The Property of Chaotic Orbits with Lower Positions of Numerical Solutions in the Logistic Map

Abstract

In this paper, we introduce an iterative method with lower positions of true numerical solutions located in the real orbit in order to investigate the property of the logistic map. The basic structure of the logistic map is presented, which consists of the root gene position, the common gene position and the individual gene position. The ergodicity and randomness of the logistic map are dependent on the individual gene position. We find that the lower positions of the true numerical solutions in the real orbits have the property of a half-life.

1. Introduction

Chaotic systems have been widely studied in physics, mathematics, biology, information technology, and so on. When the real chaos is realized in the discretized one with the finite computational precision, the continuous chaotic systems may collapse in digital computers, due to the sensitivity of chaotic systems to initial conditions and control parameters. If the precision of a numerical solution in iterating chaotic maps is greater than the specified precision of the computer, the value of the numerical solution will be truncated for the next iteration.

The truncation errors of arithmetic and round-off errors in numerical solutions are inevitable. Computation errors in digital computers are introduced into iterations of chaotic maps at every discrete step. The discrete orbits with the finite precision are different from the theoretical ones after several iterations [1,2]. Therefore, the computational precision has close links with the ergodicity and other dynamical properties of chaotic systems. More unseen phenomena hidden in the continuous and discrete chaos deserve further exploration. There are many research reports of theoretical and experimental analysis for the precision-dependent behavior of chaotic systems [3–7]. The theory and approach are used to study the problem, which include the unstable periodic orbit in the attractor, the shadowing lemma, the number theory, the statistical analysis of probability theory and the combination of these.

Unstable periodic orbits in the chaotic attractor play a key role in determining dynamics [8]. An important problem is to estimate the effect of the finite precision on chaotic orbits. Since discretized chaotic iterations with finite precision are constrained in the limited discrete space, every chaotic orbit will eventually become periodic [9]. Grebogi [10] studied the effect of the finite precision on the average period of chaotic maps.

The shadowing lemma is used to create a bridge between the true trajectory and the pseudo-trajectory. To explain in what sense the noisy trajectory reflects the true dynamics of the actual system, the procedure of containment and refinement was developed to show that some true trajectories remain close to the noisy one for long times [11].

The approach of number theory was used to reveal the precision-dependent behavior [12]. A rational number is converted to the binary equivalent. The authors made the point that increasing precision does not certainly lead to improving accuracy. Although the computation errors are deterministic, it is difficult to deal with them during iterations of chaotic maps.

Moreover, statistical experiments are used to explore this issue [13–15]. Li [13] proposed a series of dynamical indicators for the 1D piecewise linear chaotic map, which were used to investigate computable and measurable indicators of the dynamical properties of the digital chaotic system.

The Lyapunov exponent is the main indicator of chaos in dynamical systems. The combination of statistical analysis, the attractor size and Lyapunov exponent is used to study the relationship among computation errors, the attractor and the dynamical expansion. Shi [16] found that the relation on the round-off error α, the attractor size γ and the maximum of local dynamical expansion β in a driven or coupled logistic map system satisfies αβ < γ. Therefore, computation errors play important roles in iterations of chaotic maps. However, computation errors make the discretized chaotic orbits deviate from the continuous ones in a complex and intractable manner.

The numerical solutions with computation errors might interfere with the nature of nonlinear dynamics. The approaches mentioned above exploit approximations, which derive from the upper positions of numerical solutions. The lower positions of the numerical solutions in the orbit are completely ignored due to the floating-point arithmetic of the computer. However, some natural characteristics of the logistic map actually hide in lower positions of true numerical solutions in the real chaotic orbits. Liu [17] reported the property of the logistic map with scalable precision.

In this paper, we introduce the iterative method with lower positions of numerical solutions to explore the precision-dependent behavior of chaotic maps. As we will see later, the method can more accurately reflect the basic structure of true numerical solutions in a real chaotic orbit of the logistic map. In comparison with the traditional existing approaches to analyze chaotic systems, the basic structure of the logistic map is independent of the computational errors in computers. To the best of our knowledge, the basic structure of the logistic map is reported for the first time. It can really reflect the property of real chaotic orbits of the logistic map. It is able to help us to deeply understand the property of chaos.

The rest of this paper is organized as follows. Section 2 describes the property of real chaotic orbits of the logistic map. Section 3 introduces the iterative method, which gets lower positions of true numerical solutions in real chaotic orbits by using the finite computation precision. Section 4 presents the basic structure of the logistic map and the natural characteristic of lower positions of true numerical solutions in the real chaotic orbits. The last section concludes the paper.

2. Real Orbits of the Logistic Map

The logistic map is defined as:

$$x_{n+1}=a{x}_n(1-x_n),n=0,1,2,\cdots$$

(1)

where x_n is a real number in the interval (0, 1), n acts as the n-th iteration and x₀ is an initial value of x_n. The control parameter a is a real number in the interval (3, 4).

Assume that a real number can be described as τ = r₀.r₁r₂ ⋯ r_i ⋯ r_n, r_i ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, i.e., $\tau ={\displaystyle {\sum }_{i=0}^n(r_i\times {10}^{-i})}$. When r_n ≠ 0, n is the precision of τ, r_n is called the last position of τ. When r₀ = 0 or r₀ = 3, τ ∈ (0, 1) or τ ∈ (3, 4), respectively.

A true numerical solution (TNS) in a real chaotic orbit of the logistic map is computed in absolute accuracy. A real chaotic orbit of the logistic map consists of the true numerical solutions. We take an example to show a real chaotic orbit of the logistic map.

When x₀ = 0.1 and a = 3.9, the true numerical solutions in a real orbit are listed as follows.

• x₁ = 0.351.
• x₂ = 0.8884161.
• x₃ = 0.386618439717081.
• x₄ = 0.9248640249724621386134396738121.
• x₅ = 0.271013185108376366200014086640124882498446354809446978470185001.
• x₆ = 0.7705036505625782420100288501800724673002582158090841219337408507214354984983318832966012404717170683366990906868712244550569961.
• x₇ = 0.689628322626042399346162417789631574331673792279087727295587249959700088953695903028503904227316732329007279116817939583585410360955275942688611619532554186417689465749874585911734826353859703733156498976808571405387443824495937854094564832442251733880681.

We can observe that the last two positions of the true numerical solutions have regularity, which is as follows.

$$\begin{array}{l}\begin{array}{r}{x}_1=0.351\\ x_2=0.\cdots \underset{¯}{61}{x}_3=0.\cdots 81x_4=0.\cdots 21x_5=0.\cdots 01\end{array}\\ x_6=0.\cdots \underset{¯}{61}{x}_7=0.\cdots 81\end{array}$$

Note that the last two positions of the true numerical solutions become periodic from x₂ to x₆. We can deduce that the last two positions of the true numerical solution x₉ in the real chaotic orbit are equal to 01.

The precision of the true numerical solution x₉ is equal to 1023. We provide the upper ten positions and the lower ten positions of the true numerical solution x₉. The middle positions of the true numerical solution x₉ are omitted.

The true numerical solution x₉ is equal to 0. $\underset{Upper10positions}{\underbrace{5379486456}}\cdots \underset{Lower10positions}{\underbrace{48652074\underset{¯}{01}}}$.

The real chaotic orbits of the logistic map are not periodic. The lower positions of true numerical solutions in the real orbits have regularity. An interesting question is: what is the characteristic of the lower positions of the true numerical solutions in real chaotic orbits?

The precision of the true numerical solutions of x₇, x₁₀ and x₂₀ is equal to 255, 2 047 and 2 097 151, respectively. The precision difference is close to 1000 times after 10 iterations.

When the values of the last position of x₀ and a belong in the integer set Z^d = {1, 2, 3, 4, 6, 7, 8, 9}, the relationship between the iteration and the precision satisfies:

$$L{x}_{n+1}=2\times L{x}_n+L_a$$

(2)

where L_xn and L_a are the precision of x_n and a, respectively. Equation (2) implies that the greater the iteration is, the larger the precision of the true numerical solutions in the real orbit will be.

Computing systems of computers perform the floating-point computation. A common standardized format for the floating-point computation is called the IEEE 754 format. It comes in three forms, all of which are very similar in procedure: single precision (32-bit), double precision (64-bit) and extended precision (80-bit).

With the double precision of the fraction significantly appearing in the memory format, the total precision is approximately 15 decimal digits. The principle of the floating-point arithmetic in computers is that the upper digit positions are stored in the memory.

On the one hand, when the precision of a true numerical solution exceeds the maximum precision with double precision in computers, the lower positions of the true numerical solution will be truncated. The lower positions of the numerical solution could be difficult to obtain by using the arithmetic operation of floating-point computation. Therefore, the floating-point arithmetic in computers cannot get the lower positions of true numerical solutions in the real orbits of the logistic map. This is the reason why the property of the lower positions of the numerical solutions in the orbit is completely ignored.

On the other hand, according to Equation (2), the precision of true numerical solutions in the real orbit could tremendously vary after several iterations. The finite computational precision has an influence on the dynamics of chaotic systems. The computational errors will gradually accumulate, since the precision in computers is limited. Furthermore, the true numerical solutions in a real chaotic orbit of the logistic map are obtained when the computing procedure does not include any computational error.

The problem is to obtain the lower positions of true numerical solutions in the real orbits by using the finite computational precision in computers.

3. Iterative Method with Lower Positions

We use an iterative method with lower positions of numerical solutions in order to overcome the limit of precision. Every step of the computing procedure does not include the accumulated round-off error. In other words, the logistic map is computing in absolute precision without computational errors.

Assume that we need to get the lower p_n positions starting from the last position of true numerical solutions in the real orbits, where p_n is a positive integer. In order to guarantee the absolute precision, the logistic map can be divided into two parts for computation. The iterative method with lower positions is described as following:

• Step 1: computing μ = x_n(1 − x_n) in absolute accuracy;
• Step 2: computing ξ = a μ in absolute accuracy;
• Step 3: if the total number of positions of ξ is less than or equal to p_n, all positions of the numerical solution will be returned for the next iteration; otherwise, the lower p_n positions starting from the last position are returned.

We take an example to explain the iterative method with lower positions of numerical solutions. Assume x₀ = 0.1 and a = 3.9; we need to get the lower 12 positions of true numerical solutions in the real orbit, i.e., p_n = 12.

The computing procedure is shown as follows.

$\widehat{x_1}=a{x}_0(1-x_0)=3.9\times 0.1\times (1-0.1)=0.351$. The total number of positions of $\widehat{x_1}$ is less than p_n. All positions will be returned for the next iteration.

Thus, $x_1^l=0.351$.

$\widehat{x_2}=a{x}_1^l(1-x_1^l)=3.9\times 0.351\times (1-0.351)=0.8884161$. $\widehat{x_2}$ has the same situation as $\widehat{x_1}$.

Thus, $x_2^l=0.8884161$.

$\widehat{x_3}=a{x}_2^l(1-x_2^l)=3.9\times 0.8884161\times (1-0.8884161)=0.386618439717081$. The totalbnumber of positions of $\widehat{x_3}$ is greater than p_n. The lower twelve positions of $\widehat{x_3}$ are returned.

Thus, $x_3^l=0.618439717081$.

$\widehat{x_4}=a{x}_3^l(1-x_3^l)=3.9\times 0.618439717081\times (1-0.618439717081)=0.9202909303293134396738121$. $\widehat{x_4}$ has the same situation as $\widehat{x_3}$.

Thus, $x_4^l=0.134396738121$.

The true numerical solution x₄ in the real orbit is equal to $0.9248640249724621386\underset{Lower12positions}{\underbrace{134396738121}}$.

The iterative method can get the lower p_n positions of the corresponding true numerical solution of x₄.

$\widehat{x_5}=a{x}_4^l(1-x_4^l)=3.9\times 0.134396738121\times (1-0.134396738121)=0.4537035941233978470185001$.

Thus, $x_5^l=0.978470185001$.

It is clear that $x_5^l$ gets the lower twelve positions of the corresponding true numerical solution of x₅. $\widehat{x_6}=a{x}_5^l(1-x_5^l)=0.0821585000539244550569961$.

Thus, $x_6^l=0.244550569961$.

$\widehat{x_7}=a{x}_6^l(1-x_6^l)=0.7205077959017251733880681$.

Thus, $x_7^l=0.251733880681$.

It is obvious that the iterative method can get the lower p_n positions of the corresponding true numerical solution in the real orbit of the logistic map. The precision of the true numerical solution of x₇ is equal to 255. In the computing procedure, we use the precision of 30 decimal digits. The iterative method with lower positions exploits the finite computational precision and solves the difficulty of getting the lower positions of the corresponding true numerical solutions with large precision.

4. Properties of the Logistic Map

Each orbit of the logistic map consists of x₁, x₂,⋯, x_i,⋯, x_n. When x_i is computed in absolute accuracy, real chaotic orbits are nonperiodic.

The iterative methods for observing orbits with computational errors are difficult with respect to getting the real chaotic orbits, because the accumulated round-off errors at every step of iterations could crumple the hidden regularity of the lower positions of true numerical solutions in the real chaotic orbit.

We use the lower p_n positions of the true numerical solutions instead of x_i in order to investigate the property of the lower positions of the logistic map.

We compute the logistic map by using the iterative method with the lower p_n positions. This means that a chaotic orbit consists of the lower p_n positions of true numerical solutions. For instance, assume that x₀ = 0.1, a = 3.9 and p_n = 2; the orbit is shown as follows.

$$\begin{array}{r}{x}_1=0.51\\ \begin{array}{l}{x}_2=\underset{¯}{0.61}{x}_3=0.81x_4=0.21x_5=0.01\\ x_6=\underset{¯}{0.61}{x}_7=0.81\cdots \end{array}\end{array}$$

It is clear that the chaotic orbit becomes periodic. The length of period is equal to four. This implies that the lower two positions of x_i in the real orbit belong in the set {61, 81, 21, 01}. Furthermore, we can precisely compute every TNS with lower positions. For example, the lower two positions of the TNS of x₁₀₂ are equal to 61.

The orbit includes two parts:

• Part 1: x₁, x₂,⋯, x_d,
• Part 2: x_d+1, x_d+2,⋯,x_d+c.

The Part 1 and Part 2 are called the delay part and cycle part, respectively. c is the length of the cycle part. We focus on the cycle part of orbits in the logistic map.

We increase the value of p_n in order to further explore the property of lower positions. Assume that x₀ = 0.1, a = 3.9 and p_n = 3; the lower p_n positions of true numerical solutions are listed as follows.

$$\begin{array}{r}{x}_1=0.351\\ \begin{array}{rrrr}{x}_2=0.\underset{¯}{161}& x_3=0.081& x_4=0.121& x_5=0.001\\ x_6=0.961& x_7=0.681& x_8=0.321& x_9=0.401\\ x_{10}=0.761& x_{11}=0.281& x_{12}=0.521& x_{13}=0.801\\ x_{14}=0.561& x_{15}=0.881& x_{16}=0.721& x_{17}=0.201\\ x_{18}=0.361& x_{19}=0.481& x_{20}=0.921& x_{21}=0.601\\ x_{22}=0.\underset{¯}{161}& x_{23}=0.081& x_{24}=0.121& x_{25}=0.001\end{array}\end{array}$$

The length of the period is equal to 20 for x₀ = 0.1, a = 3.9 and p_n = 3. Figure 1 shows the structure of the lower four positions of TNSs for x₀ = 0.1 and a = 3.9.

The position located in the last positions of TNSs is called the root gene position, when the value of the root gene positions in the cycle part is invariable. For example, when x₀ = 0.1, a = 3.9 and p_n = 4, the root gene position is the last position of TNSs. In Figure 1, the root gene position is located in the center. Its value is equal to one.

When the period of the lower positions is equal to four, the positions are called the common gene position (CGP). For example, in Figure 1, the last two positions are the common gene positions, i.e., “21”, “81”, “01” and “61”. Others are called the individual gene position (IGP).

In Figure 1, every individual gene position varies in the set Z^odd = {1, 3, 5, 7, 9} and the set Z^even = {0, 2, 4, 6, 8}. For example, the individual gene position with the common gene positions “81” is “081”, “681”, “281”, “881” and “481”, which belong in the set Z^even. The individual gene position with the common gene positions “21” is “121”, “321”, “521”, “721” and “921”, which belong in the set Z^odd.

Note that every iteration with the individual gene position moves from one common gene position to another. For example, for x₀ = 0.1, a = 3.9 and p_n = 3, x₁₇ = 0.201 with the individual gene position “2” and the common gene positions “01” moves to x₁₈ = 0.361 with the common gene positions “61”. In Figure 1, x₉ = 0.7401 with the individual gene positions “74” and the common gene positions “01” moves to x₁₀ = 0.2761 with the common gene positions “61”.

We take another example to investigate the property of lower positions. Assume that x₀ = 0.3 and a = 3.4; the true numerical solutions are listed as follows.

• x₁ = 0.714.
• x₂ = 0.6942936.
• x₃ = 0.721649989796736.
• x₄ = 0 682⋯5727374336.
• x₅ = 0.736⋯1171009536.
• x₆ = 0.660⋯4300199936.
• x₇ = 0.762⋯4470260736.

We provide the upper three positions and the lower ten positions of the true numerical solutions from x₄ to x₇. Some middle positions are omitted, because the situation has no effect on observing lower positions of the true numerical solutions. As seen above, the lower three positions of TNSs are listed as follows.

$$\begin{array}{r}{x}_1=0.714\\ \begin{array}{l}{x}_2=\underset{¯}{0.936}{x}_3=0.736x_4=0.336x_5=0.536\\ x_6=\underset{¯}{0.936}{x}_7=0.736\cdots \end{array}\end{array}$$

Note that the regularity occurs in the lower three positions. The length of period is equal to four. For x₀ = 0.3 and a = 3.4, the root gene positions are the last two positions, i.e., “36”. The common gene positions are “936”, “736”, “336” and “536”.

We increase the value of lower p_n positions. Assume that x₀ = 0.3, a = 3.4 and p_n = 4. The lower four positions of TNSs are listed as follows.

$$\begin{array}{r}{x}_1=0.714x_2=0.2936\\ \begin{array}{rrrr}{x}_3=0.\underset{¯}{6736}& x_4=0.4336& x_5=0.9536& x_6=0.9936\\ x_7=0.0736& x_8=0.2336& x_9=0.5536& x_{10}=0.1936\\ x_{11}=0.4736& x_{12}=0.0336& x_{13}=0.1536& x_{14}=0.3936\\ x_{15}=0.8736& x_{16}=0.8336& x_{17}=0.7536& x_{18}=0.5936\\ x_{19}=0.2736& x_{20}=0.6336& x_{21}=0.3536& x_{22}=0.7936\\ x_{23}=0.\underset{¯}{6736}& x_{24}=0.4336& x_{25}=0.9536& x_{26}=0.9936\end{array}\end{array}$$

As seen above, the individual gene position with the common gene positions “736” is “6736”, “0736”, “4736”, “8736” and “2736”, which belong in the set Z^even. The individual gene position with the common gene positions “936” is “1936”, “3936”, “5936”, “7936” and “9936”, which belong in the set Z^odd.

Assume that c_pn=η is the length of the period with the lower p_n positions, where p_n is equal to η. From

Table 1. The length of period in the iterative method with lower positions for the initial conditions of x₀ = 0.1 and a = 3.9.

pn	Length of period	${\mathrm{c}}_{{\mathrm{p}}_{\mathrm{n}}=\eta +1}/{\mathrm{c}}_{{\mathrm{p}}_{\mathrm{n}}=\eta }$
1	1
2	4	${\mathrm{c}}_{p_n=2}/{\mathrm{c}}_{p_n=1}=4$
3	20	${\mathrm{c}}_{p_n=3}/{\mathrm{c}}_{p_n=2}=5$
4	100	${\mathrm{c}}_{p_n=4}/{\mathrm{c}}_{p_n=3}=5$
5	500	${\mathrm{c}}_{p_n=5}/{\mathrm{c}}_{p_n=4}=5$
6	2 500	${\mathrm{c}}_{p_n=6}/{\mathrm{c}}_{p_n=5}=5$
7	12 500	${\mathrm{c}}_{p_n=7}/{\mathrm{c}}_{p_n=6}=5$
8	62 500	${\mathrm{c}}_{p_n=8}/{\mathrm{c}}_{p_n=7}=5$
9	312 500	${\mathrm{c}}_{p_n=9}/{\mathrm{c}}_{p_n=8}=5$
10	1 562 500	${\mathrm{c}}_{p_n=10}/{\mathrm{c}}_{p_n=9}=5$
11	7 812 500	${\mathrm{c}}_{p_n=11}/{\mathrm{c}}_{p_n=10}=5$
12	39 062 500	${\mathrm{c}}_{p_n=12}/{\mathrm{c}}_{p_n=11}=5$
13	195 312 500	${\mathrm{c}}_{p_n=13}/{\mathrm{c}}_{p_n=12}=5$
14	976 562 500	${\mathrm{c}}_{p_n=14}/{\mathrm{c}}_{p_n=13}=5$
15	4 882 812 500	${\mathrm{c}}_{p_n=15}/{\mathrm{c}}_{p_n=14}=5$

, we can see that when the length of the period is greater than four and the individual gene position increases by one digit position, the new length of period will be five-times as long as the original length of period.

In order to further verify the regularity, we take values of the control parameter of a ∈ {3.1, 3.2, 3.3, 3.4, 3.6, 3.7, 3.8, 3.9}, and initial values of x₀ ∈ {0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, 0.9}.

When a is specified, the value of x₁ is the same for the initial values of x₀ = 0.1 and x₀ = 0.9, x₀ = 0.2 and x₀ = 0.8, x₀ = 0.3 and x₀ = 0.7, x₀ = 0.4 and x₀ = 0.6, respectively. Therefore, we give, for example, (x₀ = 0.1, a) and omit (x₀ = 0.9, a).

The length of the period is summarized in

Table 2. The length of the period in the iterative method with lower positions for the precision 1 of the control parameter and initial value x₀. c¹, c² and c³ represent the length of period for initial conditions in Group 1, Group 2 and Group 3, respectively.

pn	c1	c2	c3
1	1	1	1
2	1	1	4
3	4	1	20
4	20	4	100
5	100	20	500
6	500	100	2 500
7	2 500	500	12 500
8	12 500	2 500	62 500
9	62 500	12 500	312 500
10	312 500	62 500	1 562 500
11	1 562 500	312 500	7 812 500
12	7 812 500	1 562 500	39 062 500
13	39 062 500	7 812 500	195 312 500
14	195 312 500	39 062 500	976 562 500
15	976 562 500	195 312 500	4 882 812 500

. Experimental results by the initial conditions are divided into three groups in the form of (x₀, a):

• Group 1 = {(0.1, 3.1), (0.4, 3.2), (0.1, 3.3), (0.2, 3.3), (0.3, 3.4), (0.3, 3.8)};
• Group 2 = {(0.2, 3.1), (0.3, 3.7)};
• Group 3 = {(0.1, 3.2), (0.1, 3.4), (0.1, 3.6), (0.1, 3.7), (0.1, 3.8), (0.2, 3.2), (0.2, 3.4), (0.2, 3.6), (0.2, 3.7), (0.2, 3.8), (0.2, 3.9), (0.3, 3.1), (0.3, 3.2), (0.3, 3.3), (0.3, 3.6), (0.3, 3.9), (0.4, 3.1), (0.4, 3.3), (0.4, 3.4), (0.4, 3.6), (0.4, 3.7), (0.4, 3.8), (0.4, 3.9)}.

For x₀ = 0.2 and a = 3.1 in Group 2, the lower four positions of TNSs are listed as follows.

$$\begin{array}{r}{x}_1=0.496\\ \begin{array}{l}{x}_2=\underset{¯}{0.9504}{x}_3=0.3504x_4=0.1504x_5=0.7504\\ x_6=\underset{¯}{0.9504}{x}_7=0.3504\cdots \end{array}\end{array}$$

For x₀ = 0.2, a = 3.1 and p_n = 4, the root gene positions are the last three positions, i.e., “504”. When the length of the period is equal to four, it is called the critical point of the period.

From Table 2, the situation is similar in that the new length of the period is five-times as long as the original length of the period when the individual gene position increases by one digit position. This implies that the length of the period will enlarge for the same initial conditions and control parameter when the individual gene positions increase.

When lower p_n positions are specified, the less the number of the root gene position is, the larger the length of period will be. For example, assume that the lower positions are set to four. For x₀ = 0.2, a = 3.1 and p_n = 4 in Group 2, the length of the period is equal to four. The root gene position occupies three digital positions, i.e., “504”. For x₀ = 0.3, a = 3.4 and p_n = 4 in Group 1, the length of the period is equal to 20. The root gene position occupies two digital positions, i.e., “36”. For x₀ = 0.1, a = 3.9 and p_n = 4, the length of the period is equal to 100. The root gene position occupies a digital position, i.e., “1”.

For a = 3.9, x₀ = 0.2 and p_n = 4, the lower four positions of the true numerical solutions in the real orbit become the period starting from x₂ to x₁₀₂. The length of period is equal to 100. The individual gene positions with the common gene positions are shown in

Table 3. The individual gene positions with the same common gene positions for x₀ = 0.2, a = 3.9 and p_n = 4. CGP, common gene position.

CGP =“36”	CGP =“56”	CGP =“96”	CGP =“76”
x2 = 0.0336	x3 = 0.7056	x4 = 0.1696	x5 = 0.9776
x6 = 0.3136	x7 = 0.4656	x8 = 0.4896	x9 = 0.8176
x10 = 0.7936	x11 = 0.6256	x12 = 0.6096	x13 = 0.2576
x14 = 0.4736	x15 = 0.1856	x16 = 0.5296	x17 = 0.2976
x18 = 0.3536	x19 = 0.1456	x20 = 0.2496	x21 = 0.9376
x22 = 0.4336	x23 = 0.5056	x24 = 0.7696	x25 = 0.1776
x26 = 0.7136	x27 = 0.2656	x28 = 0.0896	x29 = 0.0176
x30 = 0.1936	x31 = 0.4256	x32 = 0.2096	x33 = 0.4576
x34 = 0.8736	x35 = 0.9856	x36 = 0.1296	x37 = 0.4976
x38 = 0.7536	x39 = 0.9456	x40 = 0.8496	x41 = 0.1376
x42 = 0.8336	x43 = 0.3056	x44 = 0.3696	x45 = 0.3776
x46 = 0.1136	x47 = 0.0656	x48 = 0.6896	x49 = 0.2176
x50 = 0.5936	x51 = 0.2256	x52 = 0.8096	x53 = 0.6576
x54 = 0.2736	x55 = 0.7856	x56 = 0.7296	x57 = 0.6976
x58 = 0.1536	x59 = 0.7456	x60 = 0.4496	x61 = 0.3376
x62 = 0.2336	x63 = 0.1056	x64 = 0.9696	x65 = 0.5776
x66 = 0.5136	x67 = 0.8656	x68 = 0.2896	x69 = 0.4176
x70 = 0.9936	x71 = 0.0256	x72 = 0.4096	x73 = 0.8576
x74 = 0.6736	x75 = 0.5856	x76 = 0.3296	x77 = 0.8976
x78 = 0.5536	x79 = 0.5456	x80 = 0.0496	x81 = 0.5376
x82 = 0.6336	x83 = 0.9056	x84 = 0.5696	x85 = 0.7776
x86 = 0.9136	x87 = 0.6656	x88 = 0.8896	x89 = 0.6176
x90 = 0.3936	x91 = 0.8256	x92 = 0.0096	x93 = 0.0576
x94 = 0.0736	x95 = 0.3856	x96 = 0.9296	x97 = 0.0976
x98 = 0.9536	x99 = 0.3456	x100 = 0.6496	x101 = 0.7376
x102 = 0.0336	x103 = 0.7056	x104 = 0.1696	x105 = 0.9776

.

The numerical solution represents 0.r₁r₂r₃r₄ for a = 3.9, x₀ = 0.2 and p_n = 4. The root gene position is the digit position r₄, which is equal to six. The combined digit positions of r₃ and r₄ are the common gene positions, which are equal to “36”, “56”, “96” and “76”. The digit positions r₁ and r₂ stand for the first position and the second position of the individual gene positions.

The randomness of IGPs of the true numerical solutions in the real orbit is shown in

Table 4. Stochastic individual gene positions of the true numerical solutions in the real orbit of the logistic map.

r1\r2	0	1	2	3	4	5	6	7	8	9
0	x92	x29	x71	x2	x80	x93	x47	x94	x28	x97
1	x63	x46	x36	x41	x19	x58	x4	x25	x15	x30
2	x32	x49	x51	x62	x20	x13	x27	x54	x68	x17
3	x43	x6	x76	x61	x99	x18	x44	x45	x95	x90
4	x72	x69	x31	x22	x60	x33	x7	x14	x8	x37
5	x23	x66	x16	x81	x79	x78	x84	x65	x75	x50
6	x12	x89	x11	x82	x100	x53	x87	x74	x48	x57
7	x3	x26	x56	x101	x59	x38	x24	x85	x55	x10
8	x52	x9	x91	x42	x40	x73	x67	x34	x88	x77
9	x83	x86	x96	x21	x39	x98	x64	x5	x35	x70

. The permutation and combination values of r₁ and r₂ range from 00 to 99. It is obvious that the combined individual gene positions of r₁ and r₂ are different from each other and impinge on all points in the given space. In fact, the property of the logistic map derives from the individual gene positions. In other words, the individual gene positions exhibit the ergodicity and randomness of the logistic map.

Table 5. The property of the half-life of the individual gene positions. c_max represents the maximum length of period. c_IGP represents the length of the period with the same common gene position. IGP, individual gene position.

Number of IGP	cmax	cIGP	cIGP/cmax
1	101	5	50.0%
2	102	25	25.0%
3	103	125	12.5%
4	104	625	6.25%
5	105	3 125	3.125%
6	106	15 625	1.5625%
7	107	78 125	0.78125%
8	108	390 625	0.390625%
⋯	⋯	⋯	⋯
n	10n	5n	1/2n

shows the property of the half-life of the logistic map. For example, the first column of Table 4 shows the length of the period with the common gene position “36” equal to 25. The same situation happens in the second, third and fourth columns of Table 4. From Table 5, we can see that when the individual gene positions increase, the length of the period with the same common gene positions has the property of the half-life. The reason is that each digital position of the individual gene positions varies in the sets Z^odd and Z^even.

The property of the half-life is the nature of the logistic map, which is independent of computational errors in computers. Therefore, the numerical solutions with computational errors could interfere with the nature of nonlinear dynamics.

5. Discussion and Conclusions

In summary, we introduce the iterative method with lower positions of numerical solutions in order to explore the property of the real orbits in the logistic map. The iterative method can observe the property of the lower positions of true numerical solutions in the real chaotic orbit of the logistic map.

Every real chaotic orbit includes the same root gene position, which is constant and invariable in iterations of the logistic map. The root gene position represents the basic characteristic of a real chaotic orbit. It seems that the gene of a parent is passed on to their offspring. In the evolution, they could possess the same characteristic.

When the length of the period of lower p_n positions is larger than the critical point of the period, the individual gene position will appear. The ergodicity and randomness of the logistic map depend on the individual gene positions. The basic structure of the logistic map is presented, which reflects the nature of the logistic map.

The interesting finding is that the lower positions of the true numerical solutions in the real orbits of the logistic map exhibit the property of the half-life. The property of the half-life is independent of computation errors. In other words, it is the natural characteristic of the logistic map.

These observations can help us to deeply understand chaos. The greatest mystery in this topic still remains open. It may be worthwhile to reveal whether other chaotic systems have this property.