Fractal Newton basins

The dynamics of complex cubic polynomials have been studied extensively in the recent years. The main interest in this work is to focus on the Julia sets in the dynamical plane, and then is consecrated to the study of several topics in more detail. Newton's method is considered since it is the main tool for finding solutions to equations, which leads to some fantastic images when it is applied to complex functions and gives rise to a chaotic sequence.


Introduction
Isaac Newton discovered what we now call Newton's method around 1670.Although Newton's method is an old application of calculus, it was discovered relatively recently that extending it to the complex plane leads to a very interesting fractal pattern.
We have focused on complex analysis, that is, studying functions holomorphic on a domain in the complex plane or holomorphic mappings.It is a very exciting field, in which many new phenomena wait to be discovered (and have been discovered).It is very closely linked with fractal geometry, as the basins of attraction for Newton's method have fractal character in many instances.We are interested in geometric problems, for example, the boundary behavior of conformal mappings.In mathematics, dynamical systems have become popular for another reason: beautiful pictures.Principally, we have focused on Julia sets and the iteration of rational functions.The Newton's method is one of the most important iterative algorithms for finding the zeroes of a polynomial or, more generally, a function meromorphic on the complex plane.
We first discuss some general properties of the method for rational functions with complex roots, including convergence criteria and rates of convergence, periodic points and basins of attractions of the roots.Next, we present results of Newton's method for one-dimensional and two-dimensional cases.Applying Newton's method to cubic function, whose roots are known as the 3th roots of unity, we show that the boundaries of the basins of attraction of each root are typically fractal in nature, and that they correspond to Julia sets in the complex plane.We conclude with a discussion of the implications of the fractal structure on the utility of Newton's method as a root-finding algorithm.
A Newton basin is just the set of initial guesses that lead to one solution or root.What is interesting is that Newton basins are fractals!and have beautifully intricate boundaries.In fact, they are kinds of Julia sets.
We refer to the surveys [2][3][4] for the general view of the dynamics of complex maps.We will use some concepts and facts of this theory without extra explanations.
Recall that one of the most important iteration schemes is Newton's method for finding the roots of algebraic equations equivalently, solutions to equations of the form Not only is the method easy to comprehend, it is a very efficient way and very helpful tool for finding roots as long as the function converges.In general, Newton's method works impressively fast (with quadratic convergence).But it requires a good initial guess, which normally needs to be close to one of the roots.Otherwise, strange things may happen.
Historically the survey relative to basins of attraction of a map generated by the Newton's algorithm related to the quadratic equation resolution with complex variable has been studied initially by Schroeder (1871).The cubic case developed by Cayley in 1879 [2].Specifically, Cayley wished to characterize the global basins of attraction for each root, more precisely, these basins of attraction are the initial conditions that Newton's method carries to each root.This task of identifying such basins for complex polynomials is known as "Cayley's problem."The works of Cayley have been reconsidered by Gaston Julia, who creates the Julia theory.
In this work we examine the chaotic behavior of a function of cubic type First, we study the geometry of the Julia set relative to the map (1.1) and we calculate its fractal dimension (Figure 1.1).By construction, the Julia set is the set of points for which Newton's method fails to find a root.

Notations, definitions and fundamental results
We denote C the complex plane; we define the Riemann sphere C as (C = C ∪ {∞}).
Let us consider the complex map z k+1 = g z k .
(2.1) Definition 2.1.We define the set of wandering points by (2.2) Definition 2.2.The Fatou set Ᏺ(g) of nonwandering points is defined by (2.3) Definition 2.3.The sets D g (∞) and Ᏺ(g) have a common boundary which is the Julia set (g), then Definition 2.4.The basin of attraction Ꮽ(ω) of an attractive fixed point ω, associated with a function g is (2.5) Definition 2.5.A point α is a fixed point of g if g(α) = α, and p-periodic if g p (α) = α, for all p ∈ N. The set {g k (α)} k=0,...,p−1 is called orbit or cycle.

The Newton's method
Isaac Newton devised an algebraic procedure for a rapidly convergent sequence of successive approximations to a root of the equation f (z) = 0, z ∈ C. Newton's derivation is equivalent to the modern iterative procedure described by the equation.Solving the polynomial equation with real coefficients in C given here by Let f be a polynomial and let z k be the kth iterate under Newton's method of the point z 0 .Then we say that z 0 is an approximate zero of f .
The basic idea is to construct a recurrent sequence {z k } k=0,2,... such as f (z k ) → 0. Assume that the image z k is known; the solution is the following image z k+1 , then replacing in (3.3) and taking into account that we obtain the well-known equation as Newton's sequence

Convergence property
The Newton's algorithm converges globally if for all z ∈ C a subsequence of {z k } k∈N * which tends to z * such that | f (z * )| 0. We test the efficiency of the algorithm by the exam maps that it generated.The notion of convergence speed permits to qualify the asymptotic behavior of a mapping.

Order of convergence.
We are interested by the following rate superlinear of order γ.In particular for γ = 2 we have a quadratic convergence, and Theorem of Newton-Kantorovich [1].Suppose that f is twice continuously differentiable and call z * a zero of f then if f (z * ) = 0 and z 0 is sufficiently close of z * .The map converges to z * and we have a quadratic convergence.
The final result that we know on the importance of linking fractal geometry with numerical analysis is probably: "the global convergence pattern of Newton's method is predictable and consists of few (four) convergence states," related to the residual measure.At least for complex polynomials, we have this description.
(i) Stationary residual with constant magnitude (usually 1-the exact magnitude depends on the distance between root of the function and relevant root of the first derivative).(ii) A sharp increase in residual when the iterates reach a singularity (root of the first derivative).Assuming ultimate convergence, this has to be followed by the next state.(iii) Linear convergence with a predictable rate (n/(n − 1) for polynomials of degree n).This is the only type of convergence possible once the residual is larger than the stationary value.(For residuals slightly larger than the stationary value, linearity is only an approximation.)(iv) Quadratic convergence once the iterates are sufficiently close to the root (the residual is smaller than the stationary value)-the well-known Newton-Kantorovich convergence result.

Application on the cubic case "Cayley problem"
In 1879, Cayley asked the following question: given an initial input z 0 , to which root will Newton's method converge.The answer was only fully understood recently and leads to a beautiful fractal pattern.The other good thing about knowing the structure of the underlying Newton fractal is that we can tell which regions are bad for a starting guess (namely, the ones that have singular-Julia-points in them).Next, we discuss Newton's method applied to the simple cubic equation (5.1) The Newton's equation is expressed in the form more explicitly (5.3) We note by z + = x + + iy + and z = x + iy instead of z k+1 = x k+1 + iy k+1 and z k = x k + iy k .If we replace in (5.3), we separate the real part of the imaginary part and after a simple calculation, we have which is simply the projection of the formula (5.3) on the real and imaginary axes.

Study of the one-dimensional Newton map.
Consider first the restriction of Newton's algorithm in the one-dimensional space, we now specialize to the real case and it suffices to replace in (5.3) z + and z by x + and x then (5.5) In the following Table 5.1, one summarized results of the algorithm of Newton with different initial points (k max represents the maximal number of iterations to reach the solution x * = 1.0).It is clear (from (5.5)) that the algorithm fails only for x = 0, then 0 belongs to Julia set as well as all x such that x → 0. Now the idea to construct the Julia set is to take images of the point at the origin by the inverse algorithm of Newton.Let us come back to (5.5) to solve this last with regard to x, we obtain that x < 1 there is a unique solution, which is where the inverse formula of Newton's algorithm.Then we can construct the Julia set (see Figure 5.1) as union of points {χ k } k=0,1,2,... defined by (5.7) The map (5.7) does not admit a fixed point because we can verify that for u < 1 we have (5.8) We remark that this last inequality implies the decrease of the mapping defined in (5.24) and that {χ k } k=0,1,2,... ⊂ R − .Let us examine now the behavior of the map {χ k } at the infinity while calculating the quotient (5.9)Where (5.10) Solving the extrema points we verify that (5.12) and because χ k+1 /χ k > 0 and χ k < 0 for all k ≥ 0 then lim k→+∞ χ k = −∞ (5.13) the numerical tests (Table 5.2) validate the result (5.13).

Modification of Newton algorithm and global convergence. Notice that the algorithm
(5.15) converges only for any x close of the root, otherwise one does not have a global convergence.For that we can interpret the quantity (1 − x 3 )/3x 2 as a direction, the long of which we do a displacement λ to force the global convergence.A new algorithm is proposed therefore (5.16) Besides, the bifurcation diagram (see Figure 5.6) permits us to affirm that in the interval ]0,2] there is convergence, on the other hand in the interval ]2,7] there is divergence of cyclic type due to a succession of period doubling and apparition of chaos.

The two-dimensional map.
It is easy to calculate the three roots of (1.3): (5.17  then 3) as solutions of the nonlinear system obtained by considering z = x + iy and f (z) = f 1 (x, y) + i f 2 (x, y).Let us pass now to the crucial construction of f schematized in Figure 5.7.Apply now to the map (5.17) the Newton algorithm by the following form where X = (x, y), and J −1 (X) is the Jacobian inverse f evaluated in X.Such that the Jacobian is given by which is invertible, the algorithm (5.18) is well defined and converges locally to the solution of (5.17), its eigenvalues are then then J(x, y) is singular for (x, y) = (0,0), elsewhere it does not have a definite sign and its inverse is expressed by (5.23)If we replace in (5.18) we obtain the formula (5.24) Remarks 5.1.We show three sets A 1 , A 2 , A 3 and their boundaries, such that A i = {z; f n (z) → z i } with some properties (see Figure 5.8).
(1) Symmetry: if you rotate the z-plane by 120 • , then A 1 becomes A 2 , A 2 becomes A 3 , and A 3 becomes A 1 .
(2) The closure of A 1 ∪ A 2 ∪ A 3 is equal to the whole z-plane.Thus any point z is either in A 1 , A 2 , A 3 or their boundaries.(3) The boundaries of A 1 , A 2 , and A 3 are the same and very exotic.If we denote ∂A = boundary of the set A, then ∂A 1 ∪ ∂A 2 ∪ ∂A 3 is called the Julia set and denote it as (ᏺ).(4) 0 and its preimages are in (ᏺ).Notice that f (0) = ∞.Hence, 0 is the hole leading to infinity.Clearly 0 / ∈ ∂A 1 ∪ ∂A 2 ∪ ∂A 3 .Thus 0 ∈ (ᏺ).We then conclude that all the preimages of 0 are also in (ᏺ).Note that these preimages of 0 are countable.
(5) Other aperiodic points which wander about in (ᏺ) forever must also be in (ᏺ).
These points are uncountable and make up the majority of (ᏺ).(6) There is a dense orbit in (ᏺ), the map (5.3) is chaotic on (ᏺ).
It is clear the first element of the Julia set (ᏺ) relative to the map (5.3) is the point (0,0), we can obtain the others elements by solving with regard to z the equation of Newton 2z 3 − 3z + z 2 + 1 = 0. (5.25) This last admits three complex solutions given by where (5.27) While projecting on real and imaginary axes, we have the following formulas in R 2 : (5.28) (5.29) with (5.31) In Table 5.3, we remark the behavior of the Newton algorithm with an initial condition in the Julia set (case of nonconvergence) whereas for a neighboring point the qualitative behavior changes.We now show how to construct the Julia set with the help of the inverse Newton algorithm defined by formulas (5.28), (5.29), and (5.30).

Figure 5
Figure 5.7.Solutions of the cubic equation.

2 Figure 5
Figure 5.8.The figure shows the basins of attraction for Newton's method applied to the equation z 3 − 1 = 0.

Table 5 .
2. Algorithm of Newton and its inverse.

Table 5 .
3. Cases of convergence and of nonconvergence for Newton algorithm.