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

Authors: Wang Xingyuan; He Yijie; Sun Yuanyuan
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## 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'k-periodic regions (k=3,4,5,6) for the index numbers α=−25,−24,…,−1 , all of which have highly numerical accuracy.


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## Body

## 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 mappingz←zα+c(α=2) utilizing computer graphics technologies [1]. 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 α∈ℝ and found there existed orderly structure within the M-sets [2–16]. For example, Álvarez et al. studied the location and number of each periodic region in M-sets [9]; Buchanan et al. studied the location of periodic region of the general M-sets with α=-2 [11]; 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 [15]; The  author studied the structure and distribution of the general M-sets with integer index number [16].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 [15]).
Letfc(z)=zα+c for α∈ℤ with z,c∈ℂ, then the general M-sets is defined to be the set
(2.1)M={c∈ℂ̂,limk→∞fck(0)≠∞}.Definition 2.2 (see [15]).
The sets defined byPm={c∈ℂ̂:c=reiϕm,0≤r,ϕm=mπ/|α-1|} for m∈{1,2,…,2|α-1|} are called the rays of symmetry. The set P1 is called the principal ray of symmetry. As is shown in Figure 1(b).General M-sets and their periodic region.
(a)
α=-1(b)
Primary symmetric axisP1 and section S1(c)
α=-2(d)
Partial enlargement of little block in Figure1(c)(e)
α=-3(f)
Partial enlargement of little block in Figure1(e)Definition 2.3.
The setS1={c∈ℂ̂:c=reiθ,0≤r,0<θ≤π/|α-1|} is called the principal symmetric sector, as is shown in Figure 1(b).Theorem 2.4.
In theℂ̂ parameter plane, M is symmetric about Pm.Proof.
Letc=ρeiϕ with ρ≥0,0≤ϕ<2π,i=-1. For all k∈ℕ, there exists the following recursive relations:
(2.2)Tk+1(ϕ)=T1(ϕ)(1+Tk(ϕ))α,
where T1(ϕ)=cα-1=ρα-1ei(α-1)ϕ. Using the mathematical induction, for all k∈ℕ and m∈{1,2,…,2|α-1|},
(2.3)Tk(ϕ)¯=Tk(-ϕ)=Tk(2ϕm-ϕ)
exists. So we obtain fck+1(0)=c(1+Tk) since
(2.4)|fck+1(0)|=ρ|1+Tk(ϕ)|=ρ|1+Tk(ϕ)|¯=ρ|1+Tk(ϕ)¯|=ρ|1+Tk(-ϕ)|=ρ|1+Tk(2ϕm-ϕ)|=|c*(1+Tk(2ϕm-ϕ))|=|fc*k+1(0)|
with c*=ρei(2ϕm-ϕ), we have c*∈M whenever c∈M. This completes the proof.Definition 2.5 (see [15]).
An attractingk-periodic region is denoted by Mk′ and is defined as a region of the set
(2.5){c∈ℂ:thereexistz0∈ℂsuchthatfck(z0)=z0,|ddzfck(z)|z=z0<1}.Definition 2.6.
Whenα is a positive, if c0satisfying fc0k(0)=0 and c0∈Mk′, then c0  is called the center of an attracting k-periodic region.Definition 2.7.
Whenα is a negative, c0 is the center of a stable k-periodic region only if ∞ belongs to this k-periodic region and c0 satisfies fc0k(0)=0.


## 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 equationfck(0)=0 which is a polynomial of c. The equation can be written as(3.1)fck(0)=c(1+gk(c)),
where gk is a recursive function defined as(3.2)gk+1(c)=cα-1(1+gk(c))α,g1(c)=0,k≥1.
The c in (3.2) has a degree of αk-1-1 and thus will encounter a difficulty in obtaining accurate solutions as α and k increase. But the transformation w=1+c|α-1| reduces the degree by |α-1|.LetHk(w)=1+gk(c).(1) When α is a positive integer, (3.2) can be written as(3.3)Hk+1(w)=1+(w-1)Hk(w)α,k≥1,
where H1(w)=1 and w=1+cα-1.Definition 3.1 (see [15]).
Letki(1≤i≤ν) is a integer smaller than k and satisfying 1<k1<k2<⋯<kν<k, let
(3.4)Fk(w)={Hk(w),ifk=prime,αisapositiveinteger,Hk(w)∏i=1νFki(w),ifk≠prime,αisapositiveinteger.(2) When α is a negative integer, (3.2) can be written as(3.5)Hk+1(w)=1+1(w-1)Hk(w)α,k≥2,
where H2(w)=1 and w=1+c1-α.So we can solve the roots of (3.4) or (3.5) instead of (3.1). If α is a positive integer, we can use (3.4) to calculate the center of periodic region. If α is a negative integer, we can use (3.5) to calculate the center of periodic region.


### 3.2. Numerical Algorithm

Through solving the roots ofFk(ω)=0 or Hk(w)=0, we can solve the roots of 1+gk(c)=0. Let ωj(j=1,2,…,r) with Im(ωj)≥0 be a root of Fk(ω)=0orHk(w)=0. The transformation ωj=1+c|α-1| with the symmetry of M-sets yields
(3.6)cj=ρj1/|α-1|ei(θj/|α-1|),
where ρj=(Re(ωj)-1)2+Im(ωj)2. If Re(ωj)-1≥0, thenθj=tan-1(Im(ωj)/(Re(ωj)-1)); If Re(ωj)-1<0,  then θj=π+tan-1(Im(ωj)/(Re(ωj)-1)).Among thesecj values, we select the ones in the interior of the primary symmetric S1 and the ones on the primary symmetric axis P1. 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 [15].(1)
Setn,k,ε1,ε2, and Digits(number of precision digits).(2)
ConstructFk(w).(3)
ConstructSolveCenter(k) that does the following. (1) Using the solve function in maple to find the approximate roots of Fk(w)=0; (2) Select the roots with nonnegative imaginary parts, and compute r and ∥Fk(w)∥=max{|Fk(w)|}.(4)
ConstructNewton(f,w,N) that does the following. (1) Take the initial values w of about three decimal digits of accuracy from the results of solve to precede the Newton sequence {wm}; (2) Set the maximum iteration number N; (3) Set ε satisfying ∥Re(wm+1-wm)∥<ε and ∥Im(wm+1-wm)∥<ε.(5)
ConstructDefCenter(k) finding coordinate of the periodic region’s center: (1) Reset Digits to a higher number NtDigits; (2) If k=2, then w*=0; If k≠2, then call Newton(f,w,N) to get refined roots w*; (3) Compute c*=(w*-1)1/|n-1|, m1,m2, and ∥Fk(w)∥.(6)
CallSolveCenter(k), Compute residual error Fk(wj)(j=1,2,…,r) and ∥Fk(w)∥=max{|Fk(wj)|}.(7)
If∥Fk(w)∥<ε1 is not met, increase Digits and call SolveCenter(k); If ∥Fk(w)∥<ε1 is met, then do the following. (1) Increase NtDigits; (2) Call DefCenter(k) to get refined roots w*, and calculate ∥Fk(w*)∥; (3) If ∥Fk(w*)∥<ε2 is not met, increase NtDigits and call DefCenter(k); If ∥Fk(w*)∥<ε2 is met, the check the convergence of the sequence ηm=∥em+1/em2∥, where em=wm-w*: If ηm is not convergent, increase NtDigits and call DefCenter(k); If ηm is convergent, then accept c*=(w*-1)1/|n-1| as the desired solution and terminate the entire procedure.The above algorithm is appropriate forα as a positive integer; if α is a negative integer, we can take Hk(w) instead of Fk(w). The algorithm is achieved by the maple. Tables 1 and 2 list typical coordinates of periodic region’ centers for α=-1,-2,-5,-10,-25 and 3≤k≤6; 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 ∥Fk(w)∥ and the values of asymptotic error η=limm→∞∥em+1/em2∥. The parameters are defined as ε=0.5×10-48,ε1=0.5×10-3, and ε2=0.5×10-90 in the experiment.Table 1
Typical coordinates(x,y)of periodic regions’ centers in general M-sets forα=-1,-2.
αkxy301401.414213562373095048801688724209698078569−1500.618033988749894848204586834365638117720301.618033988749894848204586834365638117720601.732050807568877293527446341505872366942−230.50000000000000000000000000000000000000000.86602540378443864676372317075293618347140.36278131512316317038180607006038976716780.628355669829974126342078405158097669672340.68912038624460509620066759635407427904181.1935915215071448965120983537662956296380.40422369054045645729947488500272535685630.70013596963906955206824739838366368042460.58656017165781273575624348449142539291710.522746915457430578220722108192078300644650.64247544302096045090558860862911229429411.1128001099276267580897196758860619084730.77969250461670800793962162131565997577731.350467032276769692645172849662184857212Table 2
Typical coordinates(x,y)of periodic regions’ centers in general M-sets for α=-5,-10,-25.
αkxy30.86602540378443864676372317075293618347140.50000000000000000000000000000000000000000.93969262078590838405410927732473146993620.342020143325668733044099614682259580763041.0947126868235232133169282478295587029520.63203266442285961289304785813126999073360.79672572154316425199802221164198960733700.39538143335487720602107635997971996321670.73608649419613244141936257312740478163870.42497973557098496801989644431689321130360.78563385873275829654012496567095514801810.41202357795354036549381007292560159553410.80044460015700648821324169099598526860470.40920178958502381277860286512844408090010.80941195338729723126085240043193833531540.40182330972082824012577483084041472303550.81458133210908400573510120618189967205710.38791526348940744587357940036583749455260.81499585149285542237881662131452770735390.33994484265909586882751073319632684014990.90441258638982323792016818205880986671130.3455698635303133428003524080560280276196−550.99933627157433937204274409986552247919400.28359265320184014635115385194800409012300.92551170374321018874687273856810427415970.36200333968285058083351240082167345562020.94519463791578739071779171581145536498620.36565670954961407607752854405499859099330.96946417411071806235750273204475623161330.35600809619273321501213413898717773688341.1165805263635739718341975272357038888000.57424128635076257184583863487651529160701.0699556212951390645555342638336420055090.59621261229057323511748631267101376132081.1939918283979600014277260517131557811300.689351503535775659153177564685195833875530.95949297361449738989036805706632769906240.28173255684142969771141791534661689903570.89785746537763395433886934054632088948180.26363473871736567271625969438593245346220.91049890794072368857535172278731234378320.2367915193754428000032888830923803306147−100.93345009469497818080398233437944152875030.212349777932438891240904303421824982017840.97276229975460395206434351733014355686230.19184898330487450709127543381271846544261.0497096766982957303425277368992509627280.18914621275124175876323933955602537539491.1303125246684178119085201638105391854970.331889702542698634888399612172249243244530.99270887409805399280075164949252017934360.12053668025532305334906768745254358227360.96643231951328303613083803633786768285690.11489371648221880205058014962890323690930.96762382832768494272727232586570425469280.11013021408853351285105334367338315640120.96944330162262172914959157669916542904100.10543290186309469126703880185748580834630.97193987839238336928950023222102803177350.10080109564523965924639603357890594366670.97519595063148995162340218452561676614120.096238469227241493769929386439446125749270.97934149758788674464868365315974254426960.09175563494485486517255996206353137944357−2540.98458125197181142600815050852829763954980.087375862542128762161220557898390279791790.99124887323652560715212868889737313675700.083148797420661064415343880134249370985130.99992551708381000204443864904940747092960.079188101383950865861306939317213884790831.0117375349376348694676507318892839873470.075798068515360973186702231946195620541471.0292586914587439269935937119634410234800.074061018049173251559835402348685458868781.0593964767386416436749423083898690777200.08045677368903782206077846838233504471072Table 3
Residual errors and asymptotic error constant forα=-1,-2-5,-10,-25.
αk∥Hk(w)∥η−130not available40not available50.100000e-992.4472136 atc = 0.61803398874989484820i60not available−230not available40.892511e-1002.7888544 atc = 0.36278131512316317038 + 0.6283556698299741263i50.150000e-993.4739187 atc = 0.64247544302096045091 + 1.112800109927626758i−530not available40.101663e-977.9352875 atc = 0.79672572154316425199 + 0.39538143335487720602i50.8609622e-9878.011649 atc = 0.81458133210908400573 + 0.38791526348940744587i−1030not available40.188919e-9816.988191 atc = 0.91049890794072368857 + 0.23679151937544280000i−2530not available40.375119e-9839.779704 atc = 0.97193987839238336928 + 0.10080109564523965924i


### 3.3. Numerical Results

Now, we study the relationship between the numbers of roots onP1 and in the interior of S1. According to the Rotation symmetry, we consider that c is and only is on P1 and in the interior of S1.LetW={w∈C:Im(w)≥0}, we select roots having only nonnegative imaginary part and suppose r is the number of all such roots. Let d denotes the number of roots for Hk(w)=0 in the complex plane ℂ, m1 denotes the number of centers lying in the interior of S1, m2 denotes the number of centers lying on P1, and Nk(α) denotes the total number of periodic regions’ centers lying in the complex plane ℂ. Then, the following relation holds [15]:(3.7)m1+m22=d2=r-m22.Next we discuss the calculation of periodic regions’ centers and the relation amongm1,m2,d and Nk(α) when α is a positive integer and a negative integer, respectively.(1) If α is a positive integer,The general M-sets with positive integer index number are similar to flowers combined withα-1 major petals, and stable region is embedded in unstable region. If α=1, the complex mapping f 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 α=1, the complex structure appear immediately [3].Theorem 3.2.
Ifα is a positive integer, then the relation m1+m2/2=Nk(α)/(2(α-1)) (or d=Nk(α)/(α-1)) yields the following [15].(1)
Ifk=2, then m1=0 and m2=1 for any α;(2)
Ifk≥3, then m1=Nk(α)/(2(α-1)) and m2=0 for all odd α;(3)
Ifk≥3, then m1≤(1/2){Nk(α)/(α-1)-1} and m2≥1 for all even α.(4)
Ifk=3, then m1=α/2 and m2=1 for all even α;(5)
Ifk=4, then m1=α(α+1)/2-1 and m2=2 for all even α;(6)
Ifk=5,6,7,8,9,10, then m2=3,5,9,16,28,51, respectively, for all even α.We can get the following from Theorem3.2. Statement (1) implies that a two-periodic region’s center lies on P1; Statements (1) and (2) show that only a two-periodic region’s center lies on P1 when α is odd. Statements (2) and (3) show that every k-periodic region’s center with k≥2 lies on P1 when α is even. Statements (4) and (5) show that for any even integer α≥2 one three-periodic and two four-periodic periodic region’s centers lie on P1, respectively.(2) If α is a negative integerThe general M-sets with negative integer index number have planetary configuration consisting of a central planet with|α-1| 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 α being a positive integer.Theorem 3.3.
Ifα is a negative integer, then the relation m1+m2/2=Nk(α)/(2|α-1|)  (or d=Nk(α)/|α-1|)) yields the following:(1)
Ifα=-2 and k≥3, then Nk(α)=Nk(-α);(2)
Ifα≤-3 and k≥3, then Nk(-α)=|α+1|Nk(α), namely Nk(α) with negative integerαis equal to d with -α;(3)
Ifk=2 or 3, then m1=0,m2=1,d=1,Nk(α)=|α-1| for all α;(4)
Ifk=4, then m2=2 and m1=|α|/2-1 for |α| being even; m2=1,m1=(|α|-1)/2 for  |α| being odd;(5)
Ifk=4, then d=|α| for all α.We can get the following from Theorem3.2. Statement (3) implies that one two-periodic and one three-periodic region’s center lies on P1; Statement (4) state that two four-periodic region’s center lies on P1 when |α| is even, as is shown in Figure 1(d); When |α| is odd, only one four-periodic region’s center lies on P1, as is shown in Figure 1(f). Table 4 shows the relationship among m1,m2,d and Nk(α) when α is a negative integer, which testifies Theorem 3.3 and (3.7). The numbers in Table 4 are m1/m2. The corresponding d and Nk(α) satisfy Theorem 3.2.Table 4
(m1/m2) versus (α,k).
αk2345678910−12/12/12/14/24/26/36/38/48/40/10/10/10/20/20/30/30/40/4−23/13/16/215/527/963/21120/40252/840/10/10/21/32/56/912/1628/28−34/14/112/340/10116/280/10/11/14/214/1−45/15/120/485/170/10/11/27/3−56/16/130/5156/260/10/12/112/2−67/17/142/6259/370/10/12/217/3−78/18/156/7400/500/10/13/124/2−89/19/172/8585/650/10/13/231/3−910/10/190/9820/820/10/14/140/2−1011/111/1110/100/10/14/2−1112/112/1121/120/10/15/1−1213/113/1156/120/10/15/2−1314/114/1182/130/10/16/1−1415/115/1210/140/10/16/2−1516/116/1240/150/10/17/1−1617/117/1272/160/10/17/2−1718/118/1306/170/10/18/1−1819/119/1342/180/10/18/2−1920/120/1380/190/10/19/1−2021/121/1420/200/10/19/2−2122/122/1462/210/10/110/1−2223/123/1506/220/10/110/2−2324/124/1552/230/10/111/1−2425/125/1600/240/10/111/2−2526/126/1650/250/10/112/1


## 3.1. Calculation Method

The center of periodic region can be located by numerically solving the governing equationfck(0)=0 which is a polynomial of c. The equation can be written as(3.1)fck(0)=c(1+gk(c)),
where gk is a recursive function defined as(3.2)gk+1(c)=cα-1(1+gk(c))α,g1(c)=0,k≥1.
The c in (3.2) has a degree of αk-1-1 and thus will encounter a difficulty in obtaining accurate solutions as α and k increase. But the transformation w=1+c|α-1| reduces the degree by |α-1|.LetHk(w)=1+gk(c).(1) When α is a positive integer, (3.2) can be written as(3.3)Hk+1(w)=1+(w-1)Hk(w)α,k≥1,
where H1(w)=1 and w=1+cα-1.Definition 3.1 (see [15]).
Letki(1≤i≤ν) is a integer smaller than k and satisfying 1<k1<k2<⋯<kν<k, let
(3.4)Fk(w)={Hk(w),ifk=prime,αisapositiveinteger,Hk(w)∏i=1νFki(w),ifk≠prime,αisapositiveinteger.(2) When α is a negative integer, (3.2) can be written as(3.5)Hk+1(w)=1+1(w-1)Hk(w)α,k≥2,
where H2(w)=1 and w=1+c1-α.So we can solve the roots of (3.4) or (3.5) instead of (3.1). If α is a positive integer, we can use (3.4) to calculate the center of periodic region. If α is a negative integer, we can use (3.5) to calculate the center of periodic region.


## 3.2. Numerical Algorithm

Through solving the roots ofFk(ω)=0 or Hk(w)=0, we can solve the roots of 1+gk(c)=0. Let ωj(j=1,2,…,r) with Im(ωj)≥0 be a root of Fk(ω)=0orHk(w)=0. The transformation ωj=1+c|α-1| with the symmetry of M-sets yields
(3.6)cj=ρj1/|α-1|ei(θj/|α-1|),
where ρj=(Re(ωj)-1)2+Im(ωj)2. If Re(ωj)-1≥0, thenθj=tan-1(Im(ωj)/(Re(ωj)-1)); If Re(ωj)-1<0,  then θj=π+tan-1(Im(ωj)/(Re(ωj)-1)).Among thesecj values, we select the ones in the interior of the primary symmetric S1 and the ones on the primary symmetric axis P1. 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 [15].(1)
Setn,k,ε1,ε2, and Digits(number of precision digits).(2)
ConstructFk(w).(3)
ConstructSolveCenter(k) that does the following. (1) Using the solve function in maple to find the approximate roots of Fk(w)=0; (2) Select the roots with nonnegative imaginary parts, and compute r and ∥Fk(w)∥=max{|Fk(w)|}.(4)
ConstructNewton(f,w,N) that does the following. (1) Take the initial values w of about three decimal digits of accuracy from the results of solve to precede the Newton sequence {wm}; (2) Set the maximum iteration number N; (3) Set ε satisfying ∥Re(wm+1-wm)∥<ε and ∥Im(wm+1-wm)∥<ε.(5)
ConstructDefCenter(k) finding coordinate of the periodic region’s center: (1) Reset Digits to a higher number NtDigits; (2) If k=2, then w*=0; If k≠2, then call Newton(f,w,N) to get refined roots w*; (3) Compute c*=(w*-1)1/|n-1|, m1,m2, and ∥Fk(w)∥.(6)
CallSolveCenter(k), Compute residual error Fk(wj)(j=1,2,…,r) and ∥Fk(w)∥=max{|Fk(wj)|}.(7)
If∥Fk(w)∥<ε1 is not met, increase Digits and call SolveCenter(k); If ∥Fk(w)∥<ε1 is met, then do the following. (1) Increase NtDigits; (2) Call DefCenter(k) to get refined roots w*, and calculate ∥Fk(w*)∥; (3) If ∥Fk(w*)∥<ε2 is not met, increase NtDigits and call DefCenter(k); If ∥Fk(w*)∥<ε2 is met, the check the convergence of the sequence ηm=∥em+1/em2∥, where em=wm-w*: If ηm is not convergent, increase NtDigits and call DefCenter(k); If ηm is convergent, then accept c*=(w*-1)1/|n-1| as the desired solution and terminate the entire procedure.The above algorithm is appropriate forα as a positive integer; if α is a negative integer, we can take Hk(w) instead of Fk(w). The algorithm is achieved by the maple. Tables 1 and 2 list typical coordinates of periodic region’ centers for α=-1,-2,-5,-10,-25 and 3≤k≤6; 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 ∥Fk(w)∥ and the values of asymptotic error η=limm→∞∥em+1/em2∥. The parameters are defined as ε=0.5×10-48,ε1=0.5×10-3, and ε2=0.5×10-90 in the experiment.Table 1
Typical coordinates(x,y)of periodic regions’ centers in general M-sets forα=-1,-2.
αkxy301401.414213562373095048801688724209698078569−1500.618033988749894848204586834365638117720301.618033988749894848204586834365638117720601.732050807568877293527446341505872366942−230.50000000000000000000000000000000000000000.86602540378443864676372317075293618347140.36278131512316317038180607006038976716780.628355669829974126342078405158097669672340.68912038624460509620066759635407427904181.1935915215071448965120983537662956296380.40422369054045645729947488500272535685630.70013596963906955206824739838366368042460.58656017165781273575624348449142539291710.522746915457430578220722108192078300644650.64247544302096045090558860862911229429411.1128001099276267580897196758860619084730.77969250461670800793962162131565997577731.350467032276769692645172849662184857212Table 2
Typical coordinates(x,y)of periodic regions’ centers in general M-sets for α=-5,-10,-25.
αkxy30.86602540378443864676372317075293618347140.50000000000000000000000000000000000000000.93969262078590838405410927732473146993620.342020143325668733044099614682259580763041.0947126868235232133169282478295587029520.63203266442285961289304785813126999073360.79672572154316425199802221164198960733700.39538143335487720602107635997971996321670.73608649419613244141936257312740478163870.42497973557098496801989644431689321130360.78563385873275829654012496567095514801810.41202357795354036549381007292560159553410.80044460015700648821324169099598526860470.40920178958502381277860286512844408090010.80941195338729723126085240043193833531540.40182330972082824012577483084041472303550.81458133210908400573510120618189967205710.38791526348940744587357940036583749455260.81499585149285542237881662131452770735390.33994484265909586882751073319632684014990.90441258638982323792016818205880986671130.3455698635303133428003524080560280276196−550.99933627157433937204274409986552247919400.28359265320184014635115385194800409012300.92551170374321018874687273856810427415970.36200333968285058083351240082167345562020.94519463791578739071779171581145536498620.36565670954961407607752854405499859099330.96946417411071806235750273204475623161330.35600809619273321501213413898717773688341.1165805263635739718341975272357038888000.57424128635076257184583863487651529160701.0699556212951390645555342638336420055090.59621261229057323511748631267101376132081.1939918283979600014277260517131557811300.689351503535775659153177564685195833875530.95949297361449738989036805706632769906240.28173255684142969771141791534661689903570.89785746537763395433886934054632088948180.26363473871736567271625969438593245346220.91049890794072368857535172278731234378320.2367915193754428000032888830923803306147−100.93345009469497818080398233437944152875030.212349777932438891240904303421824982017840.97276229975460395206434351733014355686230.19184898330487450709127543381271846544261.0497096766982957303425277368992509627280.18914621275124175876323933955602537539491.1303125246684178119085201638105391854970.331889702542698634888399612172249243244530.99270887409805399280075164949252017934360.12053668025532305334906768745254358227360.96643231951328303613083803633786768285690.11489371648221880205058014962890323690930.96762382832768494272727232586570425469280.11013021408853351285105334367338315640120.96944330162262172914959157669916542904100.10543290186309469126703880185748580834630.97193987839238336928950023222102803177350.10080109564523965924639603357890594366670.97519595063148995162340218452561676614120.096238469227241493769929386439446125749270.97934149758788674464868365315974254426960.09175563494485486517255996206353137944357−2540.98458125197181142600815050852829763954980.087375862542128762161220557898390279791790.99124887323652560715212868889737313675700.083148797420661064415343880134249370985130.99992551708381000204443864904940747092960.079188101383950865861306939317213884790831.0117375349376348694676507318892839873470.075798068515360973186702231946195620541471.0292586914587439269935937119634410234800.074061018049173251559835402348685458868781.0593964767386416436749423083898690777200.08045677368903782206077846838233504471072Table 3
Residual errors and asymptotic error constant forα=-1,-2-5,-10,-25.
αk∥Hk(w)∥η−130not available40not available50.100000e-992.4472136 atc = 0.61803398874989484820i60not available−230not available40.892511e-1002.7888544 atc = 0.36278131512316317038 + 0.6283556698299741263i50.150000e-993.4739187 atc = 0.64247544302096045091 + 1.112800109927626758i−530not available40.101663e-977.9352875 atc = 0.79672572154316425199 + 0.39538143335487720602i50.8609622e-9878.011649 atc = 0.81458133210908400573 + 0.38791526348940744587i−1030not available40.188919e-9816.988191 atc = 0.91049890794072368857 + 0.23679151937544280000i−2530not available40.375119e-9839.779704 atc = 0.97193987839238336928 + 0.10080109564523965924i


## 3.3. Numerical Results

Now, we study the relationship between the numbers of roots onP1 and in the interior of S1. According to the Rotation symmetry, we consider that c is and only is on P1 and in the interior of S1.LetW={w∈C:Im(w)≥0}, we select roots having only nonnegative imaginary part and suppose r is the number of all such roots. Let d denotes the number of roots for Hk(w)=0 in the complex plane ℂ, m1 denotes the number of centers lying in the interior of S1, m2 denotes the number of centers lying on P1, and Nk(α) denotes the total number of periodic regions’ centers lying in the complex plane ℂ. Then, the following relation holds [15]:(3.7)m1+m22=d2=r-m22.Next we discuss the calculation of periodic regions’ centers and the relation amongm1,m2,d and Nk(α) when α is a positive integer and a negative integer, respectively.(1) If α is a positive integer,The general M-sets with positive integer index number are similar to flowers combined withα-1 major petals, and stable region is embedded in unstable region. If α=1, the complex mapping f 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 α=1, the complex structure appear immediately [3].Theorem 3.2.
Ifα is a positive integer, then the relation m1+m2/2=Nk(α)/(2(α-1)) (or d=Nk(α)/(α-1)) yields the following [15].(1)
Ifk=2, then m1=0 and m2=1 for any α;(2)
Ifk≥3, then m1=Nk(α)/(2(α-1)) and m2=0 for all odd α;(3)
Ifk≥3, then m1≤(1/2){Nk(α)/(α-1)-1} and m2≥1 for all even α.(4)
Ifk=3, then m1=α/2 and m2=1 for all even α;(5)
Ifk=4, then m1=α(α+1)/2-1 and m2=2 for all even α;(6)
Ifk=5,6,7,8,9,10, then m2=3,5,9,16,28,51, respectively, for all even α.We can get the following from Theorem3.2. Statement (1) implies that a two-periodic region’s center lies on P1; Statements (1) and (2) show that only a two-periodic region’s center lies on P1 when α is odd. Statements (2) and (3) show that every k-periodic region’s center with k≥2 lies on P1 when α is even. Statements (4) and (5) show that for any even integer α≥2 one three-periodic and two four-periodic periodic region’s centers lie on P1, respectively.(2) If α is a negative integerThe general M-sets with negative integer index number have planetary configuration consisting of a central planet with|α-1| 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 α being a positive integer.Theorem 3.3.
Ifα is a negative integer, then the relation m1+m2/2=Nk(α)/(2|α-1|)  (or d=Nk(α)/|α-1|)) yields the following:(1)
Ifα=-2 and k≥3, then Nk(α)=Nk(-α);(2)
Ifα≤-3 and k≥3, then Nk(-α)=|α+1|Nk(α), namely Nk(α) with negative integerαis equal to d with -α;(3)
Ifk=2 or 3, then m1=0,m2=1,d=1,Nk(α)=|α-1| for all α;(4)
Ifk=4, then m2=2 and m1=|α|/2-1 for |α| being even; m2=1,m1=(|α|-1)/2 for  |α| being odd;(5)
Ifk=4, then d=|α| for all α.We can get the following from Theorem3.2. Statement (3) implies that one two-periodic and one three-periodic region’s center lies on P1; Statement (4) state that two four-periodic region’s center lies on P1 when |α| is even, as is shown in Figure 1(d); When |α| is odd, only one four-periodic region’s center lies on P1, as is shown in Figure 1(f). Table 4 shows the relationship among m1,m2,d and Nk(α) when α is a negative integer, which testifies Theorem 3.3 and (3.7). The numbers in Table 4 are m1/m2. The corresponding d and Nk(α) satisfy Theorem 3.2.Table 4
(m1/m2) versus (α,k).
αk2345678910−12/12/12/14/24/26/36/38/48/40/10/10/10/20/20/30/30/40/4−23/13/16/215/527/963/21120/40252/840/10/10/21/32/56/912/1628/28−34/14/112/340/10116/280/10/11/14/214/1−45/15/120/485/170/10/11/27/3−56/16/130/5156/260/10/12/112/2−67/17/142/6259/370/10/12/217/3−78/18/156/7400/500/10/13/124/2−89/19/172/8585/650/10/13/231/3−910/10/190/9820/820/10/14/140/2−1011/111/1110/100/10/14/2−1112/112/1121/120/10/15/1−1213/113/1156/120/10/15/2−1314/114/1182/130/10/16/1−1415/115/1210/140/10/16/2−1516/116/1240/150/10/17/1−1617/117/1272/160/10/17/2−1718/118/1306/170/10/18/1−1819/119/1342/180/10/18/2−1920/120/1380/190/10/19/1−2021/121/1420/200/10/19/2−2122/122/1462/210/10/110/1−2223/123/1506/220/10/110/2−2324/124/1552/230/10/111/1−2425/125/1600/240/10/111/2−2526/126/1650/250/10/112/1


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

Letfck(0)=hk(c)/hk-1(c)|α|, then(4.1)hk(c)=hk-2(c)|α|2+chk-1(c)|α|(k≥2),
where h0(c)=0 and h1(c)=1. The coordinates of periodic regions’ centers can be obtained by solving the equation hk(c)=0 with c satisfying fck(0)=0.Tables5 and 6 list the expressions of centers’ coordinates from two-periodic to five-periodic region.Table 5
Expression ofhk(c) when α=-1.
khk(c)2c3c2+14c3+2c5c4+3c2+1Table 6
Expression ofhk(c) when α=-2.
khk(c)2c3c3+14c7+3c4+c5c15+7c12+5c9+12c6+5c3+1For the M-set constructed from the complex mappingf:z←z-1+c (shown in Figure 1(a)), one-periodic region’s center is ∞ for f1(0)=∞. 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, h0(c)=0 in Table 5 deduces c=0. Similarly, three-periodic region’s centers can be located by the roots of c2+1=0.For the M-set constructed from the complex mappingf:z←z-2+c (shown in Figure 1(c)), one-periodic region is a huge area whose boundary is defined by c=21/3e-iθ-2-2/3e2iθ. 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 c3+1=0.The discussion above indicates thatk-periodic region’s centers can be located by the roots of hk(c)=0. The results of centers’ coordinates are almost the same as the results obtained by the first method described in Section 2 when α is a negative integer.


## 5. Conclusions

(1) 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 α is a negative integer which is described in Section 2.(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 [17]. In addition, the methods we proposed are helpful for solving polynomial equations, especially of high degree.(3) 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 [18]. 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 [16], 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.



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*Source: 653816-2010-08-16.xml*