singular
Perturbations in the McMullen Domain. With Robert L. Devaney. March 2007.
-Singular Perturbations of zn with Multiple Poles. International Journal of Bifurcations and Chaos, 18 (2008) pp. 1085-1100. We study the dynamics of the family of complex maps given by fλ(z)=zn + λ / ((z - a)da (z - b)db) where n ≥2 is an integer and λ is an arbitrarily small complex parameter. The poles a and b are such that |a|,|b| ≠ 0,1 and their orders are da, db ≥ 1. We focus on the topological characteristics of the Julia set and the Fatou set of fλ. We prove that despite the large amount of possibilities there are only four different cases that correspond to different positions and orders of the poles a and b. Preprint [pdf]
-Evolution of the McMullen Domain for Singularly Perturbed Rational Maps. With Robert L. Devaney. Top. Proc. 32 (2008) pp. 301-320. (E-published on August 15, 2008). In this paper we consider families of maps of the form zn + C / (z - a)d where a and C are complex and |a| is not equal to one. Also, n is greater or equal than 2 and d is greater or equal to 1. We are especially interested in the case where a is close to 0. For when a = 0, and 1/n +1/d<1 there is a McMullen domain in the parameter space, but as soon as a becomes nonzero, this domain moves away from the origin and the Julia sets become very different. In the McMullen domain, the Julia sets are Cantor sets of concentric circles about the origin; when a is nonzero, the Julia set consists of infinitely many circles, only one of which surrounds the others, and uncountably many point components. To appear in Topology Proceedings. Preprint [pdf] See animations related to this paper here.
-The McMullen Domain: Rings Around the Boundary. With Robert L. Devaney. Transactions of the American Mathematical Society 359, Number 7, July 2007, 3251-3273. In this paper we prove that the McMullen domain for the family of rational maps zn + λ/zn when n>2 is surrounded by infinitely many disjoint simple closed curves Ck converging down to the boundary of the domain as k tends to infinity. On Ck , there are precisely (n-2) nk-1 +1 parameters that are superstable (i.e., for which a critical orbit is periodic), and the same number of parameter values for which the critical orbits land on the pole at 0 (i.e., parameters that are centers of Sierpinski holes). Preprint [pdf]
-Living
in Critters’ world. Revista
Ciencias Exatas e Naturais, Vol. 7,
n° 1, pp.9-34, Jan/Jun 2005, Universidade Estadual do Centro-Oeste, UNICENTRO,
Guarapuava/Iratí, Brasil.
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. We show the kind of oscillators and
particles we can construct. We also make them collide with each other and with
barriers of different kind. Finally, we make experiments to capture the
diffusion properties of this rule and we find the relationship between the
diffusion coefficient and the density. [pdf]
-Prediction of the mean path of random walkers. With Horacio A. Caruso. Revista Ciencias Exatas e Naturais, Vol. 4, n° 2, pp.123-135, Jul/Dez 2002, Universidade Estadual do Centro-Oeste, UNICENTRO, Guarapuava/Iratí, Brasil. A random walker is allowed to walk on a surface along two orthogonal directions; the possibility to select either of them is a specific function of the number of steps he has performed. Once he has chosen one of the two axis he is also given a possibility to choose a forward or a backward step; this possibility may also be a specific function of the number of steps. In spite of the wide variety of possibilities that may be assigned we herein describe a theoretical method through which the mean path of a large number of walkers may be found. Several numerical examples for random walkers verify the theoretical results for their mean path. [pdf]
-Inertial random walkers on a surface. With Horacio A. Caruso. Revista Ciencias Exatas e Naturais, Vol. 4, n° 1, pp.21-41, Jan/Jun 2002, Universidade Estadual do Centro-Oeste, UNICENTRO, Guarapuava/Iratí, Brasil. The classical random walker found in the literature has steps along four orthogonal directions, i.e., North, South, East and West, each one with the same possibility to be chosen. We propose a new type of random walker with changing possibilities. They are encouraged to make a new step in the same direction of the previous step. [pdf]
-Sequences of Complex Numbers Resembling the Fibonacci Series. With Horacio A. Caruso. Revista Ciencias Exatas e Naturais, Ano 2, n° 1, pp.49-59, Jul/Dez 2000, Universidade Estadual do Centro-Oeste, UNICENTRO, Guarapuava/Iratí, Brasil. Each term of the classical Fibonacci sequence of numbers is the sum of the two previous terms of the sequence. If instead of the sum the third term is an addition or a subtraction of the two previous terms, one of them multiplied by a constant, new and rich sequences are obtained. Some of the properties of these sequences are herein studied by means of numerical procedures, incorporating the condition that each term, including the constant, are complex numbers. [pdf]
-Self-Avoiding Random Walkers With Steps Along Eight Possible Directions (Octopus Walkers). With Horacio A. Caruso. Revista Ciencias Exatas e Naturais, Ano 2, n° 1, pp.61-71, Jul/Dez 2000, Universidade Estadual do Centro-Oeste, UNICENTRO, Guarapuava/Iratí, Brasil. Different types of random walkers are studied by assuming that they have eight different possibilities to make steps in pre-determined directions. The walkers are not allowed to step on already visited places in the plane (self-avoiding walkers). The main variables chosen to depict their behavior are the number of steps and the distance they reach before they are trapped. A measure of how the walkers are diffused, and index of how the paths of the walkers are folded and the effect of restricting the region where the walkers may walk are also studied. [pdf]
Education:
-Using Flip Camcorders
for Active Classroom Metacognitive Reflection in Higher Education. With Jace Hargis. July, 2009.
The Center for Teaching and Learning provided Flip camcorders to a group of ten
new faculty members who were asked to use this tool in their classroom
instruction. The classes included mathematics, political science, computer
engineering, psychology, business, music and dance. The qualitative results
indicated that all faculty members enjoyed the experience and found innovative
methods to integrate the camera into their classroom teaching which resulted in
more engagement and positive student outcomes. Several faculty members
developed methods, procedures and assessment rubrics and guidelines for using
the Flip in assignments, which are shared in this work. To appear in Journal of Active Learning in Higher
Education. Preprint [doc]
Online publications
-Mathematica notebooks for Iterated Function Systems. April 2005. This is a set of five Mathematica notebooks to study Iterated Function Systems (IFS's). There is an Introduction, the Backward Iteration Algorithm, Affine transformations, Random Sequences and Conclusions. We explain the concept of an IFS and show how to use IFS's to plot the Julia sets of complex functions. The notebook on affine transformations shows examples of IFS's using contractions. The notebook on random sequences shows that some random sequences can be studied using IFS's. We also show how to make the animations shown in the web page. This work received Honorable Mention in the DSWeb Tutotial Contest of 2005. The files can be downloaded from: http://math.bu.edu/people/smarotta/IFS.htm and from http://www.dynamicalsystems.org/tu/tu/.
-A First Course in Chaotic
Dynamical Systems: Experiments with Mathematica. With Robert L. Devaney.
December 2003. A set of Mathematica
notebooks that are meant to accompany the book ‘A First Course in Chaotic
Dynamical Systems’ by Robert L. Devaney. This notebooks are used by students
taking Chaotic Dynamical Systems MA 471 (671) in the Department of Mathematics
and Statistics at
Recreational mathematics
-La Oveja Voraz y el Campo Circular. Lecturas
Matemáticas, Publicación de la Sociedad Colombiana de Matemáticas, Vol. 24,
n° 1, pp. 55-60, 2003. De manera amena se discute un problema conducente a una
ecuación diferencial. [pdf]