Research Article | Open Access

Volume 2010 |Article ID 653816 | https://doi.org/10.1155/2010/653816

Wang Xingyuan, He Yijie, Sun Yuanyuan, "Accurate Computation of Periodic Regions' Centers in the General M-Set with Integer Index Number", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 653816, 12 pages, 2010. https://doi.org/10.1155/2010/653816

# Accurate Computation of Periodic Regions' Centers in the General M-Set with Integer Index Number

Revised24 Feb 2010
Accepted07 Jun 2010
Published16 Aug 2010

#### Abstract

This paper presents two methods for accurately computing the periodic regions' centers. One method fits for the general M-sets with integer index number, the other fits for the general M-sets with negative integer index number. Both methods improve the precision of computation by transforming the polynomial equations which determine the periodic regions' centers. We primarily discuss the general M-sets with negative integer index, and analyze the relationship between the number of periodic regions' centers on the principal symmetric axis and in the principal symmetric interior. We can get the centers' coordinates with at least 48 significant digits after the decimal point in both real and imaginary parts by applying the Newton's method to the transformed polynomial equation which determine the periodic regions' centers. In this paper, we list some centers' coordinates of general M-sets' -periodic regions for the index numbers , all of which have highly numerical accuracy.

#### 1. Introduction

According to the idea of complex dynamic system theory presented by Julia and Fatou, the famous mathematician Mandelbrot constructed and studied the M-sets of complex mapping utilizing computer graphics technologies . During the last 20 years, people have researched the embedded-layer relationship and distribution of the bifurcation sequence and topological rule of periodic trajectories in the general M-sets with and found there existed orderly structure within the M-sets . For example, Álvarez et al. studied the location and number of each periodic region in M-sets ; Buchanan et al. studied the location of periodic region of the general M-sets with ; Geum and Kim analyzed the quantitative relationship of each periodic region in the general Mandelbrot sets with positive integer index number, and calculated the coordinates of periodic regions’ centers ; The author studied the structure and distribution of the general M-sets with integer index number .

M-sets consist of different-period regions which constitute the fractal structures of the M-sets. The analysis on the stability of the maps uncovered new and unexpected algebraic properties of the periodic regions. The centers of the periodic regions are determined by the transformed polynomials we worked on. The motivation for computing the periodic regions’ centers is provided by the need to consider the locations of the periodic regions and fractal structures of the M-sets, which are useful to understand the inner infinite structures of the M-sets.

On the basis of above research, we study the periodic region’s centers in the general M-sets with integer index number, determine the relationship between each periodic region’s number in the general M-sets with positive index number and negative index number, and then present a new method of calculating the coordinates of periodic regions’ centers in the general M-sets with negative integer index number. Our research has good prospects in physics, information science, and other fields.

#### 2. Periodic Region Theory of General M-Sets

Definition 2.1 (see ). Let for with , then the general M-sets is defined to be the set

Definition 2.2 (see ). The sets defined by for are called the rays of symmetry. The set is called the principal ray of symmetry. As is shown in Figure 1(b).

Definition 2.3. The set is called the principal symmetric sector, as is shown in Figure 1(b).

Theorem 2.4. In the parameter plane, M is symmetric about .

Proof. Let with . For all , there exists the following recursive relations: where . Using the mathematical induction, for all and , exists. So we obtain since with , we have whenever . This completes the proof.

Definition 2.5 (see ). An attracting -periodic region is denoted by and is defined as a region of the set

Definition 2.6. When is a positive, if satisfying and , then is called the center of an attracting -periodic region.

Definition 2.7. When is a negative, is the center of a stable -periodic region only if belongs to this -periodic region and satisfies .

#### 3. Calculation of Periodic Regions’ Centers in General M-sets with Integer Index Number

##### 3.1. Calculation Method

The center of periodic region can be located by numerically solving the governing equation which is a polynomial of . The equation can be written as where is a recursive function defined as The in (3.2) has a degree of and thus will encounter a difficulty in obtaining accurate solutions as and increase. But the transformation reduces the degree by .

Let .

() When is a positive integer, (3.2) can be written as where and .

Definition 3.1 (see ). Let is a integer smaller than and satisfying , let

() When is a negative integer, (3.2) can be written as where and .

So we can solve the roots of (3.4) or (3.5) instead of (3.1). If is a positive integer, we can use (3.4) to calculate the center of periodic region. If is a negative integer, we can use (3.5) to calculate the center of periodic region.

##### 3.2. Numerical Algorithm

Through solving the roots of or , we can solve the roots of . Let with be a root of or. The transformation with the symmetry of M-sets yields where . If , then; If , then .

Among these values, we select the ones in the interior of the primary symmetric and the ones on the primary symmetric axis . Then by the rotation symmetry, we can get the center’s coordinate of each periodic region.

Using the Newton method, the following algorithm can locate the center of periodic region .()Set , and (number of precision digits).()Construct .()Construct SolveCenter(k) that does the following. () Using the function in to find the approximate roots of ; () Select the roots with nonnegative imaginary parts, and compute and .()Construct Newton that does the following. () Take the initial values of about three decimal digits of accuracy from the results of to precede the Newton sequence ; () Set the maximum iteration number ; () Set satisfying and .()Construct DefCenter(k) finding coordinate of the periodic region’s center: () Reset to a higher number NtDigits; () If , then ; If , then call Newton to get refined roots ; () Compute , , and .()Call SolveCenter(k), Compute residual error and .()If is not met, increase Digits and call SolveCenter ; If is met, then do the following. () Increase NtDigits; () Call DefCenter(k) to get refined roots , and calculate ; () If is not met, increase and call DefCenter(k); If is met, the check the convergence of the sequence , where : If is not convergent, increase NtDigits and call DefCenter ; If is convergent, then accept as the desired solution and terminate the entire procedure.

The above algorithm is appropriate for as a positive integer; if is a negative integer, we can take instead of . The algorithm is achieved by the . Tables 1 and 2 list typical coordinates of periodic region’ centers for and ; the accuracy is of 48 precision digits, but only the first 40 precision digits were printed for the tabulation. Table 3 shows the residual error defined by and the values of asymptotic error . The parameters are defined as ,, and in the experiment.

 k 3 0 1 4 0 1.414213562373095048801688724209698078569 −1 5 0 0.6180339887498948482045868343656381177203 0 1.618033988749894848204586834365638117720 6 0 1.732050807568877293527446341505872366942 −2 3 0.5000000000000000000000000000000000000000 0.8660254037844386467637231707529361834714 0.3627813151231631703818060700603897671678 0.6283556698299741263420784051580976696723 4 0.6891203862446050962006675963540742790418 1.193591521507144896512098353766295629638 0.4042236905404564572994748850027253568563 0.7001359696390695520682473983836636804246 0.5865601716578127357562434844914253929171 0.5227469154574305782207221081920783006446 5 0.6424754430209604509055886086291122942941 1.112800109927626758089719675886061908473 0.7796925046167080079396216213156599757773 1.350467032276769692645172849662184857212
 k x y 3 0.8660254037844386467637231707529361834714 0.5000000000000000000000000000000000000000 0.9396926207859083840541092773247314699362 0.3420201433256687330440996146822595807630 4 1.094712686823523213316928247829558702952 0.6320326644228596128930478581312699907336 0.7967257215431642519980222116419896073370 0.3953814333548772060210763599797199632167 0.7360864941961324414193625731274047816387 0.4249797355709849680198964443168932113036 0.7856338587327582965401249656709551480181 0.4120235779535403654938100729256015955341 0.8004446001570064882132416909959852686047 0.4092017895850238127786028651284440809001 0.8094119533872972312608524004319383353154 0.4018233097208282401257748308404147230355 0.8145813321090840057351012061818996720571 0.3879152634894074458735794003658374945526 0.8149958514928554223788166213145277073539 0.3399448426590958688275107331963268401499 0.9044125863898232379201681820588098667113 0.3455698635303133428003524080560280276196 −5 5 0.9993362715743393720427440998655224791940 0.2835926532018401463511538519480040901230 0.9255117037432101887468727385681042741597 0.3620033396828505808335124008216734556202 0.9451946379157873907177917158114553649862 0.3656567095496140760775285440549985909933 0.9694641741107180623575027320447562316133 0.3560080961927332150121341389871777368834 1.116580526363573971834197527235703888800 0.5742412863507625718458386348765152916070 1.069955621295139064555534263833642005509 0.5962126122905732351174863126710137613208 1.193991828397960001427726051713155781130 0.6893515035357756591531775646851958338755 3 0.9594929736144973898903680570663276990624 0.2817325568414296977114179153466168990357 0.8978574653776339543388693405463208894818 0.2636347387173656727162596943859324534622 0.9104989079407236885753517227873123437832 0.2367915193754428000032888830923803306147 −10 0.9334500946949781808039823343794415287503 0.2123497779324388912409043034218249820178 4 0.9727622997546039520643435173301435568623 0.1918489833048745070912754338127184654426 1.049709676698295730342527736899250962728 0.1891462127512417587632393395560253753949 1.130312524668417811908520163810539185497 0.3318897025426986348883996121722492432445 3 0.9927088740980539928007516494925201793436 0.1205366802553230533490676874525435822736 0.9664323195132830361308380363378676828569 0.1148937164822188020505801496289032369093 0.9676238283276849427272723258657042546928 0.1101302140885335128510533436733831564012 0.9694433016226217291495915766991654290410 0.1054329018630946912670388018574858083463 0.9719398783923833692895002322210280317735 0.1008010956452396592463960335789059436667 0.9751959506314899516234021845256167661412 0.09623846922724149376992938643944612574927 0.9793414975878867446486836531597425442696 0.09175563494485486517255996206353137944357 −25 4 0.9845812519718114260081505085282976395498 0.08737586254212876216122055789839027979179 0.9912488732365256071521286888973731367570 0.08314879742066106441534388013424937098513 0.9999255170838100020444386490494074709296 0.07918810138395086586130693931721388479083 1.011737534937634869467650731889283987347 0.07579806851536097318670223194619562054147 1.029258691458743926993593711963441023480 0.07406101804917325155983540234868545886878 1.059396476738641643674942308389869077720 0.08045677368903782206077846838233504471072
 −1 3 not available 4 not available 5 2.4472136 at c = 0.61803398874989484820 6 not available −2 3 not available 4 2.7888544 at c = 0.36278131512316317038 + 0.6283556698299741263 5 3.4739187 at c = 0.64247544302096045091 + 1.112800109927626758 −5 3 not available 4 7.9352875 at c = 0.79672572154316425199 + 0.39538143335487720602 5 78.011649 at c = 0.81458133210908400573 + 0.38791526348940744587 −10 3 not available 4 16.988191 at c = 0.91049890794072368857 + 0.23679151937544280000 −25 3 not available 4 39.779704 at c = 0.97193987839238336928 + 0.10080109564523965924
##### 3.3. Numerical Results

Now, we study the relationship between the numbers of roots on and in the interior of . According to the Rotation symmetry, we consider that is and only is on and in the interior of .

Let , we select roots having only nonnegative imaginary part and suppose is the number of all such roots. Let denotes the number of roots for in the complex plane , denotes the number of centers lying in the interior of , denotes the number of centers lying on , and denotes the total number of periodic regions’ centers lying in the complex plane . Then, the following relation holds :

Next we discuss the calculation of periodic regions’ centers and the relation among ,, and when is a positive integer and a negative integer, respectively.

() If is a positive integer,

The general M-sets with positive integer index number are similar to flowers combined with major petals, and stable region is embedded in unstable region. If , the complex mapping degenerates into linear mapping, the general M-sets is only a circular which is a trivial structure without complex boundary and self-similar structure. Once deviating from , the complex structure appear immediately .

Theorem 3.2. If is a positive integer, then the relation (or ) yields the following .()If , then and for any ;()If , then and for all odd ;()If , then and for all even .()If , then and for all even ;()If , then and for all even ;()If , then , respectively, for all even .

We can get the following from Theorem 3.2. Statement () implies that a two-periodic region’s center lies on ; Statements () and () show that only a two-periodic region’s center lies on when is odd. Statements () and () show that every -periodic region’s center with lies on when is even. Statements () and () show that for any even integer one three-periodic and two four-periodic periodic region’s centers lie on , respectively.

() If is a negative integer

The general M-sets with negative integer index number have planetary configuration consisting of a central planet with major satellite structures, and the unstable region is embedded in the stable region, as is shown in Figure 1. The numbers and locations of the periodic region’s centers in the general M-sets with negative integer index number can be calculated using the same method as being a positive integer.

Theorem 3.3. If is a negative integer, then the relation (or )) yields the following:()If and , then ;()If and , then , namely with negative integeris equal to with ;()If or , then for all ;()If , then and for being even; , for being odd;()If , then for all .

We can get the following from Theorem 3.2. Statement () implies that one two-periodic and one three-periodic region’s center lies on ; Statement () state that two four-periodic region’s center lies on when is even, as is shown in Figure 1(d); When is odd, only one four-periodic region’s center lies on , as is shown in Figure 1(f). Table 4 shows the relationship among ,, and when is a negative integer, which testifies Theorem 3.3 and (3.7). The numbers in Table 4 are . The corresponding and satisfy Theorem 3.2.

 2 3 4 5 6 7 8 9 10 −1 2/1 2/1 2/1 4/2 4/2 6/3 6/3 8/4 8/4 0/1 0/1 0/1 0/2 0/2 0/3 0/3 0/4 0/4 −2 3/1 3/1 6/2 15/5 27/9 63/21 120/40 252/84 0/1 0/1 0/2 1/3 2/5 6/9 12/16 28/28 −3 4/1 4/1 12/3 40/10 116/28 0/1 0/1 1/1 4/2 14/1 −4 5/1 5/1 20/4 85/17 0/1 0/1 1/2 7/3 −5 6/1 6/1 30/5 156/26 0/1 0/1 2/1 12/2 −6 7/1 7/1 42/6 259/37 0/1 0/1 2/2 17/3 −7 8/1 8/1 56/7 400/50 0/1 0/1 3/1 24/2 −8 9/1 9/1 72/8 585/65 0/1 0/1 3/2 31/3 −9 10/1 0/1 90/9 820/82 0/1 0/1 4/1 40/2 −10 11/1 11/1 110/10 0/1 0/1 4/2 −11 12/1 12/1 121/12 0/1 0/1 5/1 −12 13/1 13/1 156/12 0/1 0/1 5/2 −13 14/1 14/1 182/13 0/1 0/1 6/1 −14 15/1 15/1 210/14 0/1 0/1 6/2 −15 16/1 16/1 240/15 0/1 0/1 7/1 −16 17/1 17/1 272/16 0/1 0/1 7/2 −17 18/1 18/1 306/17 0/1 0/1 8/1 −18 19/1 19/1 342/18 0/1 0/1 8/2 −19 20/1 20/1 380/19 0/1 0/1 9/1 −20 21/1 21/1 420/20 0/1 0/1 9/2 −21 22/1 22/1 462/21 0/1 0/1 10/1 −22 23/1 23/1 506/22 0/1 0/1 10/2 −23 24/1 24/1 552/23 0/1 0/1 11/1 −24 25/1 25/1 600/24 0/1 0/1 11/2 −25 26/1 26/1 650/25 0/1 0/1 12/1

#### 4. Calculation of Periodic Regions’ Centers in General M-sets with Negative Integer Index Number

Let , then where and . The coordinates of periodic regions’ centers can be obtained by solving the equation with satisfying .

Tables 5 and 6 list the expressions of centers’ coordinates from two-periodic to five-periodic region.

 2 3 4 5
 2 3 4 5

For the M-set constructed from the complex mapping (shown in Figure 1(a)), one-periodic region’s center is for . Two-periodic region’s center reaches the one-periodic region’s center by one iteration, so the origin is the center of two-periodic region, that is, in Table 5 deduces . Similarly, three-periodic region’s centers can be located by the roots of .

For the M-set constructed from the complex mapping (shown in Figure 1(c)), one-periodic region is a huge area whose boundary is defined by . Two-periodic stable region divides the area outside of one-periodic region into three parts. One-periodic region’s center is . Two-periodic region’s center is the origin. Similarly, three-periodic region’s centers can be located by the roots of .

The discussion above indicates that -periodic region’s centers can be located by the roots of . The results of centers’ coordinates are almost the same as the results obtained by the first method described in Section 2 when is a negative integer.

#### 5. Conclusions

() In this paper, we proposed two methods for calculating the periodic regions’ centers of the general M-sets. The first method fits for calculating the periodic regions’ centers in the general M-sets with integer index number, which is to transform the polynomial equation that governs the periodic regions’ centers, obtain high precision of the coordinates by the simple method, and analyze the relationship between the number of each periodic regions’ centers on the principal symmetric axis and in the principal symmetric interior, then comparatively analyze the relation of periodic regions’ number in general M-sets with opposite integer index number. The second methods as discussed in Section 3 suits for calculating the periodic regions’ centers in the general M-sets with negative integer index number, which also transforms the polynomial equation. The results of centers’ coordinates obtained by the second method are almost the same as that of the first method when is a negative integer which is described in Section 2.

() The investigation of the periodic regions’ centers of the M-sets can help us explore the distribution of periodic regions of the M-sets, which can further help us to study the fractal structures of the M-sets. The centers of the periodic regions are located as the roots of certain polynomials, which are shown to coincide with solutions of the Douady and Hubbard formula . In addition, the methods we proposed are helpful for solving polynomial equations, especially of high degree.

() This research is some inspiration for the people studying on the difficult problems in their professional and interdisciplinary fields.

As a classical example of physics, Brownian movement is the most simple and typical random movement. The Langevin equation can depict the rule of a charged particle under the circularly successive influence of the impulse functions. However, it is difficult to visually depict the trajectory and dynamics of these systems with many random variables. If we construct the complex general M-sets using the rules of Langevin equation, the fractal structure characteristics of the general M-sets can reveal the changing rule of the particle velocity visually . This study makes it possible to depict complex Brownian movement more accurately.

In addition, the theories of M-sets have potential applications on image processing. We have known that general M-sets are illustrated dictionary of the corresponding general Julia sets , which means a single point on the M-sets can represent the huge amount image data of the general Julia sets with manifold shapes and complicated structures. This research provides the technology support for determining rapidly the coordinates of the points on the M-sets by the Julia images. On the basis of the above research results, future work includes establishing dictionary of fractal compression and studying the corresponding coding algorithm to improve the transmission and memory of the information, which could provide the new thoery for fractal compression technology.

#### Acknowledgment

This research is supported by the National Natural Science Foundation of China (nos. 60573172, 60973152), the Superior University Doctor Subject Special Scientific Research Foundation of China (no. 20070141014), and the Natural Science Foundation of Liaoning province (no. 20082165).

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