# Using the Logistic Map as Compared to the Cubic Map to Show the Convergence and the Relaxation of the Period–1 Fixed Point

Authors: Patrick Akwasi Anamuah Mensah; William Obeng-Denteh; Ibrahim Issaka; Kwasi Baah Gyamfi; Joshua Kiddy K. Asamoah
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this paper, we employ the logistic map and the cubic map to locate the relaxation and the convergence to the periodic fixed point of a system, specifically, the period—1 fixed point. The study has shown that the period—1 fixed point of a logistic map as a recurrence has its convergence at a transcritical bifurcation having its power-law fit with exponentβ=−1 when α=1 and μ=0. The cubic map shows its convergence to the fixed point at a pitchfork bifurcation decaying at a power law with exponent β=−1/2α=1 and μ=0. However, the system shows their relaxation time at the same power law with exponents and z=−1.


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## Body

## 1. Introduction

### 1.1. Preliminary

In dynamical systems, the most frequent map or function that has been extensively studied is the one-dimensional logistic map [1]. In many studies, the effects of the control parameters when changed shows a change in behavior (asymptotic behavior) of the orbits or trajectories of the map. The transition process of the logistic map shows the cascades of period-doubling bifurcation leading to chaos [2, 3]. Reference [4] asserted that the structural changes in the trajectory or orbit of a given system are termed bifurcation and it was first used in a work by Henri Poincare. In a dynamical system, bifurcation appears when there is a change in parameters that affects the structural system [5]. Bifurcation is very important in dynamics since the structural change of a system in its behavior or nature is core in its studies. The study of stability (attractors) and instability (repeller) happens when there is bifurcation [6]. In the dynamical system, the bifurcation diagram helps one to understand the behavior of the system either in its fixed points, stability, periodicity, etc., for instance, when the parameter in the system is varied the system changes affecting its stability. The system may be in equilibrium states when the bifurcation of the system is in one dimension, hence local bifurcation. Local bifurcation as stated by [7, 8] occurs when the points in the neighborhood are in equilibrium, and there are three main types/forms of bifurcation are Saddle-node, pitchfork, and transcritical see [6, 9]. Within the neighborhood [0, 1] of a system, the bifurcation of the map is dependent on the parameter as we keep on varying and iterating it through/within the neighborhood, see [10, 11]. When the parameter of the system passes its critical value, the type of bifurcation is flipped as a result of a loss of stability of the periodic orbit [10]. The flip bifurcation as cited by [9, 11] is locally supercritical when the period of the parameter value is double with stable periodic orbits. It is locally subcritical when the periodic orbit is unstable with twice the period of the parameter value as the critical values show a new one.A point exhibits some sort of recurrence behavior when the dynamical system returns the point to itself or to a neighborhood of itself, in a particular way. The simplest example of recurrent dynamical behavior is the fixed point. In a dynamical system, under iteration, a fixed point not only comes back, it never goes away [12].The study of logistic function as a quadratic map shows that it does not stay fixed in its transformation as the control parameterλ keeps alternating or changing [13, 14]. The logistic function is a nonlinear system which is a type of difference equation and a quadratic in nature. It is given as follows.Xn+1=λxn−λxn2, where n=0,1,2,…, gives the discrete time and λ∈1.4 is the control parameter and X∈0,1. Also, the study of the cubic map is a difference equation whose nonlinearity is cubic and it is defined by; Let f:X⟶X be a continuous map defined as follows. Xn+1=fxn=αxn−xn3 for all α∈1,4 and X∈0,1 see [15, 16].The main goal of this paper is to investigate and compare the relaxation and convergence of the period–1 fixed point of the logistic map and the cubic map. In this study, we will consider the convergence of the period-1 fixed point which is the simplest example of periodic-like recurrent and finally look at the relaxation of both the logistic map and the cubic map as they go through the iteration process.


## 1.1. Preliminary

In dynamical systems, the most frequent map or function that has been extensively studied is the one-dimensional logistic map [1]. In many studies, the effects of the control parameters when changed shows a change in behavior (asymptotic behavior) of the orbits or trajectories of the map. The transition process of the logistic map shows the cascades of period-doubling bifurcation leading to chaos [2, 3]. Reference [4] asserted that the structural changes in the trajectory or orbit of a given system are termed bifurcation and it was first used in a work by Henri Poincare. In a dynamical system, bifurcation appears when there is a change in parameters that affects the structural system [5]. Bifurcation is very important in dynamics since the structural change of a system in its behavior or nature is core in its studies. The study of stability (attractors) and instability (repeller) happens when there is bifurcation [6]. In the dynamical system, the bifurcation diagram helps one to understand the behavior of the system either in its fixed points, stability, periodicity, etc., for instance, when the parameter in the system is varied the system changes affecting its stability. The system may be in equilibrium states when the bifurcation of the system is in one dimension, hence local bifurcation. Local bifurcation as stated by [7, 8] occurs when the points in the neighborhood are in equilibrium, and there are three main types/forms of bifurcation are Saddle-node, pitchfork, and transcritical see [6, 9]. Within the neighborhood [0, 1] of a system, the bifurcation of the map is dependent on the parameter as we keep on varying and iterating it through/within the neighborhood, see [10, 11]. When the parameter of the system passes its critical value, the type of bifurcation is flipped as a result of a loss of stability of the periodic orbit [10]. The flip bifurcation as cited by [9, 11] is locally supercritical when the period of the parameter value is double with stable periodic orbits. It is locally subcritical when the periodic orbit is unstable with twice the period of the parameter value as the critical values show a new one.A point exhibits some sort of recurrence behavior when the dynamical system returns the point to itself or to a neighborhood of itself, in a particular way. The simplest example of recurrent dynamical behavior is the fixed point. In a dynamical system, under iteration, a fixed point not only comes back, it never goes away [12].The study of logistic function as a quadratic map shows that it does not stay fixed in its transformation as the control parameterλ keeps alternating or changing [13, 14]. The logistic function is a nonlinear system which is a type of difference equation and a quadratic in nature. It is given as follows.Xn+1=λxn−λxn2, where n=0,1,2,…, gives the discrete time and λ∈1.4 is the control parameter and X∈0,1. Also, the study of the cubic map is a difference equation whose nonlinearity is cubic and it is defined by; Let f:X⟶X be a continuous map defined as follows. Xn+1=fxn=αxn−xn3 for all α∈1,4 and X∈0,1 see [15, 16].The main goal of this paper is to investigate and compare the relaxation and convergence of the period–1 fixed point of the logistic map and the cubic map. In this study, we will consider the convergence of the period-1 fixed point which is the simplest example of periodic-like recurrent and finally look at the relaxation of both the logistic map and the cubic map as they go through the iteration process.


## 2. Main Work and Results

Under this section, we will investigate the convergence and the relaxation using the logistic map and the cubic map.


### 2.1. The Convergence of Period-1 Fixed Point Using the Logistic Function

#### 2.1.1. Logistic Map

The logistic map is defined as follows:(1)Xn+1=αxn−xn2,(2)Letfxn=Xn+1,Thenfxn=αxn−xn2,(3)iffxn=xn.For the solution of the map, equations (1) and (3).(4)Thenαxn−xn2=xn.By solving them algebraically, the solutions of the map are(5)xn=0,xn=α−1α.For the stability of the system.(a)
Letα=1 implies x0=1−1/1=0 as the initial.Then,limx0⟶0fx=limx0⟶0xn−xn2,n=0,1,2,…Table1 shows the Iteration of lim fx with α=1 and. x0=0.(b)
Letα=1 implies x0=0.1 as the initial.Then,limx0⟶0fx=limx0⟶0xn−xn2,n=0,1,2,…Table2 shows the Iteration of lim fx with α=1 and x0=0.1.Table 1
Iteration of limfx with α=1 and x0=0.
n01234xn00000Table 2
Iteration of limfx with α=1 and x0=0.1.
n01234xn0100000.090000.0819000.0751920.069538Figure1 shows the linear stability graph of the logistic map when α=1, x0=0and0.1.Figure 1
Linear stability graph of the logistic map whenα=1, x0=0and0.1.An example/type of a bifurcation is seen when the control parameterα=1 both (fixed points) solutions of the system meets at x=0 and alters their stableness [17]. The structural changes of the system at α=1 is where transcritical bifurcation occurs and the stability of the fixed point occurs there.Figure2 shows a subcritical bifurcation of a logistic map when the parameter is exactly 1. The vertical lines in both Figures 2(a) and 2(b) show that for the logistic map transcritical bifurcation occurs at α=1 and in the dynamical system it is the most common one. Figure 2(c) shows the convergence of period-1 fixed point having transcritical bifurcation when α=1 and period doubling starting at α=3 and beyond.Figure 2
Bifurcation diagram of the logistic mapα=1 with the initial value 0and0.1.
(a)(b)(c)Figure3 shows that at α=±1, it is very difficult to determine the linear stability of the period–1 point (fixed point) since it is very marginal. This confirms work in [17, 18], that fixed point is marginal at α=1 and α=−1 and its linear stability cannot be determined.Figure 3
The cobweb diagram of the logistic map.


### 2.2. The Relaxations of the Fixed Point Using the Logistic Map

Considering two basic hypotheses,(1)
Ifμ=0 then x decayed algebraically so that (6)xn,μ=0∝nβ,whereβ is a critical exponent and depends on the type of bifurcation.(2)
Atμ≠0, the orbit is at equilibrium exponentially through relaxation such that(7)xn,μ∝e−n/r,whereτ∝μz is the relaxation time and z is a critical exponent.Now, we look at the theoretical argumentation on the characterization of the logistic function/map, for the transcritical bifurcation.Letα=1, equation (1) can be written as follows:(8)dxdn≅xn+1−xn=xn+1−xnn+1−n.Then, the approximation of equation (8) is(9)dxdn=−x2.Taking a very close/nearby value to the point only make the limit ofxn of equation (9) very valid.(10)Then,dx−x2=dn.Taking the integration of equation (10) on both sides.(11)−∫x0xdx−x2=∫0ndn,⇒−−1xx0x=n0n,⇒1x−1x0=n,x=1n+1/x0.Asn increases to the condition n>1/x0,we obtain the following equation:(12)xn∝1n,xn∝n−1.By comparingxn∝n−1 to xn,μ=0∝nβ. The critical exponent at α=1 is. β=−1when. α>1Similarly, we subtractxn from both sides of equation (1) to obtain the following equation:(13)dxdn≅xn+1−xn=xn+1−xnn+1−n,(14)Then,dxdn=xα−1−αx2.Taking anyx value that is very close to the fixed point, equation (14) becomes;(15)dxdn=μx,(16)⇒dxx=μdn.Taking the Integral of equation (16) on both sides;(17)∫x0xdxx=μ∫0ndn,lnxx0x=μn0n,lnx−lnx0=μn−μ0,lnxx0=μn,xx0=eμn,xn=x0eμn.Comparingxn,μ∝e−n/r to xn=x0eμn. The critical exponent z=−1.


### 2.3. Numerical Simulation of the Logistic Map

The numerical simulation of the logistic map as shown in Figure4 indicates that in Figure 4(a) at μ=0 the convergence to the fixed point where the power law fit gives β=−0.99997≅−1 and Figure 4(b) shows a slope of z=−0.994≅−1 at μ≠0.Figure 4
The convergence to the period fixed point using the logistic mapμ=0andμ≠0.
(a)(b)Clearly, the results for the arguments and the numerical simulations in Figure4 are the same that β=−0.99997≅−1 and z=−0.994≅−1.Finally, for a logistic map the growth or decay of a perturbation of its periodic orbit is an exponential with the same rate for the periodic-1 point (fixed point) of other maps especially, the Poincare map, see [18].


### 2.4. The Convergence of Period-1 Fixed Point Using the Cubic Function

#### 2.4.1. Cubic Map: the Cubic Map is a Difference Equation Whose Nonlinearity is Cubic

Letf:X⟶X be a continuous map defined as follows: (18)Xn+1=fxn=αxn−xn3,where α∈1,4 and X∈0,1.For the fixed point of the map, we equations and (3) and (18).(19)Letαxn−xn3=xn,⟹xn3−αxn+xn=0,xnxn2−α+1=0,Thenxn=0.Also,xn2−α+1=0,⇒xn2=α−1.Then,xn=α−1,xn=−α−1.Hence,xn=0, xn=α−1 and xn=−α−1 are the solutions of the map.


#### 2.4.2. The Stability of the Cubic Map

(20)iffxn=αxn−xn3.We take the derivative of the map.(21)That isf′xn=α−3xn2.(1)
For attractor/stable condition,(22)α−3xn2<1.Whenxn=0,(23)α−302<1.⇒α<1 or −1<α<1 is where the map is stable or attracting.Whenxn=α−1,(24)⇒α−3α−12<1,⇒α−3α−1<1,−2α+3<1.Solving this absolute systematically, we obtained1<α<2 as where the map is attracting or stable.(2)
For repulsive/unstable condition,(25)α−3xn2>1,α−3α−12>1,⇒α−3α−1>1,−2α+3>1.Then,(26)−2α+3>1,−2α>1−3,−2α>−2,∴α<1.Also,(27)2α+3>1,2α>1+3,2α>4,∴α>2.Hence, the cubic map is repulsive when.α<1orα>2.Figure5 shows the bifurcation diagram of the cubic map and that within the interval of the parameter [2, 3]. The system exhibits a pitchfork bifurcation right at α=1 and beyond that the system experiences period doubling bifurcation. At α=1, there is a loss of stability but regain it back as keeps oscillating.Figure 5
The bifurcation diagram of the cubic map whenRc=α=1.


### 2.5. The Relaxations of the Fixed Point Using the Cubic Map

Now, we look at the theoretical argumentation on the characterization of the cubic function/map, for the pitchfork bifurcation.When.α=1Equation (18) can be written as follows:(28)dxdn≅xn+1−xn=xn+1−xnn+1−n.Then, the approximation of equation (28) is(29)dxdn=−x3.Taking a very close/nearby value to the point only make the limit ofxn of equation (29) very valid.(30)Then,dx−x3=dn.Taking the integration of equation (30) on both sides,(31)−∫x0xdx−x3=∫0ndn,⇒−−12x2x0x=n0n,⇒12x2−12x02=n,x2=12n+1/x02,x=12n+1/x02.Asn increases to the condition 2n>1/x02,(32)x≅12n,x≅12n,x≅12n−1/2.By comparing equations (6)–(32), the critical exponent at α=1 is β=−1/2,whenα>1,Similarly, we subtractxn from both sides of equation (18) to obtain the following equation:(33)dxdn≅xn+1−xn=xn+1−xnn+1−n.Then, by Taking any value that is very close to the fixed point, equation (33) becomes;(34)dxdn=μx−x3.Similarly, taking the integral of equation (34) on both sides and going through the same procedure as used for the logistic map, equation (34) is end up like that of equation (9). Then, by comparing, the critical exponent z=−1.


### 2.6. Numerical Simulation of the Cubic Map

The numerical simulation of the logistic map as shown in Figure6 indicates that in Figure 6(a) at μ=0 the convergence to the fixed point where the power law fit gives β=−0.497≅−1/2 and Figure 6(b) shows a slope of z=−0.9927≅−1 at μ≠0.Figure 6
The convergence to the period fixed point using the cubic mapμ=0andμ≠0.
(a)(b)Clearly, the results for the arguments and the numerical simulations in Figure6 are the same that β=−0.497≅−1/2 and z=−0.9927≅−1.In summary, there is an algebraic decay of the system whenμ=0 but when μ≠0 the decay is not algebraic in nature but rather characterized by a relaxation time [19]. The system is unstable when the parameter of the logistic map and cubic map is α=1 with the initial value x=0, there is a relaxation time that depends on the μ in showing the stability of the bifurcation.


## 2.1. The Convergence of Period-1 Fixed Point Using the Logistic Function

### 2.1.1. Logistic Map

The logistic map is defined as follows:(1)Xn+1=αxn−xn2,(2)Letfxn=Xn+1,Thenfxn=αxn−xn2,(3)iffxn=xn.For the solution of the map, equations (1) and (3).(4)Thenαxn−xn2=xn.By solving them algebraically, the solutions of the map are(5)xn=0,xn=α−1α.For the stability of the system.(a)
Letα=1 implies x0=1−1/1=0 as the initial.Then,limx0⟶0fx=limx0⟶0xn−xn2,n=0,1,2,…Table1 shows the Iteration of lim fx with α=1 and. x0=0.(b)
Letα=1 implies x0=0.1 as the initial.Then,limx0⟶0fx=limx0⟶0xn−xn2,n=0,1,2,…Table2 shows the Iteration of lim fx with α=1 and x0=0.1.Table 1
Iteration of limfx with α=1 and x0=0.
n01234xn00000Table 2
Iteration of limfx with α=1 and x0=0.1.
n01234xn0100000.090000.0819000.0751920.069538Figure1 shows the linear stability graph of the logistic map when α=1, x0=0and0.1.Figure 1
Linear stability graph of the logistic map whenα=1, x0=0and0.1.An example/type of a bifurcation is seen when the control parameterα=1 both (fixed points) solutions of the system meets at x=0 and alters their stableness [17]. The structural changes of the system at α=1 is where transcritical bifurcation occurs and the stability of the fixed point occurs there.Figure2 shows a subcritical bifurcation of a logistic map when the parameter is exactly 1. The vertical lines in both Figures 2(a) and 2(b) show that for the logistic map transcritical bifurcation occurs at α=1 and in the dynamical system it is the most common one. Figure 2(c) shows the convergence of period-1 fixed point having transcritical bifurcation when α=1 and period doubling starting at α=3 and beyond.Figure 2
Bifurcation diagram of the logistic mapα=1 with the initial value 0and0.1.
(a)(b)(c)Figure3 shows that at α=±1, it is very difficult to determine the linear stability of the period–1 point (fixed point) since it is very marginal. This confirms work in [17, 18], that fixed point is marginal at α=1 and α=−1 and its linear stability cannot be determined.Figure 3
The cobweb diagram of the logistic map.


## 2.1.1. Logistic Map

The logistic map is defined as follows:(1)Xn+1=αxn−xn2,(2)Letfxn=Xn+1,Thenfxn=αxn−xn2,(3)iffxn=xn.For the solution of the map, equations (1) and (3).(4)Thenαxn−xn2=xn.By solving them algebraically, the solutions of the map are(5)xn=0,xn=α−1α.For the stability of the system.(a)
Letα=1 implies x0=1−1/1=0 as the initial.Then,limx0⟶0fx=limx0⟶0xn−xn2,n=0,1,2,…Table1 shows the Iteration of lim fx with α=1 and. x0=0.(b)
Letα=1 implies x0=0.1 as the initial.Then,limx0⟶0fx=limx0⟶0xn−xn2,n=0,1,2,…Table2 shows the Iteration of lim fx with α=1 and x0=0.1.Table 1
Iteration of limfx with α=1 and x0=0.
n01234xn00000Table 2
Iteration of limfx with α=1 and x0=0.1.
n01234xn0100000.090000.0819000.0751920.069538Figure1 shows the linear stability graph of the logistic map when α=1, x0=0and0.1.Figure 1
Linear stability graph of the logistic map whenα=1, x0=0and0.1.An example/type of a bifurcation is seen when the control parameterα=1 both (fixed points) solutions of the system meets at x=0 and alters their stableness [17]. The structural changes of the system at α=1 is where transcritical bifurcation occurs and the stability of the fixed point occurs there.Figure2 shows a subcritical bifurcation of a logistic map when the parameter is exactly 1. The vertical lines in both Figures 2(a) and 2(b) show that for the logistic map transcritical bifurcation occurs at α=1 and in the dynamical system it is the most common one. Figure 2(c) shows the convergence of period-1 fixed point having transcritical bifurcation when α=1 and period doubling starting at α=3 and beyond.Figure 2
Bifurcation diagram of the logistic mapα=1 with the initial value 0and0.1.
(a)(b)(c)Figure3 shows that at α=±1, it is very difficult to determine the linear stability of the period–1 point (fixed point) since it is very marginal. This confirms work in [17, 18], that fixed point is marginal at α=1 and α=−1 and its linear stability cannot be determined.Figure 3
The cobweb diagram of the logistic map.


## 2.2. The Relaxations of the Fixed Point Using the Logistic Map

Considering two basic hypotheses,(1)
Ifμ=0 then x decayed algebraically so that (6)xn,μ=0∝nβ,whereβ is a critical exponent and depends on the type of bifurcation.(2)
Atμ≠0, the orbit is at equilibrium exponentially through relaxation such that(7)xn,μ∝e−n/r,whereτ∝μz is the relaxation time and z is a critical exponent.Now, we look at the theoretical argumentation on the characterization of the logistic function/map, for the transcritical bifurcation.Letα=1, equation (1) can be written as follows:(8)dxdn≅xn+1−xn=xn+1−xnn+1−n.Then, the approximation of equation (8) is(9)dxdn=−x2.Taking a very close/nearby value to the point only make the limit ofxn of equation (9) very valid.(10)Then,dx−x2=dn.Taking the integration of equation (10) on both sides.(11)−∫x0xdx−x2=∫0ndn,⇒−−1xx0x=n0n,⇒1x−1x0=n,x=1n+1/x0.Asn increases to the condition n>1/x0,we obtain the following equation:(12)xn∝1n,xn∝n−1.By comparingxn∝n−1 to xn,μ=0∝nβ. The critical exponent at α=1 is. β=−1when. α>1Similarly, we subtractxn from both sides of equation (1) to obtain the following equation:(13)dxdn≅xn+1−xn=xn+1−xnn+1−n,(14)Then,dxdn=xα−1−αx2.Taking anyx value that is very close to the fixed point, equation (14) becomes;(15)dxdn=μx,(16)⇒dxx=μdn.Taking the Integral of equation (16) on both sides;(17)∫x0xdxx=μ∫0ndn,lnxx0x=μn0n,lnx−lnx0=μn−μ0,lnxx0=μn,xx0=eμn,xn=x0eμn.Comparingxn,μ∝e−n/r to xn=x0eμn. The critical exponent z=−1.


## 2.3. Numerical Simulation of the Logistic Map

The numerical simulation of the logistic map as shown in Figure4 indicates that in Figure 4(a) at μ=0 the convergence to the fixed point where the power law fit gives β=−0.99997≅−1 and Figure 4(b) shows a slope of z=−0.994≅−1 at μ≠0.Figure 4
The convergence to the period fixed point using the logistic mapμ=0andμ≠0.
(a)(b)Clearly, the results for the arguments and the numerical simulations in Figure4 are the same that β=−0.99997≅−1 and z=−0.994≅−1.Finally, for a logistic map the growth or decay of a perturbation of its periodic orbit is an exponential with the same rate for the periodic-1 point (fixed point) of other maps especially, the Poincare map, see [18].


## 2.4. The Convergence of Period-1 Fixed Point Using the Cubic Function

### 2.4.1. Cubic Map: the Cubic Map is a Difference Equation Whose Nonlinearity is Cubic

Letf:X⟶X be a continuous map defined as follows: (18)Xn+1=fxn=αxn−xn3,where α∈1,4 and X∈0,1.For the fixed point of the map, we equations and (3) and (18).(19)Letαxn−xn3=xn,⟹xn3−αxn+xn=0,xnxn2−α+1=0,Thenxn=0.Also,xn2−α+1=0,⇒xn2=α−1.Then,xn=α−1,xn=−α−1.Hence,xn=0, xn=α−1 and xn=−α−1 are the solutions of the map.


### 2.4.2. The Stability of the Cubic Map

(20)iffxn=αxn−xn3.We take the derivative of the map.(21)That isf′xn=α−3xn2.(1)
For attractor/stable condition,(22)α−3xn2<1.Whenxn=0,(23)α−302<1.⇒α<1 or −1<α<1 is where the map is stable or attracting.Whenxn=α−1,(24)⇒α−3α−12<1,⇒α−3α−1<1,−2α+3<1.Solving this absolute systematically, we obtained1<α<2 as where the map is attracting or stable.(2)
For repulsive/unstable condition,(25)α−3xn2>1,α−3α−12>1,⇒α−3α−1>1,−2α+3>1.Then,(26)−2α+3>1,−2α>1−3,−2α>−2,∴α<1.Also,(27)2α+3>1,2α>1+3,2α>4,∴α>2.Hence, the cubic map is repulsive when.α<1orα>2.Figure5 shows the bifurcation diagram of the cubic map and that within the interval of the parameter [2, 3]. The system exhibits a pitchfork bifurcation right at α=1 and beyond that the system experiences period doubling bifurcation. At α=1, there is a loss of stability but regain it back as keeps oscillating.Figure 5
The bifurcation diagram of the cubic map whenRc=α=1.


## 2.4.1. Cubic Map: the Cubic Map is a Difference Equation Whose Nonlinearity is Cubic

Letf:X⟶X be a continuous map defined as follows: (18)Xn+1=fxn=αxn−xn3,where α∈1,4 and X∈0,1.For the fixed point of the map, we equations and (3) and (18).(19)Letαxn−xn3=xn,⟹xn3−αxn+xn=0,xnxn2−α+1=0,Thenxn=0.Also,xn2−α+1=0,⇒xn2=α−1.Then,xn=α−1,xn=−α−1.Hence,xn=0, xn=α−1 and xn=−α−1 are the solutions of the map.


## 2.4.2. The Stability of the Cubic Map

(20)iffxn=αxn−xn3.We take the derivative of the map.(21)That isf′xn=α−3xn2.(1)
For attractor/stable condition,(22)α−3xn2<1.Whenxn=0,(23)α−302<1.⇒α<1 or −1<α<1 is where the map is stable or attracting.Whenxn=α−1,(24)⇒α−3α−12<1,⇒α−3α−1<1,−2α+3<1.Solving this absolute systematically, we obtained1<α<2 as where the map is attracting or stable.(2)
For repulsive/unstable condition,(25)α−3xn2>1,α−3α−12>1,⇒α−3α−1>1,−2α+3>1.Then,(26)−2α+3>1,−2α>1−3,−2α>−2,∴α<1.Also,(27)2α+3>1,2α>1+3,2α>4,∴α>2.Hence, the cubic map is repulsive when.α<1orα>2.Figure5 shows the bifurcation diagram of the cubic map and that within the interval of the parameter [2, 3]. The system exhibits a pitchfork bifurcation right at α=1 and beyond that the system experiences period doubling bifurcation. At α=1, there is a loss of stability but regain it back as keeps oscillating.Figure 5
The bifurcation diagram of the cubic map whenRc=α=1.


## 2.5. The Relaxations of the Fixed Point Using the Cubic Map

Now, we look at the theoretical argumentation on the characterization of the cubic function/map, for the pitchfork bifurcation.When.α=1Equation (18) can be written as follows:(28)dxdn≅xn+1−xn=xn+1−xnn+1−n.Then, the approximation of equation (28) is(29)dxdn=−x3.Taking a very close/nearby value to the point only make the limit ofxn of equation (29) very valid.(30)Then,dx−x3=dn.Taking the integration of equation (30) on both sides,(31)−∫x0xdx−x3=∫0ndn,⇒−−12x2x0x=n0n,⇒12x2−12x02=n,x2=12n+1/x02,x=12n+1/x02.Asn increases to the condition 2n>1/x02,(32)x≅12n,x≅12n,x≅12n−1/2.By comparing equations (6)–(32), the critical exponent at α=1 is β=−1/2,whenα>1,Similarly, we subtractxn from both sides of equation (18) to obtain the following equation:(33)dxdn≅xn+1−xn=xn+1−xnn+1−n.Then, by Taking any value that is very close to the fixed point, equation (33) becomes;(34)dxdn=μx−x3.Similarly, taking the integral of equation (34) on both sides and going through the same procedure as used for the logistic map, equation (34) is end up like that of equation (9). Then, by comparing, the critical exponent z=−1.


## 2.6. Numerical Simulation of the Cubic Map

The numerical simulation of the logistic map as shown in Figure6 indicates that in Figure 6(a) at μ=0 the convergence to the fixed point where the power law fit gives β=−0.497≅−1/2 and Figure 6(b) shows a slope of z=−0.9927≅−1 at μ≠0.Figure 6
The convergence to the period fixed point using the cubic mapμ=0andμ≠0.
(a)(b)Clearly, the results for the arguments and the numerical simulations in Figure6 are the same that β=−0.497≅−1/2 and z=−0.9927≅−1.In summary, there is an algebraic decay of the system whenμ=0 but when μ≠0 the decay is not algebraic in nature but rather characterized by a relaxation time [19]. The system is unstable when the parameter of the logistic map and cubic map is α=1 with the initial value x=0, there is a relaxation time that depends on the μ in showing the stability of the bifurcation.


## 3. Conclusions

It has been shown that the period-1 fixed point of a logistic map as a recurrence has its convergence at a transcritical bifurcation having its power law fit with exponentβ=−1 when α=1 and μ=0. The cubic map shows its convergence to the fixed point at a pitchfork bifurcation decaying at a power law with exponent β=−1/2 at when α=1 and μ=0. However, the system shows their relaxation time at the same power law with exponents and z=−1.



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