Reducing Chaos and Bifurcations in Newton-Type Methods

and Applied Analysis 3 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x M̃ 1 .0 ,f − ( x )


Introduction
One of the most important and usual problems in numerical analysis is finding the solutions of a nonlinear equation: where  : R → R is a   function, with  ≥ 1.To approximate the solutions of these equations, iterative methods can be used.There are many iterative methods with different properties but the most commonly used is Newton's iterative method: This method and its variants have been studied by many authors since it was proposed by Newton in 1669.It is well known that under certain assumptions related to the starting point  0 and to the function  the method provides a sequence {  } ∈N that converges quadratically to a solution of (1).
The classical third-order schemes, such as Halley or Chebyshev methods, evaluate second-order Fréchet derivatives.These evaluations are very time consuming for systems of equations.Indeed, let us observe that, for a nonlinear system of  equations and  unknowns, the second Fréchet derivative has  3 entries.In particular, these methods are hardly used in practice.For that reason, we are interested in the study of methods that increase the order but evaluating only first-order Fréchet derivatives.The methods consist of two (or more) steps of the Newton method having the same derivative, other main advantage of these methods relays on the fact that if we consider a system of equations only one  decomposition is necessary in each iteration.Because of these two properties the schemes are considered a real alternative to the classical Newton method.
On the other hand, we can find a rigorous and useful procedure to find the best scheme in families of Newtontype methods with frozen derivatives in [1].For most of the cases, the most efficient candidate of the family is the two-step iterative method.We will study all this family but with special attention to the two-step scheme.
In this paper, our main purpose is to study the dynamics of the discrete system defined by the methods but introducing a damping factor  ∈ (0, 1].We are interested to search chaos or sensitive dependence on initial conditions for this discrete dynamical system.In [2], Hurley and Martin showed that the classical Newton iterative method exhibits chaos for a large class of functions (see also [3]).The introduction of the damping factor not only reduces the order of convergence but also reduces the chaos and the bifurcations as we will see in Sections 3 and 4. In particular, we can utilize these schemes as starting points for the classical Newton schemes.We conclude this paper by comparing the damped method with the original one and the successive damped high-order methods that come from using more Newton's steps.
The paper is organized as follows.In Section 2 we introduce our new scheme and present the associated scaling theorem.The scaling theorem allows up to suitable change of coordinates, to reduce the study of the dynamics of iterations of general maps, to the study of specific families of iterations of simpler maps.The dynamics for polynomials of degreetwo and three, having chaotic dynamics in some cases, are presented in Sections 3 and 4, respectively.In these sections, we observe that the inclusion of the damping factor reduces the bad zone of convergence.In Section 5 we give the main conclusions drawn from the study and we summarize the benefits to include the damping factor.Finally, we propose the use of damped methods in order to find good starting points for the original methods.

A Damped Newton-Type Iterative Method
Our first goal is to analyze the dynamics of a modification of the following third-order iterative root-finding method [9]: This iterative root-finding algorithm is defined by the following iterative function: where   () =  −   () and   () = ()/  ().In other words,  +1 =   (  ).The dynamics of this method were studied in [10].Now, we introduce a damping factor  ∈ (0, 1] and we analyze its influence on the dynamics in the iterative method (4).By introducing the damping factor the rootfinding algorithm has the following form: Thus the following iterative function describes the rootfinding algorithm: where  , () =  − ()/  ().
Notice that it is easy to proof that the roots of the function  are fixed points of  , .Moreover, we have that In particular, if  is a simple root of , that is, () = 0 and   () ̸ = 0, then   , () = (1 − ) 2 .Thus every simple root of  is an attracting point of  , and for  = 1 a superattracting fixed point.If  is a multiple root of multiplicity  of , then it is an attracting fixed point for every value of .There are fixed points of  , () different from the roots.These points are called extraneous fixed points.In [10], Amat et al. showed that this method for  = 1 (without damping factor) presents chaos and bifurcations when it is applied to a one-parameter family of cubic polynomials.Chaos behavior are related to the existence of points  * such that   ( * ) = 0 and ( * ) ̸ = 0.When we apply the iterative method  , to a polynomial, we may have some problems, since we obtain a rational map, say  , () = ()/(), where  and , are polynomials, that we may suppose without common factors.The difficulty arises at those points where the evaluation of numerator is nonzero and the denominator is zero, one such points correspond to the poles of the iterative method.
In order to study the dynamics of  , by means of graphical analysis we consider it as a map M, on [0, 1[.For this, let  : R → ]0, 1[ be given by , where  , is the iterative root-finding method introduced above for a map  : R → R. We may extend M, to maps from [0, 1] into itself.We use the same notation for this extension.Note that the extended function has fixed points at  = 0 and at  = 1 and that these fixed points are repelling.
Now we have the following useful result.
The Scaling Theorem allows up to suitable change of coordinates, to reduce the study of the dynamics of iterations  , , to the study of specific families of iterations of simpler maps.For example, each quadratic polynomial () =  2 +  + , with  ̸ = 0, that we suppose that is monic, by an affine change of variables reduces to one of the following polynomials: depending on the number of real roots of () = 0 (two, one (double), or not real roots).Therefore the study of the dynamics of  , , when it is applied to a quadratic polynomial, reduces to the study of the dynamics of  ,  where  = +, 0, −, up to affine change of variables.On the other hand, the study of the dynamics of the iterative method M, reduces to the study of the dynamics of the interval map M,  .Similarly, any cubic polynomial reduces to one of the simplest cubic polynomials: The last one is an appropriate rescaling that puts  , inside the conjugacy class.

Quadratic Polynomials
Following the analysis of Section 2, we have to study three cases.
It is clear that M1.0, − restricted to the interval  has an invariant Cantor set Δ − of zero Lebesgue measure of nonescaping points; that is, Δ − is the set of points whose orbits remain in the interval  under iteration by M1.0, − .In other words, these points are not attracted to one of the superattracting fixed points of M1.0, − (note that when  = 1 the fixed points are superattracting).
From the above analysis, we have that the orbit of any point  ∈ [0, 1]\(Δ − ∪  1 ,  2 ) is attracted to one of the superattracting fixed points of M1.0, − .Consequently, the orbit of any point in R \ ( −1 (Δ − ) ∪ { 3 ,  4 }) is attracted to one of the fixed points of  1.0, − .Thus we can define the "bad zone" as BZ = Δ − ∪ { 1 ,  2 } since if  ∈ BZ, then it is not attracted by any of the attracting points of M, − .Lemma 2. The "bad zone" of the damping two-step Newtonlike method defined in (6) applied to quadratic polynomials with two different roots decreases with the damping parameter .
Proof.We begin by calculating So we have that Thus  is an interval centered in 0. Δ − decreases when  decreases.Taking into account that the functions (/√8 + (−4 + ))() and arctan() are increasing functions, it follows that  decreases as  so with smaller  we have smaller BZ.
Thus we have seen analytically and graphically, in Figures 3, 4, 5, and 6, that the size of (Δ − ∪ { 1 ,  2 }) decreases with the damping factor  and as a consequence we can state that the chaotic zone decreases with .

Cubic Polynomials
In this section we consider cubic polynomials.As we have seen in Section 2, by an affine change of coordinates, every cubic polynomial  reduces to one of the simplest  0 ,  − ,  + or to a member of the one-parameter family of cubic polynomials   .Therefore, we analyze the dynamics of these simpler cubic polynomials.
Case 4 ( + () =  3 + ).In this case,  has only one real root at  = 0, and the method (6) has the following form: Since   + () = 1+3 2 > 0 for all , it follows that  + has no critical points; hence,   + , () is a global homeomorphism from R into itself and its dynamics is trivial.
Case 6 ( 0 () =  3 ).The function  0 only has one triple root at  = 0. We have that (6) has the following form: which is a linear contraction for every value of  ∈ (0, 1].The only fixed point is attracting, but not superattracting and the dynamics of   0 , is trivial. Case 7 (  () =  3 +  + 1).We have from (6) that We are going to analyze in this section the dynamics of    , () depending on the two parameters  and .Our main purpose is to compare the results by fixing the parameter  for different values of the damping parameter  and to study how it changes the dynamics.We begin by studying the case  < 0. In this case, our function   has two critical points:  +,− = ±√−/3.Let be  * = −3/ 3 √ 4 ≈ −1.88988.Notice that   * has a double root at the positive critical point  + .Hence, this value of the parameter will be very significant in the study of the dynamics.
We have three different possibilities.
(i) If  <  * , the polynomial   has three real roots.
(ii) If  =  * , the polynomial   has one positive double real root and a simple one which is negative.
(iii) If  >  * , the polynomial   has only one real root.
For  <  * the iterative method    , () has two vertical asymptotes.   , () has 7 fixed points, the 3 real roots of the polynomial (attracting fixed points), and the extraneous fixed points (repelling).See Figure 9 as an example.As we can observe in Figures 10,and 11 if the damping factor decreases, then extraneous fixed points tend to be closer to the asymptotes.
For  * ≤  < 0 the iterative method    , () has two vertical asymptotes at  − and  + , as we can see in Figure 12.Notice that lim  →  ± −    ,0 () = ∞ and lim  →  ± +    ,0 () = ∞.   , () has 3 fixed points, the one which corresponds to the real solution of the polynomial   is attracting and two repelling extraneous fixed points.Again we obtain that when the damping factor decreases the chaotic zone also decreases.Now we are going to consider the simplest case which corresponds to  = 0, and we have This map has one attracting point at  1 = −1 which corresponds to the root of  0 .Moreover it has another two extraneous fixed points which are the real solutions of the equation: As we see in Figure 13 both extraneous fixed points are repelling.Now if we focus our attention on the damping factor, we obtain that when  decreases, the extraneous fixed points tend to be closer to the asymptote  = 0.This behavior can be observed in Figures 14 and 15.Notice that  = 0 is the critical point of  0 .The last study is related with  > 0. Notice that for, every value of  ̸ = 0,    , presents a maximum which increases when  decreases.Moreover there exists a value   that depends on the value of the damping factor, such that, for each  >   ,    , has only one fixed point (the root of the function) and the dynamics is trivial.In Table 1, it shows the different approximations of   for some values of . Figure 16 illustrates this case.
When  =   ,    ,1 presents another fixed point which is a saddle node (as Amat et al. showed in [10]) and when the parameter  decreases, it appears bifurcations.If we now see Figures 17, 18, 19, and 20, it is clear that the problems decrease with ; even for small values ( < 0.2) they seem they disappear.
Finally, we conclude with the following conjecture: if  = /2  with ,  ∈ N, then    , has only three fixed points.

Concluding Remarks
In this work we have analyzed the dynamics of a new damped third-order Newton-like method.We have stated the following conclusions drawn from the study (i) The connected component of the basin of attraction increases when the damping factor decreases.
(ii) The sizes of the chaotic and the bifurcation zones decrease when the damping factor decreases.
(iii) The nonconvergence zone decreases when the damping factor decreases.
In addition, we have performed a similar study including more Newton's steps.The conclusions remain valid.Figures 21,22,23,and 24 show examples of the behavior with more Newton's steps.Therefore, we can propose the use of damped methods in order to find good starting points for the original methods.We would like to analyze these hybrid methods in a forthcoming paper.We are also interested to analyze the situation of nonfixed damping factors.

Table 1 :
Approximations of   for some values of .