Abstract
The dynamical systems of trigonometric functions are explored, with a focus on and the fractal image created by iterating the Newton map, , of . The basins of attraction created from iterating are analyzed, and some bounds are determined for the primary basins of attraction. We further prove - and -axis symmetry of the Newton map and explore the nature of the fractal images.
1. Introduction
Newton’s method is a very well-known iterative procedure for finding roots. Rather than applying Newton’s method to one point at a time, we can instead consider the Newton map:of the function . Iterating from a given starting value yields the familiar sequence produced by Newton’s method. Iterating the function yields a sequence of maps with fascinating dynamics.
The dynamics of the Newton map have been studied for the cases when the original function is a polynomial or rational function (see Dwyer et al. [1]; Dwyer et al. [2]). In this paper, we consider the case .
We first present an outline of the primary definitions and theorems, with an emphasis on the concepts used in this paper. Next, we describe the Newton maps of and before moving to the much more complex case of . We illustrate the symmetries of the Newton map for and provide bounds for the basins of attraction.
1.1. Preliminaries in Complex Analysis
An expression of the form, where and are real numbers, is a complex number and is the set of all complex numbers. We call the real part, denoted as , and the imaginary part, denoted as . The modulus (or absolute value) of , , is a real number which measures the distance from the origin, and is the complex conjugate of .
Each complex number in can be identified with the unique point in the plane . We can establish polar coordinates, and , for , , , and is the angle between the positive real axis and the line segment from 0 to in the counterclockwise direction. Hence, the complex number can be written in the polar form , and using Euler’s equation we obtain .
A complex-valued function assigns to each in the domain exactly one complex number . Just as decomposes into real and imaginary parts, each complex-valued function can be written as , where and are each real-valued functions. In essence, is a pair of real functions of two real variables that maps regions from its domain in the complex plane onto its range in another copy of the complex plane.
The derivative of a complex function is defined by an extension of the definition of the real case. If is an open set in the complex plane and , then is differentiable at a point if exists. The value of this limit is denoted as and is called the derivative of
It is possible for a complex function to be differentiable solely at isolated points, thus analyticity, a property defined over open sets, is a stronger condition. A complex-valued function is said to be analytic on an open set if it has a derivative at every point of . If is analytic on ℂ, then it is said to be entire.
The complex number is a zero (or root) of the function if it is a solution to the equation .
Singularities of functions can result in particularly interesting fractal images. Let denote the ball of radius about . Then, we have the following: a function has an isolated singularity at if there exists such that is defined and analytic in but not in. That is, is an analytic in some neighborhood of but not at itself. In addition, if , then is called a pole of . The point is called a removable singularity if there is an analytic function such that for . If an isolated singularity is neither a pole nor a removable singularity, it is called an essential singularity.
A function is said to be meromorphic in a domain if at every point of it is either analytic or has a pole. In particular, we regard analytic functions on as being special cases of meromorphic functions, and in this paper, we consider only analytic and meromorphic functions.
1.2. Preliminaries in Complex Dynamical Systems
For an analytic function and given in ℂ, the orbit of is the sequence of iterates where means and is the application of the function to the value . The initial value is called the seed value, and dynamics is interested in the fate of orbits, that is, the behavior of as . Do they converge, diverge, cycle, or behave chaotically? A fixed point occurs when . If for some , and are distinct points, then is a periodic point with period, defined as the smallest for which . The set is called a n-cycle for . If the orbit of contains preliminary values before settling at either a fixed point ( for some ) or a periodic orbit ( for some , where is the period of the periodic orbit), then is called an eventually fixed point or eventually periodic, respectively.
For example, consider the complex function . Fixed points of occur when , and thus the fixed points are 0 and 1. Seed values and have orbits , and , respectively. Since each orbit lands on the fixed point after several iterations, these are all eventually fixed points. Similarly
and are periodic fixed points of period two. If , then as , and if , then as.
Theorem 1. Let be a (real or complex) continuous function from its domain set to itself. Suppose and are both in the domain set and , then
Proof. Since the sequence and is continuous, we must have
1.3. Newton’s Method
For a real-valued function , , with root , we define the Newton map of as
The sequence of iterates converges to the sought-after root, , for an initial guess “close enough” to . Note that when , ; therefore, the convergence of to a root of can be thought of as a convergence to a fixed point of . Henceforth, we will be concerned with the orbits for various analytic or meromorphic functions and seed values and explore the fractal nature of the images created by these iterations.
Example 1. Consider the function , which has zeros at and , and Newton map . For seed value, the iterates are which are clearly converging to root . Experiments with different values of will reveal that seed values which are closer to the root will produce orbits that converge to , and likewise, points closer to will iterate to under . Note that if we choose , Newton’s method will fail. Analytically, we can see this because leads to a zero in the denominator of and informally we see that the point lies directly halfway between roots and and is thus pulled equally in both directions by each root. That is, separates those points on the real line which will iterate to and those which will iterate to , so it seems our method should fail at this point. Figure 1 shows the dynamics for . Iterates of seed values from the green region will converge to , whereas seed values from the blue region will iterate to .
For any real or complex function and any real or complex number , we define , the attracting basin of , under the function , to be the set of all starting points whose iterates limit to the point . That is,
For any real or complex function and any real or complex number , we define , the primary (or immediate) basin of attraction of w under , to be the largest set containing that lies in the basin of attraction of . Equivalently, the primary basin of attraction is the connected component of the basin of attraction containing .
Theorem 2 (attracting property of Newton’s method). Given any real or complex analytic function with root , there exists such that all points within a distance of are necessarily in the set , where is the Newton map of . That is, for all initial values that are “close enough” to , the orbit converges to. An outline of the proof is discussed by Brilleslyper et al. [3].
Note that and there is no universal value for , as it depends heavily on the function . The value gives a lower bound on how close a seed value must be in order for guaranteed convergence under Newton’s method. Furthermore, as becomes more complicated, so do the dynamics of , and it may not simply be the case that all seed values close to a certain root converge to that root. Consider Figure 2, where the picture of the dynamics for is shown.
1.4. Classification of Fixed Points
The behavior of a function near its fixed points can vary, and we use this to classify fixed points. In general, if the action of is to move points closer to the fixed point, we call it attracting. Conversely, a fixed point may be repelling, in which case, no matter how close the seed is to the point, iterates of the seed will diverge from the fixed point. Formally, we have the following definition.
Definition 1. Let be a map from its domain set (a subset of either ℝ or ℂ) into itself.(a)A finite fixed point in ℂ is an attracting fixed point (of ) if there exists a neighborhood of such that for any point , we have .(b)We call ∞ an attracting fixed point of if there exists a neighborhood of ∞ such that for any point , we have . That is, the action of is to move each point in closer to ∞ (as measured by the spherical metric).(c)A finite fixed point in ℂ is called a repelling fixed point (of ) if there exists a neighborhood of such that for any point , we have.(d)We call ∞ a repelling fixed point of if there exists a neighborhood of ∞ such that for any point , we have .Observe that in the case of (a) and (b) in Definition 1, . Thus, if a fixed point is attracting, the iterates of any seed value in the neighborhood converge monotonically to the fixed point, while points in will eventually converge to . The following theorem is helpful in determining the classification of a given fixed point.
Theorem 3. Let be an analytic map on a domain ⊂ ℂ such that for some in . Then, we have the following:(a) is an attracting fixed point if and only if ,(b) is a repelling fixed point if and only if .
We note that if , is called a superattracting fixed point, and if , is a neutral fixed point. The behavior of neutral fixed points is complicated: sometimes they can exhibit both a partial attracting nature and a partial repelling nature. For example, for the complex function , we have a neutral fixed point at . For real-valued seeds, has attracting properties, but for purely imaginary-valued seeds, has a repelling nature. Indeed, for , , . Now, since is an increasing function, for all, as , and for all , as . Similarly, neutral fixed points can act in neither an attracting nor repelling fashion.
We can see that periodic points correspond exactly to fixed points of higher iterates of , so the subsequent classifications follow from Theorem 3.
Definition 2. Suppose forms a -cycle for the map . That is, . Let , then the -cycle of is called(a)Superattracting if ,(b)Attracting if ,(c)Repelling if ,(d)Neutral if .
2. Newton’s Method and Trigonometric Functions
As discussed by Alexander et al. [4], the beginning of the study of complex dynamics dates back to 1870 where Ernst Schröder studied iterative equation solving algorithms in the complex plane and hence the nature of the infinite sequence from a theoretical viewpoint. Schröder’s interest in iterations led to the cornerstones of complex dynamics: Schröder’s fixed-point theorem and fixed point classifications. Schröder used a heuristic approach in his proof of the fixed-point theorem that relied on the Taylor series expansion of about an attracting fixed point and led to an explanation as to why Newton’s method works. He also studied convergence rates. Namely, for simple roots, , of a polynomial, Newton’s method converges quadratically in a neighborhood of . For roots of multiplicity greater than, Schröder modified Newton’s method to maintain this desirable convergence, and later, developed a family of similar root-solving algorithms which would either increase the rate of convergence, or have convergence of an arbitrary order.
Schröder was successful in developing fundamental tools of complex dynamics and applying them to the Newton map of : “Schröder observed, first, that there were periodic points of of every order on the imaginary axis.His second observation was that if was on the imaginary axis, but not eventually periodic, then takes on infinitely many values” [4]. In other words, the forward orbit of consists of infinitely many distinct points. Yet, his work raised an unanswered fundamental question: how far away can an arbitrary point z be from an attracting fixed point a of a function , such that ? It was not until the emergence of independent work by Pierre Fatou and Gaston Julia near the end of WWI that solutions to this query were explored. Furthermore, Schröder was unable to extend his results to higher-degree polynomials, in particular, he failed in an attempt to understand Newton’s method for the cubic and it took more than 45 years before the dynamics of were well understood [4].
The 1918 Grand Prix des Sciences Mathematiques competition was devoted to the study of the iteration of complex functions and hence sparked a flurry of papers published by Fatou and Julia on the global behavior of iterates of complex rational functions. During this time, they were able to bring about a substantial global theory of rational dynamics, but little was known about the global iterative behavior of other kinds of complex functions [4]. Unfortunately, it was not until the onset of the personal computer that complex dynamics received considerable public attention again, and in the early 1980s the field exploded as computer-generated images of the Mandelbrot set and Newton’s method on cubic polynomials circulated widely.
As shown later, the main focus of this paper will be the iteration of the function , so a look at literature concerning the dynamics of trigonometric functions will be of use. Unfortunately, this particular set of research is more limited than that of polynomials and rational functions. Schubert [5] proves that the area of the Fatou (or stable) set of the sine function in a vertical strip of width is finite. He also references the dynamical work, both historical and recent, carried out with functions such as for real, for , and .
Devaney [6] discusses the special class of meromorphic functions whose Schwarzian derivative:is a polynomial including the family of functions . As we know, the fate of asymptotic and critical values under iteration plays a crucial role in determining dynamics, and the main property of maps with polynomial Schwarzian derivatives is that they have a finite number of asymptotic values (all of which are isolated) and no critical values. In particular, for , where , we have , has asymptotic values at and , and preserves the real axis.
The Julia set is the closure of the set of repelling periodic points, or equivalently, is the closure of the set which consists of the union of all of the preimages of the poles of . Furthermore, all the poles and their preimages are dense in the Julia set. Devaney shows that is not a fractal set, indeed, it is a smooth manifold of ℂ. If , then , and all points such that tend asymptotically to the neutral fixed point . When , has an attracting periodic cycle of period two, and points in the upper and lower half planes hop back and forth as they are attracted to the cycle. And, since for all , for [6].
Devaney’s results show that for , zero is an attracting fixed point for and breaks up into a Cantor set. In fact, the basin of zero is infinitely connected, contrasting the situation for polynomial or entire maps (such as ) in which finite attracting fixed points always have a simply connected immediate basin of attraction (Devaney [6]). A full picture of the parameter plane for the tangent family is presented by Keen and Kotus [7]; however, an extensive literature search has failed to find work on the dynamics of Newton’s method applied to trigonometric functions.
2.1. Introduction to the Dynamics of Trig Functions
This paper focuses on the dynamics of the Newton map of , so as a means of comparison, we first look at some of the basic properties and dynamics of the entire functions and .
First, has a Newton map:
For all , the fixed points of are and thus, is a superattracting fixed point of .
Similarly, has a Newton map:which has fixed points at and . Thus, is a superattracting fixed point of .
The images produced from the iteration of and look as one might expect: with strips of width about each fixed point, where points within this strip will converge (under iterations of the Newton map) to each respective root (see Figure 3(a)). However, we can see fractal properties about the boundaries and the dynamics are far from trivial. Notice that the Newton maps of and have singularities at and , respectively, which leads to the boundary behavior shown in Figure 3(b).
(a)
(b)
Now, consider the meromorphic function which yields the corresponding Newton map:
Note that is never zero, so for all and is an entire function. Furthermore, the roots of are the roots of for all ; however, the fixed points of occur at both and. Taking the derivative of the Newton map:we see that and ; thus, by Theorem 3, for all , is a superattracting fixed point of and is a repelling fixed point of . Figure 4 shows the computer-generated global dynamics of . Each root lies at the center of one of the main colored bulbs. Seed values found outside of this strip of bulbs fail to iterate to any root under .
Recall that for , we haveand is the largest connected component of containing the root .
A closer inspection of any one primary basin of attraction gives greater insight into what is happening. Continuously zooming in on the boundary of each reveals the seemingly infinite nature of the fractal image. Each bulb consists of a boundary of bulbs, and it appears that there are an infinite number of them, each consisting of seed values which converge to a different root under iterations of . Figure 5(a) shows a closer view of the dynamics about . Every seed value from will iterate to 0. Furthermore, notice that from this distance, we see no more subsets of about the boundary of . Indeed, it seems that none of the bulbs that touch contain seed values that converge to .
(a)
(b)
Figure 5(b) displays a close-up of the largest bulbs stemming from in the first quadrant. Notice that this close-up imitates the broader picture of Figure 5(a) in shape and bulb placement. Namely, the hexagonal-like shape of this bulb, while slightly distorted, resembles that of with the positioning of boundary bulbs in the same general area. Moreover, these loose properties can be observed in any bulb one chooses to zoom in on.
2.2. Symmetry of
Our first exploration into the dynamics seen in Figure 5(a) is in the symmetry of the Newton map. We employ the following standard properties for all complex numbers :(i) and ,(ii) and .
We first show that is symmetric about the axis for all . Let . Then,that is, for any , if takes to , then it takes to .
In a similar manner, it can be shown that if takes to , then it takes to . Hence, is symmetric about the axis.
The symmetry of means that an exploration of the dynamics which occur in the first quadrant is sufficient to understand the global dynamics of under Newton’s method (see Figure 6).
2.3. Bounding the Primary Basins
What can we say about the basins of attraction these images present us with? We would like to, in some way, bound the different sets of seed values converging to distinct roots, and in this section, we will focus on the primary basins about roots . Recall that the function being iterated is , , and . So, for purely imaginary seed values, , we havethat is, points on the imaginary axis remain on the imaginary axis under iteration. Similarly, points on the real axis remain on the real axis since for , is in ℝ.
Now, according to Definition 1, for each attracting fixed point , there exists a neighborhood about such that all points in converge monotonically to . We will first examine sets of real points and points of the form .
Proposition 1. Along the real axis, . That is, for all, , we have
Proof. Before proving this result, note the following:(i)For , if , then hence ,(ii)For , if , then hence ,(iii)For, if , then , and if , then ,(iv)For has period .Now, we first prove the result for . That is, we will show that for allSuppose that . Then, , , and , so by (28a), . Suppose. Then,, , and . Hence, by (28b), . Thus, for all, we have that is, .
Let us now consider the general case. For , let . Then, implies and . Thus, by the above remarks, we haveSimilarly, for , let . Then, and by the same reasoning, we haveTherefore, for all , , we haveTo place similar bounds along the imaginary axis, we define the set:and we have the following result.
Proposition 2. For all , we have .
Proof. We will first prove the result for the attracting fixed point , and then extend it to the general case . For all , we haveThus,We first consider the case where:Hence, the set satisfies the desired inequality.
Now, suppose that and let , where . Then, we haveTherefore, for the attracting fixed point, for all , we haveWe now extend this result to the general case. Let and let . Since, we have for all such that, hence by the first part we haveThus, for the attracting fixed point, for all , we haveThe proof of the last result of this section makes use of both Schwarz’s lemma and the maximum modulus principle.
Lemma 1 (Schwarz). Let , and suppose is an analytic on with(a),(b).
Then, and for all in the open disk . Moreover, if or if for some , then there is a constant , such that .
Theorem 4 (Maximum modulus). Let be a bounded open set in ℂ and suppose is a continuous function on the closure of , , and analytic in . Then,where is the boundary of .
The maximum modulus principle says that in a bounded domain, an analytic function that has a continuous extension to the boundary attains its maximum modulus on the boundary. Equivalently, if the modulus of an analytic function achieves its maximum value at some point inside the domain , then the function is constant in .
We can now show that the unit disk is contained in the primary basin of attraction.
Theorem 5. For all , where , we have
Proof. This proof will make use of the following equivalencies for all :We will first prove this result for the case, i.e., we consider and the open unit disk .
is analytic and continuous on all of ℂ, in particular, is continuous on and analytic on . Thus, by the maximum modulus principle, , attains its maximum value when . Thus, we haveNow, let us define . A direct computation shows that on the interval if and only if . This can also be seen informally by graphing either or . Therefore, takes on its maximum value at and we haveThus, the maximum of is less than for all , so certainly for all . Furthermore, , thus by Schwarz’s lemma, for all , . Note that the only point for which is when , so excluding from the domain will yield a strict inequality. That is, for all , we have , i.e., .
We now extend this to the general case where. Consider and let for . Then, we have , and by the same reasoning as above, and thus by Schwarz’s lemma,Hence,Based on Figure 4, further exploration could entail an attempt to bound monotonically converging seed values in within the ellipse , where , or equivalently, where
3. Conclusion
The fractal image created from iterating the Newton map of is symmetric about both the - and axis as well as with respect to each attracting fixed point. In general, that which can be said about the dynamics surrounding can be said about the dynamics about . Indeed, as was shown, monotonically convergent seed values are bounded inside each primary basin of attraction by a circle of radius at least one centered at . These bounds can be extended along the real axes to and , and along the line to for . This leads one to conjecture that the unit circle is a minimal bound which could be extended to the ellipse: where . Furthermore, one could possibly show that each connected component of is bounded by a scaled version of this ellipse.
Indeed, as we have highlighted by the exploration of the Newton maps of two (seemingly) simple functions ( and ), complex dynamical systems have numerous interesting properties available for study.
Data Availability
No data were produced during this research.
Conflicts of Interest
The authors declare that they have no conflicts of interest.