1. Introduction
One of the most common iterative algorithms for finding solutions to an equation is Newton’s method. Given an equation
and an initial guess
, Newton’s method attempts to locate a better approximation,
, given by

Here, the numerical technique uses information about the first derivative of
at
to obtain an improved approximation
. The process then begins anew from
, generating a sequence of numbers
intended to converge to a solution of the equation.
Interestingly, though perhaps of little surprise to those familiar with iterative algorithms, numerical methods such as this have a tendency for failing in unpredictable manners. For example, applying Newton’s method to find the roots of
leads to an entire region of the complex plane for which initial seeds eventually bounce back and forth between
and
, neither of which is a solution to
. In this case,
and
lie on a super-attracting cycle of period two for the map
. It is this failure of Newton’s method to converge on an open set of initial guesses that is investigated in this work.
The notion of studying the failure of Newton’s method applied to a complex polynomial dates back to the pioneering work of Curry, Garnett and Sullivan in 1983 [1] . Focusing on Newton’s method applied to a par- ticular family of cubic polynomials, the authors amazingly discover one of the most famous fractals of all, the Mandelbrot set, lurking throughout the parameter plane. Each parameter value in these special sets corresponds to a “bad” polynomial where Newton’s method fails on an open region in the complex plane due to the exis- tence of an extraneous attracting cycle distinct from the roots. Similar results in the case of a complex cubic were later obtained by Blanchard [2] , Head [3] , Lei [4] , Roberts and Horgan-Kobelski [5] (also verifying the phenomenon for Halley’s method), and Haeseler and Kriete [6] who applied quasiconformal surgery to prove the existence of rogue attractors for relaxed Newton’s method. The existence of Mandelbrot-like sets in the parameter plane and the connection with quadratic-like dynamics was thoroughly explained by Douady and Hubbard using their theory of polynomial-like mappings [7] .
In this work, we study Newton’s method applied to the complex quartic family
![]()
where
is a complex parameter. The symmetric placement of the four roots is motivated by the nearest-root principle, the notion that initial seeds “typically” converge to the closest root. This is precisely the case for Newton’s method applied to a complex quadratic map with two distinct roots
and
. The only points that fail to converge to either root under iteration lie on the perpendicular bisector
of the segment joining
and
. These initial seeds cannot decide which root to converge toward and consequently, remain on
for all time. For any other initial guess
, Newton’s method converges to the root located on the same side of
as
. In general, the invariance of
occurs whenever there is a line of symmetry amid the configuration of roots. For the family
, the real axis is a line of symmetry and is therefore invariant under Newton’s method. This will play a key role in studying the dynamics as it allows for a reduction to a map of one real variable.
The primary difference between the cubic and quartic cases is the number of “free” critical points. In addition to the roots of a polynomial, its inflection points also turn out to be critical points of the Newton map. Unlike the fixed roots, these points can iterate freely around the complex plane. There is only one such point for a cubic map, but two free critical points for the quartic case since the second derivative is quadratic. Not surprisingly, due to the identical numbers of critical points, our problem turns out to have many similarities with the general complex cubic. In [8] , Milnor classifies the types of hyperbolic components possible in the parameter plane for the general cubic, obtaining fractal-like figures such as Mandelbrot-like sets and tricorns as well as swallow and product configurations. The distinguishing element between these fractal-like sets is the type and behavior of the critical points. Following Milnor’s work, we are able to locate and explain the existence of these same types of fractals in the parameter plane for Newton’s method applied to
.
The main tool in our analysis is the reduction to a map in one real variable. This occurs in the cases where
is real or purely imaginary due to the invariance and symmetry described above. We apply standard arguments from real dynamical systems theory to prove that there are no extraneous attracting cycles in the case
. Using a bisection algorithm, we numerically locate an abundance of values
such that Newton’s method applied to
contains a super-attracting
-cycle. Bifurcations are explored as
varies. Whether the free critical points lie on the same or distinct periodic cycles for these special parameter values has important consequences for the resulting figures in the
-parameter plane.
2 Newton’s Method and Complex Dynamics
Let
denote the
-fold composition of
with itself. Given some
, we define the orbit of ![]()
to be the sequence of points
where
. We refer to
as the initial seed of the orbit.
A point
is said to be a periodic point if
for some
, and the smallest such
is known as the period of
. In this case, we say that
lies on an
-cycle or periodic orbit of period
. A periodic point of Period 1 is known as a fixed point.
A periodic point
with period
is said to be attracting if
and repelling if
. It is straight-forward to show that seeds in a sufficiently small neighborhood of an attracting
periodic orbit are attracted to that orbit under iteration. A periodic point
with period
is said to be
neutral if
. Finally, a periodic point
satisfying
is called super-attracting, a
title that corresponds to the rate at which nearby points converge.
Definition 1 Suppose that
is an attracting periodic orbit of period
. Then the open set
containing all points
such that
,
,
converges to some periodic point in
is called the basin of attraction for
.
We now restrict our attention to studying Newton’s method applied to a complex polynomial
. For a fixed
, this produces a rational map of the extended complex plane
denoted
and given by
![]()
Studying the convergence of Newton’s method is equivalent to investigating the orbits of initial seeds under iteration of the map
, placing our study squarely in the field of complex dynamical systems.
There are two complimentary sets used to describe the dynamics of a map in complex dynamics, the Julia set, where the interesting and chaotic behavior occurs, and its tame cousin, the Fatou set, where attracting periodic cycles and their basins of attraction lie. The Julia set for a rational map is the closure of the set of repelling periodic points [9] . This is an invariant, perfect and fractal-like set displaying sensitive dependence on initial conditions. Any neighborhood of a point in the Julia set is mapped under iteration to cover all of the extended complex plane except at most two points. For Newton’s method, this implies that arbitrarily close to any point in the Julia set are seeds that will iterate to each root of
. Thus, choosing an initial seed in the Julia set of
is not a major difficulty since a small perturbation will ensure convergence toward a root.
The roots of
and their basins of attraction are in the Fatou set. It is easy to see that if
, then
, so that the root
is a fixed point of the dynamical system
. Moreover, a short calculation gives
(1.1)
If
is a simple root of
, then it follows that
and thus
is a super-attracting fixed point. This is certainly a good property for a numerical root-finding algorithm to possess as it implies that nearby points are strongly attracted towards a root of the polynomial. However, equation (1.1) is significant for another reason, for it indicates that inflection points of
are also critical points of
. Since these inflection points are typically not roots of
(and thus not fixed), we will refer to them as the free critical points of
. In complex dynamics, it is the orbit of the critical point that governs the behavior of the underlying dynamics. In particular, we have the following important theorem of Fatou and Julia (see [9] [10] ) for rational maps on
:
Theorem 1 Every attracting cycle of a rational map attracts at least one critical point.
In the case of Newton’s method, the critical points corresponding to roots are themselves attractors. Ge- nerically, we also expect the free critical points to be attracted towards the roots. However, an intriguing situation occurs when one or more of the free critical points converges to an extraneous attracting cycle, that is, a cycle distinct from one of the roots of the polynomial. The basin of attraction for such a cycle is an entire region in
for which initial seeds never converge to a root. In this case, the roots elude detection, for a small perturbation of a failing initial seed may not lead to convergence to one of the roots. The dynamical plane for such an example is displayed in Figure 1. These “bad’’ polynomials, with elusive zeros and extraneous at- tracting cycles, are the main focus of this paper.
![]()
Figure 1. The dynamical plane for Newton’s method applied to
with
. Initial seeds are colored according to the root they converge to. Points col- ored black iterate toward a super-attracting period 2-cycle.
3. A Symmetric Fourth-Degree Family of Polynomials
We now restrict our investigation to applying Newton’s method to the family of fourth degree polynomials defined by
![]()
where
is a complex parameter. This family was briefly considered by Sutherland at the end of his doctoral thesis [11] . By expanding
, we find that
(1.2)
and thus
always has real coefficients.
For the remainder of this work, we will denote
as the complex rational map obtained by applying New- ton’s method to
. The free critical points for
, arising as the inflection points of
, are given by
(1.3)
Note that by Theorem 1, there can be at most two distinct extraneous attracting cycles. If we let
,
then the discriminant
in Equation (1.3) simplifies to
![]()
The equation
defines a hyperbola in the
-parameter plane and serves as an important boundary to the possible types of free critical points
described below and in Figure 2:
•
two real free critical points equidistant from
,
![]()
Figure 2. The hyperbola that distinguishes the
-para- meter plane in terms of the critical points of
. Within the curves, the two free critical points are real, while outside, they form a complex conjugate pair.
•
single real (repeated) free critical point at
,
•
two free critical points as a complex conjugate pair with real part
.
Using equation (1.2), we compute directly that
(1.4)
While this rational map may appear imposing, it possesses certain symmetries under conjugacy that signi- ficantly reduce the size of the
-parameter plane.
Lemma 1 For any
,
i) the real axis is invariant under
,
ii)
, and
iii)
is topologically conjugate to
via the conjugacy
.
Proof Since
is a polynomial with real coefficients, so is its derivative. This leads to all coefficients in
being real and the invariance of the real axis. To prove item (ii), note that
, since the roots of
are left unchanged when
is replaced by
. It follows that
by definition. Finally, item (iii) follows from Equation (1.4) by verifying that
![]()
as desired. ![]()
Since attracting cycles are preserved under conjugacy, the symmetries described in items (ii) and (iii) above enable us to restrict the parameter plane to the region
Being only con- cerned with those
-values for which
possesses an extraneous attracting orbit, it suffices to follow the orbit of the free critical points for
. There is an important additional symmetry when
is purely ima- ginary, as indicated by the following lemma.
Lemma 2
is conjugate to
via the conjugacy
.
Proof It is straight-forward to show that two systems arising from Newton’s method applied to polynomials whose roots differ by an affine map, are topologically conjugate (see [5] for example). Since
is an affine map and a bijection between the roots of
and the roots of
, the result follows. Alternatively, direct
substitution into Equation (1.4) shows that
. ![]()
3.1. The Case
Real
Restricting to the case
leads to some nice reductions that allow for a complete classification of the orbits of the free critical points. We will show that the orbits of
converge to one of the roots of
and con- sequently, there are no extraneous attracting cycles for
when
is real.
For
,
now contains the repeated root
in addition to the roots
and
. Because
is a repeated root, it is not super-attracting in this case. The Newton map simplifies to
![]()
Since the free critical points are real for
, we restrict the domain of the Newton map to the real axis, hence the use of the variable
as opposed to
. The free critical points are
![]()
and the two poles of
, denoted
and
, are given by
![]()
For later use, we have
(1.5)
Theorem 2 For all
, the orbits of the two free critical points under
always converge to roots of
. Specifically, we distinguish the following two cases:
1) For
for all
.
2) For
and
.
Proof Due to item (iii) of Lemma 1, it suffices to consider the case
. We begin with the case
, where a series of straight-forward estimates show that the critical points, poles and roots of
are arranged in the following manner:
(1.6)
Next, using
![]()
and the ordering described in inequality (1.6), we see that
while
. Moreover, it fol- lows from equation (1.5) that
is strictly increasing on the open interval
and that
. Thus, for
, we have
while for
, we have
(see Figure 3).
It follows that for
, the orbit of
under
is a strictly increasing sequence bounded above by the fixed point
and for
, the orbit of
is a strictly decreasing sequence bounded below by the fixed point
. Since the only other fixed points for
are
and
, standard arguments using the con- tinuity of
on
show that all points in
converge to the attracting fixed point at
.
For the case
, the Newton map reduces even further since
has the root
with multiplicity three in addition to the simple root
. In this case,
becomes
and
while
. The pole
vanishes and
. Here, the free critical point
has been transformed into an attracting fixed point with derivative
. A simple web-diagram of
shows that the orbit of
converges to the repeated root at
.
To prove the rest of item (ii), we compute that for
,
![]()
while for
,
![]()
At
, we have
(super-attracting fixed point). The major change between these two inequalities and inequality (1.6) is that the two roots
and 1 have interchanged positions. For
, the super-attract- ing fixed point 1 lies between the two poles, while for
, it is the super-attracting fixed point at
that is located between the two poles. In addition, the critical point
jumps to the other side of the pole
, be- coming a local min instead of a local max (see Figure 4). However, the dynamical behavior of the critical points is unchanged. Similar arguments as with case (i) show that the orbit of
monotonically increases as it converges to
for
. The orbit of
monotonically increases as it converges to 1 for
and monotonically decreases to 1 for
. As before, the continuity of
on the appropriate intervals in addition to the number and location of the fixed points allows us to prove these convergence results. ![]()
3.2. The Case
Pure Imaginary
Due to Theorem 2, the interesting Newton maps occur when
becomes complex. However, the reduction to a real map in the previous subsection suggests a practical approach to investigating the case where
is purely imaginary.
Consider the case
with
. By Lemma 2, we can restrict to the regime
. For ease of notation, we let
correspond to Newton’s method applied to
. In this case, the roots lie on the vertices of a rhombus in the complex plane. Based on previous work applying Newton’s method to cubics [1] [2] [5], the symmetry of this configuration suggests that Newton’s method may fail badly for certain
-values. We shall see that this is indeed the case. The driving force behind the interesting dynamics is that initial seeds on the real-
axis are equidistant from the two imaginary roots. If these seeds are far enough away from the roots at
and
, their orbits bounce around the real axis, unable to converge to a root.
For the special case
, the roots of
are the four roots of unity and as a result of this symmetry, the only “free” critical point is
of multiplicity two. However,
is also a pole for the Newton map and therefore iterates to
which is a repelling fixed point. Thus, there can be no extraneous attracting cycles for the case
.
For
, the critical points of
are real and given by
![]()
Due to the invariance of the real axis under our Newton map, this implies that any extraneous attracting cycle must lie on the real axis. As with the case
, this allows us to restrict our study to a rational map
of one real variable,
![]()
Of particular importance is the fact that
is an odd function whose critical points are symmetric about the origin. If the orbit of
lies on or converges to an attracting
-cycle, then so will its partner
. However, it may be the case that
and
lie on or converge to the same attracting orbit, a phenomenon that will be critical to understanding the larger parameter plane picture.
The poles of
are
and
, where
. Using some basic estimates for the regime
, we have
![]()
Moreover, for
,
is concave down with a local maximum at
and
. For
,
is concave up with a local minimum at the super-attracting fixed point
. It follows that each
converges to the root
under iteration of
. Since
is odd, we also have that each
con- verges to the root
. Thus, the interesting dynamics of
occurs between the poles
and
.
The following lemma captures some key information about the behavior of the critical points as
varies.
Lemma 3 For
, the orbits of the free critical points
converge to the roots
or
. Spe- cifically, we have the following:
(i) For
,
is strictly decreasing with respect to
, while
is strictly increasing with respect to
.
(ii) For
,
and
.
(iii) For
,
converges to
while
converges to
under iteration of
.
Proof Since
is an odd function and
, we see that
. Thus, to prove item (i), it suffices to show that
is decreasing with respect to
. We first compute
![]()
where
. Taking the derivative of
with respect to
and simplifying the numerator gives
![]()
which is clearly negative, proving that
is decreasing with respect to
.
To prove item (ii), letting
, we see that
. Using part (i), it follows that for
,
. The second inequality follows since
is odd.
Item (iii) follows easily from (ii) and the fact that all points in
converge to
under iteration while all points in
converge to
. ![]()
Our goal is to search for attracting cycles of
distinct from roots when
. While it is difficult to make precise calculations locating specific examples, there are three values worthy of mention.
• For
,
and
lie on the same super-attracting 2-cycle.
•
,
has
and
lying on distinct super-attracting 2-cycles.
• For
.
The first value is derived by solving the equation
for
. Since
is odd, it follows that
and thus the free critical points lie on the same super-attracting 2-cycle (see Figure 5). The calculation shows that this is the only
-value in
with this property. By Lemma 2, the same phenomenon occurs for
. We note that the constant
agrees with the value given in Exercise 10, Chapter 13 of Devaney’s undergraduate textbook [12] . The second value,
, is found by solving
for
. This leads to an even eighth-degree polynomial that can be solved using Ferrari’s formula for a quartic equation (calculated using Maple [13] ). It is the only new solution (other than its reciprocal
.) Thus,
and
are the only parameter values in
for which the critical points are pe- riodic of period two. The value
is significant in that it defines a regime
with more com- plicated dynamics for the free critical points. This will be apparent in some of our numerical investigations.
Due to the challenges of studying higher iterations of
analytically, a computer program was written in C++ to locate super-attracting periodic points numerically. For a fixed period
, the program commences with
, which is slightly greater than our upper bound of
. Using a bisection method, solutions to the equation
are located as
decreases. (Using Newton’s method to investigate Newton’s method was a bit too paradoxical for us!) The values located, shown in Table 1 up to
, are
-values where the free critical points lie on super-attracting periodic cycles. Thus, Newton’s method for these cases will fail on an open set of initial conditions in the complex plane. As the period
increases, the number of periodic orbits increases exponentially. Letting the program run exhaustively on period 16 yields 2525 periodic points. It should also be noted that error checking devices were used to ensure that poles were not mistaken for periodic points. These values were also confirmed using Maple.
From this numerical evidence, we observe some noteworthy behavior. For instance, a period-doubling cas- cade to chaos is readily apparent from our data, as shown in Table 2. These values correspond well to the universal rate of doubling given by Feigenbaum’s constant. The period doubling can also be seen in the bi- furcation diagram obtained by following the orbits of both free critical points (see Figure 6).
The value
is a key divider for much of the interesting dynamical behavior. For example, due to the structure of the graph of
, there can be no super-attracting cycles of odd period until
. For
![]()
Table 2. A table illustrating the typical period-doubling route to chaos. Each
-value corresponds to a super-attracting period
cycle for the corresponding Newton map.
, the signs of each element in the orbit of either critical point must alternate, thus making only even periodic cycles possible. This is indicated clearly in Table 1 for
and
. Moreover, for
-values close to
, the image of both free critical points is close to the pole at zero and consequently, further iteration leads to convergence to either
or
. As long as the image of the critical points is trapped close to the pole at zero, the long-term behavior will be convergence to roots. This explains the gap in the center of Figure 6.
It is important to note that in most cases, the free critical points lie on disjoint periodic orbits of the same period. Except for
and
, all the parameter values in Table 1 are for two disjoint periodic cycles. This is always the case when the period is odd.
Lemma 4 For
-values corresponding to super-attracting periodic cycles of odd period,
and
do not lie on the same orbit.
Proof Without loss of generality, suppose
for some primitive odd period
. By contradiction, suppose that
and
lie on the same periodic orbit. This implies that
for some
. However, since
is an odd function and
, this also implies that
. Then, by a simple substitution we have
![]()
However, we also have that
. Therefore, we see that
is periodic with even period
. Again, given the symmetry of
, this also implies that
is periodic with even period
. Since
and
is the least period of the orbit, it follows that
. But this contradicts the fact that
is odd. ![]()
The birth of two disjoint super-attracting period
-cycles usually arises from a pitch-fork bifurcation. In the rare case when there is a single cycle containing both free critical points on the orbit, a saddle-node bifurcation occurs (see Figure 7). These two types of bifurcations transpire remarkably close together in parameter space, with the saddle-node bifurcation taking place first. For example, when
, the successive values are
(same cycle) followed by
(different cycles). For period
, the parameter values are only
apart.
4. The General Case for ![]()
Unfortunately, once we leave the real and imaginary axes in the
-parameter plane, we are no longer able to work with the simplified versions of our Newton map. Using a computer, we follow the orbits of both free critical points under
as
varies, producing an intriguing picture of the parameter plane (indicated on the left in Figure 8). The deeper red colors indicate faster convergence to a root, while the lighter colors indicate a slower convergence. Black represents
-values where one or both critical points fails to converge to within
of a root after 100 iterations. The same color scheme is used for all figures shown in the parameter plane.
Much of the analytic work we have done up to this point will be useful in explaining the general structure and interesting figures in the parameter plane for
. Although quite intricate, there are a few phenomena that can be explained via connections with the work of Milnor on the general case of a complex cubic polynomial [8]. We find a striking similarity between the dynamical behavior of
and that of a general complex cubic-
polynomial. This is not entirely surprising, since a generic cubic map has two free critical points, exactly the case for
. In [8] , Milnor classifies four types of hyperbolic components in the parameter plane for the generic cubic based on the orbits of the critical points. These four cases are referred to as adjacent critical points, bitransitive, capture and disjoint periodic sinks. Only the second and fourth cases are relevant to our study. We adapt Milnor’s definition to our problem.
Definition 2 (Milnor [8] ) Suppose that both free critical points
converge toward attracting periodic orbits
distinct from the roots of
. Let
be the open set of all points in the basin of attraction of
.
Bitransitive: The two free critical points belong to different components
and
of
, but there exist natural numbers
and
such that
and
. We assume that
and
are primitive, so that both
and
have period
.
Disjoint Periodic Sinks: The two free critical points belong to different components
and
, where no forward image of
is equal to
and vice versa. In this case, there exist natural numbers
and
with
and
.
Milnor characterizes the types of fractal-like figures we should expect to see in the parameter plane for each case, along with a prototype map. Not surprisingly, there are different figures depending on the type and behavior of the critical points. In the bitransitive case, one finds either a swallow configuration (indicated on the right in Figure 8) or a three-pointed configuration Milnor refers to as a tricorn (indicated on the left in Figure 9). For the swallow configuration, the prototypical dynamical system is the family of real maps
with parameters
(so the model swallow lives in
). For our Newton map
, the center point of each swallow configuration appears to correspond to the situation where the two real free critical points lie on the same periodic cycle. As discussed in Section 3.2, this occurs for the parameter values
and
. Moreover, since the free critical points are real inside the hyperbolas displayed in Figure 2, we only see swallow configurations in this part of the parameter plane.
For the tricorn, the model is the complex map
. The tricorn actually contains three embedded copies of the Mandelbrot set, where the cusp of each has been stretched out over a triangular region, joining them in a peculiar fashion. The difference between the tricorn and swallow configuration is the type of critical points. For the swallow, we have real distinct critical points, while for the tricorn, we have a complex conjugate pair. Due to Lemma 2, inverting the key parameter values
and
on the imaginary axis gives a conjugate dynamical system. However, the real, symmetric free critical points are mapped to a pure-imaginary conjugate pair under the conjugacy. This explains why we see tricorns in the parameter plane centered at the
- values
and
and swallow configurations at their inverted counterparts.
In the case of disjoint periodic sinks, there is either a product configuration (two long, thin, black strips stretched across each other) or an actual copy of the Mandelbrot set itself (indicated on the right in Figure 9). The defining map for the product configuration consists of the two disjoint real functions
with
. As with the swallow configuration, this involves two real critical points as well, although now they are free to find distinct periodic cycles. Enlarging the parameter plane about the value
yields such a configuration because the free critical points of
are real and lie on two distinct super-attracting period 3-cycles. According to Milnor’s work, we see a Mandelbrot-like set at the inversion of this value,
, because the free-critical points are mapped to a pure-imaginary conjugate pair under the conjugacy
and this pair lie on distinct attracting cycles. In this case, the prototype map is simply
with
, the usual defining map for the Mandelbrot set. The disjoint periodic sink cases yield ``simpler’’ dynamical phenomenon because the orbits of the critical points stay away from each other, leading to the decoupled prototype maps.
Throughout our discussion, we see that the different possible orbits of the free critical points determine the types of fractal-like figures found in the parameter plane. Our detailed analysis of the special case restricting
to the imaginary axis has provided a useful guide explaining the locations of these figures. However, there are some exceptions worth mentioning. For example, there are disjoint periodic sink values very close to bitransitive values that are contained inside a swallow configuration. Given our limited resolution and computational resources, we are unable to numerically substantiate the claim that every parameter value corresponding to a disjoint periodic sink lies at the center of a product configuration or a Mandelbrot-like set. This discrepancy is most likely attributable to the fact that
is a cubic-like map, as introduced by Douady and Hubbard [7], and not an actual cubic polynomial. Nevertheless, after plotting the long-term behavior of each critical point separately and investigating the location of several swallow configurations, we have obtained strong numerical evidence in support of the following conjecture:
Conjecture 1 Each bitransitive
-value corresponding to the two free critical points sharing the same super-attracting
-cycle lies at the center of a swallow configuration in the parameter plane.
5. Conclusions
In conclusion, we point out one more feature of the parameter plane that can be analytically derived. The yellow diamond shaped boundary that encompasses the interesting dynamical behavior in the parameter plane (in- dicated on the left in Figure 8) is defined by those
-values where both
and
simultaneously vanish. Letting
, this occurs on the algebraic curve
defined by the equation
![]()
For any parameter value on
, one or both of the free critical points coincide with poles of
and thus map to the fixed point at
. Taking successive pre-images of these curves appears to define the sequence of intertwining yellow “leaves” that approach the real axis.
We have analytically and numerically investigated Newton’s method
applied to a highly symmetric family of fourth degree complex polynomials. Using techniques from both real and complex dynamical systems theory, we were able to study some reductions of our Newton map that shed significant light on the varied dynamical behavior of this system. Remarkably, we find quite intricate and complicated fractal-like figures throughout the parameter plane for this simple system. Milnor’s work on the complex cubic polynomial com- bined with our understanding of the dynamics for
with
purely imaginary provided some rationale for the location and types of fractals encountered.
Acknowledgements
TOB would like to thank the Department of Mathematics and Computer Science at the College of the Holy Cross for providing an enjoyable and supportive atmosphere throughout his undergraduate career. Funding for TOB was provided by a Council on Undergraduate Research Student Summer Research Fellowship in Mathe- matics and Science. GR would like to thank Paul Blanchard for pointing him in the direction of this problem and for several interesting discussions concerning this work. He would also like to thank Bruce Peckham for ongoing and insightful discussions on Newton’s method.