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

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.

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 𝑧←𝑧^𝛼+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]). Let 𝑓_𝑐(𝑧)=𝑧^𝛼+𝑐 for 𝛼∈ℤ with 𝑧,𝑐∈ℂ, then the general M-sets is defined to be the set M={𝑐∈ℂ,lim_𝑘→∞𝑓^𝑘_𝑐(0)≠∞}. (2.1)

Definition 2.2 (see [15]). The sets defined by 𝑃_𝑚={𝑐∈ℂ∶𝑐=𝑟𝑒^𝑖𝜙𝑚,0≤𝑟,𝜙_𝑚=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).

(a) 𝛼=−1

(b) Primary symmetric axis 𝑃1 and section 𝑆1

(c) 𝛼=−2

(d) Partial enlargement of little block in Figure 1(c)

(e) 𝛼=−3

(f) Partial enlargement of little block in Figure 1(e)

Figure 1 

General M-sets and their periodic region.

Definition 2.3. The set𝑆₁={𝑐∈ℂ∶𝑐=𝑟𝑒^𝑖𝜃,0≤𝑟,0<𝜃≤𝜋/|𝛼−1|} is called the principal symmetric sector, as is shown in Figure 1(b).

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

Proof. Let 𝑐=𝜌𝑒^𝑖𝜙 with 𝜌≥0,0≤𝜙<2𝜋,𝑖=√−1. For all 𝑘∈ℕ, there exists the following recursive relations: 𝑇_𝑘+1(𝜙)=𝑇₁(𝜙)(1+𝑇_𝑘(𝜙))^𝛼, (2.2) where 𝑇₁(𝜙)=𝑐^𝛼−1=𝜌^𝛼−1𝑒^{𝑖(𝛼−1)𝜙}. Using the mathematical induction, for all 𝑘∈ℕ and 𝑚∈{1,2,…,2|𝛼−1|}, 𝑇_𝑘(𝜙)=𝑇_𝑘(−𝜙)=𝑇_𝑘(2𝜙_𝑚−𝜙) (2.3) exists. So we obtain 𝑓_𝑐^𝑘+1(0)=𝑐(1+𝑇_𝑘) since |𝑓_𝑐^𝑘+1(0)|=𝜌|1+𝑇_𝑘(𝜙)|=𝜌|1+𝑇_𝑘(𝜙)|=𝜌|1+𝑇_𝑘(𝜙)|=𝜌|1+𝑇_𝑘(−𝜙)|=𝜌|1+𝑇_𝑘(2𝜙_𝑚−𝜙)|=|𝑐^∗(1+𝑇_𝑘(2𝜙_𝑚−𝜙))|=|𝑓_𝑐^𝑘+1_∗(0)| (2.4) with 𝑐^∗=𝜌𝑒^{𝑖(2𝜙𝑚−𝜙)}, we have 𝑐^∗∈M whenever 𝑐∈M. This completes the proof.

Definition 2.5 (see [15]). An attracting 𝑘-periodic region is denoted by M^′_𝑘 and is defined as a region of the set {𝑐∈ℂ∶thereexist𝑧₀∈ℂsuchthat𝑓^𝑘_𝑐(𝑧₀)=𝑧₀𝑑,|𝑓𝑑𝑧^𝑘𝑐(𝑧)|_𝑧=𝑧0<1}. (2.5)

Definition 2.6. When 𝛼 is a positive, if 𝑐₀satisfying 𝑓^𝑘_𝑐0(0)=0 and 𝑐₀∈M^′_𝑘, 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 𝑓^𝑘_𝑐0(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 equation 𝑓^𝑘_𝑐(0)=0 which is a polynomial of 𝑐. The equation can be written as𝑓^𝑘_𝑐(0)=𝑐(1+𝑔_𝑘(𝑐)), (3.1) where 𝑔_𝑘 is a recursive function defined as𝑔_𝑘+1(𝑐)=𝑐^𝛼−1(1+𝑔_𝑘(𝑐))^𝛼,𝑔₁(𝑐)=0,𝑘≥1. (3.2) The 𝑐 in (3.2) has a degree of 𝛼^𝑘−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|.

Let 𝐻_𝑘(𝑤)=1+𝑔_𝑘(𝑐).

(1) When 𝛼 is a positive integer, (3.2) can be written as𝐻_𝑘+1(𝑤)=1+(𝑤−1)𝐻_𝑘(𝑤)^𝛼,𝑘≥1, (3.3) where 𝐻₁(𝑤)=1 and 𝑤=1+𝑐^𝛼−1.

Definition 3.1 (see [15]). Let 𝑘_𝑖(1≤𝑖≤𝜈) is a integer smaller than 𝑘 and satisfying 1<𝑘₁<𝑘₂<⋯<𝑘_𝜈<𝑘, let 𝐹_𝑘𝐻(𝑤)=^𝑘𝐻(𝑤),if𝑘=prime,𝛼isapositiveinteger,_𝑘(𝑤)∏^𝜈_𝑖=1𝐹_𝑘𝑖(𝑤),if𝑘≠prime,𝛼isapositiveinteger. (3.4)

(2) When 𝛼 is a negative integer, (3.2) can be written as𝐻_𝑘+11(𝑤)=1+(𝑤−1)𝐻_𝑘(𝑤)^𝛼,𝑘≥2, (3.5) where 𝐻₂(𝑤)=1 and 𝑤=1+𝑐^1−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 of 𝐹_𝑘(𝜔)=0 or 𝐻_𝑘(𝑤)=0, we can solve the roots of 1+𝑔_𝑘(𝑐)=0. Let 𝜔_𝑗(𝑗=1,2,…,𝑟) with Im(𝜔_𝑗)≥0 be a root of 𝐹_𝑘(𝜔)=0or𝐻_𝑘(𝑤)=0. The transformation 𝜔_𝑗=1+𝑐^|𝛼−1| with the symmetry of M-sets yields 𝑐_𝑗=𝜌_𝑗^1/|𝛼−1|𝑒^{𝑖(𝜃𝑗/|𝛼−1|)}, (3.6) where 𝜌_𝑗=(Re(𝜔_𝑗)−1)²+Im(𝜔_𝑗)². If Re(𝜔_𝑗)−1≥0, then𝜃_𝑗=tan⁻¹(Im(𝜔_𝑗)/(Re(𝜔_𝑗)−1)); If Re(𝜔_𝑗)−1<0, then 𝜃_𝑗=𝜋+tan⁻¹(Im(𝜔_𝑗)/(Re(𝜔_𝑗)−1)).

Among these 𝑐_𝑗 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)Set 𝑛,𝑘,𝜀₁,𝜀₂, and 𝐷𝑖𝑔𝑖𝑡𝑠(number of precision digits).(2)Construct 𝐹_𝑘(𝑤).(3)Construct SolveCenter(k) that does the following. (1) Using the 𝑠𝑜𝑙𝑣𝑒 function in 𝑚𝑎𝑝𝑙𝑒 to find the approximate roots of 𝐹_𝑘(𝑤)=0; (2) Select the roots with nonnegative imaginary parts, and compute 𝑟 and ‖𝐹_𝑘(𝑤)‖=max{|𝐹_𝑘(𝑤)|}.(4)Construct Newton (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 {𝑤_𝑚}; (2) Set the maximum iteration number 𝑁; (3) Set 𝜀 satisfying ‖Re(𝑤_𝑚+1−𝑤_𝑚)‖<𝜀 and ‖Im(𝑤_𝑚+1−𝑤_𝑚)‖<𝜀.(5)Construct DefCenter(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 ‖𝐹_𝑘(𝑤)‖.(6)Call SolveCenter(k), Compute residual error 𝐹_𝑘(𝑤_𝑗)(𝑗=1,2,…,𝑟) and ‖𝐹_𝑘(𝑤)‖=max{|𝐹_𝑘(𝑤_𝑗)|}.(7)If ‖𝐹_𝑘(𝑤)‖<𝜀₁ is not met, increase Digits and call SolveCenter (𝑘); If ‖𝐹_𝑘(𝑤)‖<𝜀₁ is met, then do the following. (1) Increase NtDigits; (2) Call DefCenter(k) to get refined roots 𝑤^∗, and calculate ‖𝐹_𝑘(𝑤^∗)‖; (3) If ‖𝐹_𝑘(𝑤^∗)‖<𝜀₂ is not met, increase 𝑁𝑡𝐷𝑖𝑔𝑖𝑡𝑠 and call DefCenter(k); If ‖𝐹_𝑘(𝑤^∗)‖<𝜀₂ is met, the check the convergence of the sequence 𝜂_𝑚=‖𝑒_𝑚+1/𝑒²_𝑚‖, where 𝑒_𝑚=𝑤_𝑚−𝑤^∗: If 𝜂_𝑚 is not convergent, increase NtDigits and call DefCenter (𝑘); If 𝜂_𝑚 is convergent, then accept 𝑐^∗=(𝑤^∗−1)^1/|𝑛−1| as the desired solution and terminate the entire procedure.

The above algorithm is appropriate for 𝛼 as a positive integer; if 𝛼 is a negative integer, we can take 𝐻_𝑘(𝑤) instead of 𝐹_𝑘(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 ‖𝐹_𝑘(𝑤)‖ and the values of asymptotic error 𝜂=lim_𝑚→∞‖𝑒_𝑚+1/𝑒²_𝑚‖. 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.

Table 2 

Typical coordinates(𝑥,𝑦)of periodic regions’ centers in general M-sets for 𝛼=−5,−10,−25.

Table 3 

Residual errors and asymptotic error constant for 𝛼=−1,−2−5,−10,−25.

3.3. Numerical Results

Now, we study the relationship between the numbers of roots on 𝑃₁ 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.

Let 𝑊={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 𝐻_𝑘(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 𝑁_𝑘(𝛼) denotes the total number of periodic regions’ centers lying in the complex plane ℂ. Then, the following relation holds [15]:𝑚₁+𝑚₂2=𝑑2𝑚=𝑟−²2. (3.7)

Next we discuss the calculation of periodic regions’ centers and the relation among 𝑚₁,𝑚₂,𝑑 and 𝑁_𝑘(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 with 𝛼−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. If 𝛼 is a positive integer, then the relation 𝑚₁+𝑚₂/2=𝑁_𝑘(𝛼)/(2(𝛼−1)) (or 𝑑=𝑁_𝑘(𝛼)/(𝛼−1)) yields the following [15].(1)If 𝑘=2, then 𝑚₁=0 and 𝑚₂=1 for any 𝛼;(2)If 𝑘≥3, then 𝑚₁=𝑁_𝑘(𝛼)/(2(𝛼−1)) and 𝑚₂=0 for all odd 𝛼;(3)If 𝑘≥3, then 𝑚₁≤(1/2){𝑁_𝑘(𝛼)/(𝛼−1)−1} and 𝑚₂≥1 for all even 𝛼.(4)If 𝑘=3, then 𝑚₁=𝛼/2 and 𝑚₂=1 for all even 𝛼;(5)If 𝑘=4, then 𝑚₁=𝛼(𝛼+1)/2−1 and 𝑚₂=2 for all even 𝛼;(6)If 𝑘=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 Theorem 3.2. 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 integer

The 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. If 𝛼 is a negative integer, then the relation 𝑚₁+𝑚₂/2=𝑁_𝑘(𝛼)/(2|𝛼−1|) (or 𝑑=𝑁_𝑘(𝛼)/|𝛼−1|)) yields the following:(1)If 𝛼=−2 and 𝑘≥3, then 𝑁_𝑘(𝛼)=𝑁_{𝑘(−𝛼)};(2)If 𝛼≤−3 and 𝑘≥3, then 𝑁_{𝑘(−𝛼)}=|𝛼+1|𝑁_𝑘(𝛼), namely 𝑁_𝑘(𝛼) with negative integer𝛼is equal to 𝑑 with −𝛼;(3)If 𝑘=2 or 3, then 𝑚₁=0,𝑚₂=1,𝑑=1,𝑁_𝑘(𝛼)=|𝛼−1| for all 𝛼;(4)If 𝑘=4, then 𝑚₂=2 and 𝑚₁=|𝛼|/2−1 for |𝛼| being even; 𝑚₂=1,𝑚₁=(|𝛼|−1)/2 for |𝛼| being odd;(5)If 𝑘=4, then 𝑑=|𝛼| for all 𝛼.

We can get the following from Theorem 3.2. 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 𝑁_𝑘(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 𝑁_𝑘(𝛼) satisfy Theorem 3.2.

Table 4 

(𝑚₁/𝑚₂) versus (𝛼,𝑘).

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

Let 𝑓^𝑘_𝑐(0)=ℎ_𝑘(𝑐)/ℎ_𝑘−1(𝑐)^|𝛼|, thenℎ_𝑘(𝑐)=ℎ_𝑘−2(𝑐)^|𝛼|2+𝑐ℎ_𝑘−1(𝑐)^|𝛼|(𝑘≥2), (4.1) where ℎ₀(𝑐)=0 and ℎ₁(𝑐)=1. The coordinates of periodic regions’ centers can be obtained by solving the equation ℎ_𝑘(𝑐)=0 with 𝑐 satisfying 𝑓^𝑘_𝑐(0)=0.

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

Table 5 

Expression of ℎ_𝑘(𝑐) when 𝛼=−1.

Table 6 

Expression of ℎ_𝑘(𝑐) when 𝛼=−2.

For the M-set constructed from the complex mapping 𝑓∶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 mapping 𝑓∶𝑧←𝑧⁻²+𝑐 (shown in Figure 1(c)), one-periodic region is a huge area whose boundary is defined by 𝑐=2^1/3𝑒^−𝑖𝜃−2^−2/3𝑒^2𝑖𝜃. 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 that 𝑘-periodic region’s centers can be located by the roots of ℎ_𝑘(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.

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).