Generalized Mandelbrot Sets of a Family of Polynomials

In this paper, we study the general Mandelbrot set of the family of polynomials Pn(z) � z + z + c; (n≥ 2) 􏼈 􏼉, denoted by GM(Pn). We construct the general Mandelbrot set for these polynomials by the escaping method. We determine the boundaries, areas, fractals, and symmetry of the previous polynomials. On the other hand, we study some topological properties of GM(Pn). We prove that GM(Pn) is bounded and closed; hence, it is compact. Also, we characterize the general Mandelbrot set as a union of basins of attraction. Finally, we make a comparison between the properties of famous Mandelbrot set M(z2 + c) and our general Mandelbrot sets.


Introduction
In the complex dynamical systems, a most beautiful and complicated object, with a simple definition, was studied from 1980s, and it is the well-known Mandelbrot set (see Figure 1). In fact, this set has many properties, like fractals, chaos, self-similarity, and others [1][2][3]. e Mandelbrot set is used to study the dynamics of the quadratic polynomial P(z) � z 2 + c [4,5].
e Mandelbrot set, denoted by M, consists of all values of c for which the orbit of the critical point, 0, of P is bounded, i.e., M(f) � c ∈⊄: f (n) (0) is bounded ∀n ∈ N} where f (n) (z) � f(f (n− 1) (z)) [6]. Note that the critical points are important in studying the complex dynamics because of the following fact: if a function possesses an attracting periodic orbit, then the orbit of the critical points must converge to that periodic orbit [7][8][9][10].
ere are a number of works that studied the general Mandelbrot sets, like the general Mandelbrot set of (z 2 + c + Re(z)), the general Mandelbrot set of (z + c), the general Mandelbrot set of (sinz/c ), and others [11][12][13][14][15]. e general Mandelbrot set, denoted by GM(f ), is defined by GM(f) � c ∈⊄: f (n) (z cr ) is bounded, z cr is the critical point of f} [16,17].
Wang et al. emphasized that the generalized Mandelbrot set contains a lot of information to construct the Julia set, which enters the state of chaos through the solution of fractal bifurcation and triple rotational bifurcation [18]. e paper [19] studies the structure topological inflexibility and the discontinuity evolution law of the generalized M-J sets generated from the complex mapping . Moreover, it studies the physical meaning of the generalized M-J sets. e researcher has other works in this field that the reader can find through the Internet.
In this paper, we construct the general Mandelbrot sets of the family of polynomials P n (z) � z n + z + c, (n ≥2), using the escaping algorithm. Moreover, we study their areas, their boundaries, and their fractals. Some topological properties of GM(P n ) were studied. We prove that GM(P n ) is closed and bounded and hence is compact, and we find a bound of GM(P n ), ∀n ≥ 2. Also, we characterize GM(P n ) as a union of basins of attraction of the attracting periodic point of P n , ∀n ≥ 2.
Finally, we construct (plot) the general Mandelbrot sets of P n (z), n � 4,5,6,7,8,9, and make a comparison of the properties of Mandelbrot set M(z 2 + c) and our general Mandelbrot sets GM(P n ).

Definitions
(i) For any complex map g, the point z cr , for which the eigenvalues of the Jacobi matrix are equal to zero, is called the general critical point of g [20]. (ii) e general Mandelbrot set of f c (z) is defined as follows: [21]. B ′ . e general Mandelbrot set of a function can be defined also as [20] GM(f) � c ∈⊄: J c (f) is conneced . (1) (iii) A metric space (S, ρ) is said to be connected iff S is not a union P ∪ Q of any two disjoint closed sets; it is disconnected otherwise. A set A ⊆ S is called connected iff (A, ρ) is connected as a subspace of (S, ρ), i.e., iff A is not a union of two disjoint sets P, Q ≠ ∅ that are closed in (A, ρ), as a subspace of (S, ρ) [22,23]. (iv) Let c be any complex number. e smallest closed set in the complex plane that contains all repelling periodic points of a family of complex function f λ (z) is called the Julia set of f λ and is denoted as J(f ) [3].
(vi) Let f be a function and let x 0 be in the domain of f. en, More generally, f (n) (x 0 ) for the n-th iterate of x 0 for f We call the sequence f (n) (x 0 ) ∞ n�0 of iterates of x 0 the orbit of x 0 [24]. (vii) If a fixed point P of f is attracting, then all points near to P are "attracted" toward P, in the sense that their iterates converge to P. e collection of all points whose iterates converge to P is called the basin of attraction of p [1]. (viii) e basin of attraction of the period point of period k, P 0 , is the collection of all points whose iterates converges P 0 or f(P 0 ) or . . . or f (k− 1) (P 0 ) [1].

Classification of Fractals.
Fractals can be classified according to their self-similarity. ere are three types of self-similarity found in fractals. Exact self-similarity is the strongest type of self-similarity, and the fractal appears identical at different scales, for example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity. e second type is the quasi-self-similarity; here a loose form of self-similarity and the fractal appears approximately (but not exactly) identical at different scales, i.e., the set contains small copies of the entire fractal in distorted and degenerate forms. Finally, statistical self-similarity is the weakest type of self-similarity.
e fractal here has numerical or statistical measures which are preserved across scales [16,25].

2.2.
e Area. e problem of calculating the area of Mandelbrot set is not simple yet. However, there are several methods which can be used to calculate an approximated area for these sets. e methods that calculate the best approximation to that area are pixel counting method, Monte Carlo method and Grunwall method. We will use the Monte Carlo method by generating random points in the complex plane, and then these points are tested if they are in the general Mandelbrot set or not. e area of a region is then estimated from the ratio of points in the set to those outside it.

The General Mandelbrot Set of the
Polynomial P n (z) = � z n + z + c In this section, we construct the general Mandelbrot sets of the polynomials, P n (z) � z n + z + c, (n � 2, 3), denoted by GM(P n ). We determine the area, the fractal, and the boundary of GM(P n ), (n � 2,3). Also, we show that the Julia sets of P 2 and P 3 are connected for certain values of the parameter c ∈ GM(P n ) and disconnected for a value of parameters which does not belong to MG(P n ). Remark 2. For the general cases (n > 3), the same arguments, used in this section, can be applied with some more complicated details and have similar results.
3.1. e General Mandelbrot Set of P 2 , GM(P 2 ). To construct GM(P 2 ), we calculate the general critical points of P 2 (z) � z 2 + z + c . We use the notations z � x + iy, c � c re + c im , and f � u + iv. en, the equation has the following form: (2) Its Jacobi matrix is J � 2x + 1 − 2y 2y 2x + 1 .
If y � 0, then (2x + 1) 2 � 0, and this follows x � − 1/2. us, the point (− 1/2, 0) is a critical point for the polynomial e construction of GM(P 2 ) is shown in Figures 2 and 3. It is clear that the Mandelbrot set of this polynomial is similar to the original Mandelbrot set without deformations.
Let z 0 be a fixed point for the polynomial P n (z). en, z 0 is attracting if |P n ′ (z 0 )| < 1.
us, the solutions of the equation |P n ′ (z 0 )| � 1 are the boundary points of the stable (attracting) point. at is, the curve |P n ′ (z 0 )| � 1 bounds all the attracting fixed points of P n (z). Similarly, the boundary equation |P (m)′ n (z 0 )| � 1 bounds all the periodic points of period m which are attracting. In Sections 3.1.1. and 3.2.1. we use this criterion to draw the boundaries of GM (P 2 ) and GM (P 3 ), respectively, and it is clear that this criterion is true to draw the boundaries of GM (P n ), ∀ n.
3.1.1. e Boundary of the Mandelbrot Set of the Fixed Point of P 2 . For the polynomial P 2 , the fixed points may be found by solving the equation , z 1 is the first fixed point of P 2 and z 2 is the second fixed point of P 2 . e stability of each period point is determined from the derivative of the map at the point. Now, 2z where r ≥ 0 and 0 ≤ θ ≤ 2π. Substituting (6) in (3), we have One of the fixed points, say z 1 , is stable (attracting) as long as |dP c /dzz 1 | < 1. erefore, the boundary of the point of period one is given by ||dP 2 /dzz 1 |z 1 | � |2z 1 + 1| � r � 1.

3.1.2.
e Fractals of GM(P 2 ). GM(P 2 ) admits, like the original Mandelbrot set, quasi-self-similarity. Also, it is similar around the x-axis.

e Area of GM(P 2 ).
Using Monte Carlo method, the area of the general Mandelbrot set of P 2 (z) is 1.6456. Note that this area is very close to the area of the original Mandelbrot set, M(f ).

e General Mandelbrot
Set of the Polynomial P 3 (z), GM(P 3 ). Consider P 3 (z) � z 3 + z + c. Using the notations z � x + iy, c � c re + c im and f � u + iv, the equation has the following form: Let u � x 3 − 3xy 2 + x + c re and v � 3x 2 y − y 3 + y + c im .

e Area of the Polynomial GM(P 3 ).
Applying the Monte Carlo algorithm, we find that the area of P 3 (z) is 2.1952. 3 . Applying definition B′, if we take c � 0 · 2i ∈ GM(P 3 ), then we note that the Julia set J c (P 3 ) is connected ( Figure 12) and for c � 0 · 3 + 0 · 8i ∉ GM(P 3 ) then J c (P 3 ) is disconnected (Figure 13).

Topological Properties of General Mandelbrot Set of P n (z)
In this section, we study some topological properties of GM (P n ), n � 2,3, . . ..

Proposition 1.
e general Mandelbrot set of P n ,GM(P n ) is closed.
is implies that ⊄-GM(P n ) is an open set in ⊄. erefore, GM(P n ) is closed. □ Proposition 2. Let P n � z n + z + c, then there exists an integer k, such that |P (m) n (z)| ⟶ ∞ for |z| > k.
Similarly, if z 0 is a periodic point of period m, then by the same argument and using Proposition 2, we obtain the same result. us, GM(P n ) � c: |c| < k { for some k ∈ R + }. us, GM(P n ) is bounded. □ Example 1. Consider the case n � 2, i.e., P 2 (z) � z 2 + z + c.
e fixed point of P 2 satisfies z 2 + z + c � z.

Theorem 2.
e general Mandelbrot set GM(P n ) consists of the union of the basins of attraction of the attracting periodic orbits, and each basin of a periodic point contains all values of c whose orbits converge to that periodic point.
Proof. By definition, GM(P n ) � c: P (m) n (z cr ) is bounded . at is, c ∈ GM(P n ) iff the orbit of the critical points of P n (z cr ) is bounded. But the orbit of any critical point must lie in the orbit of attracting period orbit and hence in a basin of an attracting periodic point. us, GM(P n ) is composed of basins of the attracting periodic orbits, and all the values of c ∈ GM(P n ) converge to these periodic orbits. Now, we construct the general Mandelbrot set of the polynomials P n (z) � z n + z + c for n � 4,5, . . ., 9 (see Figure 14).
Note that for n ≥ 4 in previous forms, the number of arms extended in the sets equals n− 2 arm. Moreover, if we plot the GM of P n (z) � z n + z + c, then we find the same behavior. In a future work, we hope to analyze this behavior.
Finally, we give a comparison between the Mandelbrot set generated with P(z) � z 2 + c and the general Mandelbrot sets of GM(P 2 ) and GM(P 3 ) (Table 1).

Conclusions
(1) In this paper, we construct the general Mandelbrot sets of the family of polynomials P n (z) � z n + z + c, (n ≥2), using the escaping algorithm. (2) e areas, boundaries, and fractals for P n (z) are studied and calculated for n � 2,3. (3) We study some topological properties of GM(P n ). It is proved that GM(P n ) is closed and bounded and hence is compact; also, we find a bound of GM(P n ), ∀n ≥ 2. (4) GM (P n ) is characterized as a union of basins of attraction of the attracting periodic point of P n , ∀n ≥ 2. (5) Finally, we plot the general Mandelbrot sets of P n (z), n � 4,5,6,7,8,9, and make a comparison of the properties of Mandelbrot set M(z 2 + c) and our general Mandelbrot sets GM(P n ).

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e author declares that there are no conflicts of interest.