On control intervals of chaos with Fréchet derivative of certain iterations as discrete dynamical system in Banach spaces

Dynamical systems are one of the interesting concepts where iteration algorithms and chaos can be considered together. In iteration algorithms, one of the basic concepts of fixed point theory, it is well known that the behavior of the iteration mechanism is chaotic if the original transformation is taken as chaotic. One of the natural ways to transform a chaotic system into a dynamical system is through control mechanisms. In this paper, we first consider an iteration class defined on Banach spaces, which is prominent in the literature in terms of both speed and convergence rate. Then, we consider the transformation constituting the iteration class as chaotic and obtain the stability and unstability behaviors of the iterations according to the operator norm by using Gâteaux and Fréchet derivatives representing the direction‐dependent derivative. In this way, we aimed to obtain the chaos control intervals obtained with functions in real space with the help of operators in Banach spaces. Using the operator norm obtained with the Fréchet derivative, we derive interesting dynamical system intervals from a chaotic system with parameter variables of iteration algorithms. It is noteworthy that the parameter variables of the studied iteration classes are the same in some transformation classes. In addition, analytical proofs are followed by computer simulations of parameter‐dependent control intervals considering the logistic operator with a chaotic structure. At the same time, the periodic behavior of the iteration algorithms used in our study is illustrated by the Lyapunov exponent with parameter‐dependent control intervals according to operator norm. Finally, as a real‐life problem, it has been shown that the chaos of the logistic population growth model with dynamic features can be controlled by chaos control mechanisms established by fixed point iteration methods.


INTRODUCTION AND PRELIMINARIES
It is an important phenomenon that chaos is defined as uncertainty and unpredictability.Because of this challenge to scientific knowledge, chaos theory has become an interesting research field.It should be emphasized that chaotic structures appear in simple events in nature such as weather forecasting, population growth, traffic flow, genetic algorithms and the like [1][2][3][4][5][6][7][8][9][10][11].For this reason, studies on chaos theory have been carried out in many fields of science.Chaos theory started with the study of dynamical structures as the subject of mathematics has continued with the control of the determined chaos.In particular, it can be taken into account that the magnitudes with the growth rate in nature are a dynamical system.However, it is well known that the most important field of study as dynamical systems is iteration systems associated with fixed point theory.The iteration method pioneered by Picard [12] is still used in many scientific fields to solve many fixed point problems.If a chaotic operator is considered as an iteration, it is obtained that it is a chaotic dynamical system.If an iterative system is unstable and chaotic with respect to a fixed point, it is an important research topic to find the mathematical method that will allow it to become stabilized.It can be explained that the control in iteration methods with chaotic behavior is to stabilize the unstable fixed point by appropriate selection of iteration parameters.In this study, our aim is to investigate chaos control mechanisms for an iteration class defined in Banach space by using Gâteaux and especially Fréchet derivatives.Considering the usefulness of the structure of Banach spaces, the study becomes more valuable.Of course, working with operators in Banach spaces has some difficulties compared to chaos control in classical real space.The parameter ranges obtained with the intervals determined by the functions in real space are obtained with norms of continuous operators in Banach space.However, with these difficulties, controlling operators with chaotic behavior with iteration parameters by taking advantage of the rich properties of Banach spaces emerges as an important field.In real spaces, the starting concept of chaotic behavior is the growth rate, whereas in Banach spaces, the growth rate is considered based on the motion of the operator.Considering a nonlinear operator x n+1 = T (x n ) in Banach space.We can give growth rate as ‖xn+1−xn‖ ‖xn‖ ≡  n ( ‖T(xn)−xn‖ ‖xn‖ where  n ∈ (0, 1).One can find it in the following list of chaos control studies in real space.With this approach, chaos control studies in Banach spaces differ from the studies in real spaces [13][14][15][16][17][18].Another important difference is that in real spaces the control is obtained by ordinary derivatives and also, in Banach space it is obtained by Gâteaux or Fréchet derivatives.
In order to achieve this aim, we will first consider the iteration classes with chaotic operator, and then we will convert the fixed point of the system to stable using the Fréchet derivative and obtain the control ranges by selecting the appropriate parameter.One of the interesting features of fixed point iterations is that although they differ from each other in terms of convergence and speed, the control parameter intervals of them under chaotic behavior are same.Especially, it should be noted that the rate comparison and some characteristic properties of iterations occur independently of the determination of the chaos control intervals.In other words, rate and convergence that are the hereditary features of iterations continue to be inherited in the control intervals obtained with the parameters.In the following main results, we will show that the control intervals are equivalent in case of chaos over the iteration classes.In addition, using the logistic operator, we will show that iteration classes have interesting examples.Also, chaotic and dynamical structures of systems will be simulated with computer programs.Finally, as a real-life problem, it has been shown that the chaos of the logistic population growth model with dynamic features can be controlled by chaos control mechanisms established by fixed point iteration methods.Now, let us explain some basic concepts that we will need in our work.In this study, Gâteaux and Fréchet derivatives are the types of derivatives we will use to control a chaotic operator in Banach spaces.The definitions and properties of these two derivatives will be given below (for more detailed information, see the literature [19][20][21][22][23]).In addition, we have to give important concept called a unit operator.Let X be a normed space.Let I ∶ X → X is unit operator such that I(x) = x for x ∈ X and ‖I‖ = 1.To avoid confusion throughout this study, we will take the unit operator I (1) = I in place of I (1) = 1.
A useful variation of the triangle inequality is for any vectors x and .This also shows that a vector norm is a (uniformly) continuous operator.We consider real vector spaces X and Y where Y is assumed to be a normed space.Let us choose an operator T ∶ X → Y with the domain D (T) = U ⊆ X.For vectors x ∈ U and h ∈ X, we assume that x + th ∈ U for some t ∈ R. If the limit exists, then the vector DT (x) (h) ∈ Y is called the Gâteaux derivative or differential of the operator T at the vector x in the direction of the vector h.Note that in the definition of the Gâteaux derivative, the vector space X is not taken as a normed vector space.Therefore, an operator with a Gâteaux derivative need not always be continuous at that point.The derivative of an operator defined between two normed vector spaces is found with the Fréchet derivative under the condition of continuity.
Let X and Y be normed spaces and let T ∶ X → Y be a nonlinear operator.Suppose that D (T) = U ⊆ X is an open set.If a continuous linear operator T ′ (x) ∈ B (X, Y ) exists at a vector x ∈ U such that for all vectors h ∈ X, then T ′ (x) is called the Fréchet derivative of the operator T at a vector x and the operator T ′ ∶ X → B (X, Y ), which assigns a continuous operator T ′ (x) to a vector x is known as the Fréchet derivative of T. We can also give the norm of the Fréchet derivative operator as follows: Theorem 1 (Suhubi [19]).Let X and Y be normed spaces.If the operator T ∶ X → Y is Fréchet-differentiable at a vector x ∈ X, then T is continuous at x.
Theorem 2 (Suhubi[19]).Let X and Y be normed spaces.If the operator T ∶ X → Y has a Fréchet derivative at the vector x ∈ X, then it has also a Gâteaux derivative at x and these two different derivatives are equal.
Considering Theorem 2, we can say that if a nonlinear operator has a Fréchet derivative at any point, then it also has a Gâteaux derivative at the same point and both derivatives are equal to each other.Throughout the paper we will assume that the derivative of a nonlinear operator at any point is equal to the direction-dependent Gâteaux derivative at the same point and we will show the Fréchet derivative as follows: When iterations of composite structures in Banach spaces are considered as an operator, the chain rule of operators is needed to compute Fréchet derivatives of composite operators.
Theorem 3 (Chain rule [19]).Let X be a linear vector space and Y , Z be normed spaces.
Gâteaux-differentiable in X and its Gâteaux derivative at x ∈ X in the direction of the vector h is given by If X is also a normed space and the operator T is Fréchet-differentiable in X, then R is Fréchet-differentiable as well and its Fréchet derivative is given as follows: As it can be seen, the Gâteaux derivative provides a method for calculating directional changes in Banach spaces.The fact that the properties of fixed point iterations in real space such as unstability or stability of the function are obtained by ordinary derivative of the function.By using similar idea, we can obtain stability or unstability for the fixed points of an operator in Banach space by means of the Fréchet derivative.Now, the fixed point of an operator and its properties will be given.
Let X be a nonempty set and T ∶ X → X be a self mapping.A point x * ∈ X is said to be fixed point of T if For any given x * ∈ X, we define T n (x * ) inductively by T 0 (x * ) = x * and T n+1 (x * ) = T (T n (x * )) ; we call T n (x * ) the n th iterate of x * under T.
The sequence (x n ) ⊂ X given by is called the sequence of successive approximations as the Picard iteration with the initial value x 0 .Also, the trajectory of x 0 under T is the set of points x 0 , T (x 0 ) , T 2 (x 0 ) , … , T n (x 0 ) , ....The starting point x 0 for the trajectory is called the initial value of the trajectory.
Definition 1 (Devaney [24]).Let T ∶ X → X be an operator.A point x * ∈ X is a periodic point of T with period-p if T p (x * ) = x * .The point x * has prime period-p if T p (x * ) = x * and T n (x * ) ≠ x * for 0 < n < p.
Chaotic behavior of iteration algorithms is determined by chaos-structured functions in real spaces, but in Banach spaces it is determined by chaos-structured operators.In Banach space, the chaotic behavior of the operator is determined according to whether the fixed point of an operator is stable or unstable.The types of fixed points obtained by the Fréchet derivative of an operator can be classified as follows.
Definition 2 Let (X, ‖.‖).be a Banach space, and T ∶ X → X be a self-map having a fixed point x * , that is, x * = Tx * .Also, let T ′ (x * ) denote the Fréchet derivative of operator T (x) at x = x * .Then, the fixed point x * can be classified as follows: Let (X, ‖.‖) be a Banach space, and T ∶ X → X be a nonlinear operator.Consider a one-dimensional nonlinear discrete dynamical system with an initial trajectory x 0 ∈ X as for every n ∈ N. If x * ∈ X is taken as the fixed point of the operator T, the set of fixed points of this operator is represented as If T is continuous, the solution of Equation ( 6) corresponds to the solution of the equation G (x * ) = T (x * ) − x * = 0.In this case, F T is a non-empty set and G (x * ) = 0, that is, T has a fixed point.Also, if T satisfies the contraction condition, then the existence and uniqueness of the fixed point is guaranteed according to the Banach fixed point theorem [25].
Let X be a nonempty set and T ∶ X → X be a self-map.For any initial value x 0 ∈ X and let (x n ) be iteration sequence given by, where We can explain the comparison of the above iteration types in terms of convergence and speed, depending on the literature.Firstly, Picard-S [26] iteration method become prominent its convergence rate compared to some well-known iteration methods such as Picard [12], Mann [27], Ishikawa [28], Noor [29], SP [30], CR [31], and S [32].Later on, Thakur et al. [33] described a new three-step iteration method that guarantees the convergence of the nonexpansive mapping classes to the fixed point.Subsequently, Agarwal et al. [32] introduced S-iteration method given by Equation (9).Afterward, in 2011, Sahu [34] defined Normal-S iteration method given by Equation (10).Recently, in 2017, Karakaya et al. [35] demonstrated three-step iteration method defined by Equation (11).In 2018, Ullah and Arshad [36] introduced M-iteration method given by Equation (12), which converges faster than the two-step S-iteration [32] and Picard-S [26] iteration methods.
While the control of chaotic operators in Banach space is performed by Fréchet derivative, another important method for the iteration caused by the trajectory behavior between two different points is the Lyapunov exponent technique.It is important tool for Lyapunov exponent technique used in nonlinear systems to measure the sensitive dependence between two trajectories for very close initial points in Banach space.For stable periodic behavior, this method measures the rate of convergence toward the stable fixed point.It measures the rate of divergence between trajectories for chaotic behavior.Now, let us modify the Lyapunov exponent given for real space and give it for the operator in Banach spaces as follows.For fixed point iteration classes, T will be considered chaotic operator and the chaos controlling mechanisms obtained by modifying the iteration classes will be denoted by T. Also, let x and x + (0 <  < 1) be two different points for trajectories in the iteration methods in Banach space.In addition, assuming that the divergence between the two trajectories measures  ‖I‖ and the exponential growth rate as e k , where  is the Lyapunov exponent, and k is a iteration of order k.Then, we can write After taking the limit as  → 0 in Equation ( 13), we get So, we can write as follows: If the logarithm is applied to both sides of the last equation again, we get Here, T′ (k) (x) is the Fréchet derivative of T(k) .We will use the chain rule to derivative the kth-degree polynomial.From the Equation ( 14), we can write that By the logarithm of the Equation (15), we get By using Equation ( 16), we measure the convergence and divergence rates of an iteration, and we also decide whether the fixed points and periodic points of the given system are stable or unstable.In other words, we can explain when  < 0, the system is stable and when  > 0, it is also unstable.

CHAOS CONTROL INTERVALS OF THE SOME FIXED POINT ITERATION CLASSES
Consider the one-dimensional discrete dynamical system given by Equation (6).Let X be a Banach space and T ∶ X → X be a continuous and differentiable operator in the sense of Fréchet.If the operator T satisfies the contraction condition, then according to the Banach fixed point theorem [25], it has a unique fixed point on the set X, that is, there is a point x * ∈ X such that T (x * ) = x * .Let us also assume the existence of the derivative T ′ (x * ).Let T ∶ X → X be chaotic.Also, suppose that the iteration methods given by Equations ( 7) and (8) in Table 1 have chaotic behavior and we denote them with x n+1 = T (x n ).In this case, the intervals of chaos control with operator norm, which is the original approach of this work, will be discussed in the following theorems.

Picard-S iteration algorithm
Thakur iteration algorithm Karakaya iteration algorithm M-iteration algorithm Theorem 4. Let (X, ‖.‖) be a Banach space, T, I ∶ X → X be a continuous and an unit operator, respectively, and x * = Tx * .Let the chaos controlling mechanisms established by iteration methods given by Equation (7) and Equation ( 8) be denoted by x n+1 = Tx n .Then T and T have the same set of fixed points.However, let T be a chaotic operator with unstable fixed point x * ∈ X satisfying the Fréchet derivative ‖T ′ (x * )‖ > 1.In this case, the stability ranges of the control mechanisms of Picard-S [26] and Thakur [33], which transform unstable trajectories into stable (attracting), under the stability condition ‖ ‖ T′ (x * ) ‖ ‖ < 1, are equivalent.Also, there are always range of control parameters Proof.Let us defined the fixed point chaos controlling mechanism established by the Picard-S [26] iteration method given in Equation (7) as where ( n ) , ( n ) ⊂ (0, 1) are control parameter sequences.Assume that x * ∈ X is a fixed point of the operator T, that is, x * = Tx * .Substituted in control mechanism Equation (18), we obtain that x * , original system T and control mechanism T share the same set of fixed points.Now, let x * ∈ X be an unstable fixed point that provides the condition ‖T ′ (x * )‖ > 1.First of all, the Fréchet derivative must be obtained in order to ensure the condition of stability ‖ ‖ T′ (x * ) ‖ ‖ < 1 given in the theorem hypothesis.Since operator T contains composition operators, the composite structures that make up the operator T will be determined before obtaining the derivative operator.For this, in Equation ( 18), composite operators are taken as Considering Equation ( 3) and also Equation ( 19), we have From here, the Fréchet derivative of the operator H is calculated as and substituting (21) in (20), the Fréchet derivative of T is obtained as The norm of the operator T with the Fréchet derivative can be given as follows: If the stability condition ‖ ‖ T′ (x * ) ‖ ‖ < 1 given in the hypothesis of the theorem is applied, the intervals of the control parameters that stabilize the chaotic operator T are determined.Under this condition, the control parameters ( n ) , ( n ) ⊂ (0, 1) determine a dynamic range, and we obtain the following inequality using (i) of Definition 2: If Equation ( 1) is applied to Equation ( 22), we can write In the above inequality, the stability range determined by the control parameters of the control mechanism T, which stabilizes the unstable fixed point x * ∈ X of the chaotic operator T is determined as follows: Hence, using Equation ( 23), the control ranges of Picard-S [26] iteration depending on the ( n ) , ( n ) parameters can be obtained.
Fixed point chaos controlling mechanism obtained by modifying the Thakur [33] iteration method is defined by where ( n ) , ( n ) ⊂ (0, 1) are control parameter sequences.For the fixed point x * ∈ X of the operator T, if applied to similar process for previous iteration method, we find that if x * = Tx * , then T (x * ) = x * .Thus, the set of fixed points of the original operator T and the control mechanism T defined by Equation ( 24) is the same.
We assume that x * ∈ X be unstable fixed point, that is, ‖T ′ (x * )‖ > 1. Considering that the control mechanism consists of composite operators for calculating the Fréchet derivative of the control mechanism T, we can define the following operators as The Fréchet derivative of the operator T defined by Equation ( 24) at the fixed point x * ∈ X can be obtained as follows: If the Fréchet derivative of the operator H is taken, we get Substituting the Fréchet derivative of the operator H in ( 25), it can be deduced that Norm of operator T at fixed point x * ∈ X, it can be obtained as Under stability condition ‖ ‖ T′ (x * ) ‖ ‖ < 1, similar the processes of calculation in Equation ( 22) are applied to ( 26), then we get the same stability range given by Equation ( 23) such that Therefore, the stability intervals of the control mechanisms of Picard-S [26] and Thakur [33] iteration methods to stabilize a chaotic operator T are equivalent.This completes the proof.□ Similar to the stabilization process of a chaotic operator with an unstable fixed point in the above theorem, the stabilization processes of a chaotic operator with periodic properties will also be given by the following theorem.Theorem 5. Let (X, ‖.‖) be a Banach space, T, I ∶ X → X be a continuous and an unit operator, respectively, and x * = Tx * .Suppose that x * ∈ X be an periodic fixed point of period-m of T, that is, x * = T m (x * ).Let the chaos controlling mechanisms established by iteration methods given by Equations ( 7) and ( 8) for periodic trajectories with period-m be denoted by x n+1 = T(m) (x n ).Then T m and T(m) have the same set of periodic fixed points.However, let T be a chaotic operator with unstable periodic fixed point of period-m satisfying the Fréchet derivative In this case, the stability ranges of the control mechanisms of Picard-S [26] and Thakur [33] iteration methods which transform unstable periodic trajectories into stable under the stability condition Proof.Suppose that denotes the mth recurrent process of T. Define the fixed point controlling mechanism established by the Picard-S [26] iteration method given in (7) for periodic trajectories of period-m as where ( n ) , ( n ) ⊂ (0, 1) are control parameter sequences.Let x * ∈ X be an periodic fixed point of period-m of T.Then, in Equation ( 28), since x * = T m (x * ), we obtain So, operators T m and T(m) share the same set of real periodic fixed points.Now, let x * ∈ X be an unstable periodic fixed point of period-m that provides the condition In order to provide the condition of stability given in the hypothesis of the theorem, first the control mechanism T(m) will be written as composite operators and then the Fréchet derivative will be calculated.In Equation (28), composite operators are taken as The Fréchet derivative of the operator T(m) (x n ) at the fixed point x * ∈ X can be calculated as follows: In order to get the norm of T(m) with Fré chet derivative, we have Under the stability condition , if the property of normed spaces Equation ( 1) is applied, we obtain that If the Equation ( 30) is solved according to the absolute value property, the stability interval determined by the control parameters of the control mechanism T(m) of Picard-S [26] iteration method given in (7) for periodic trajectories of period-m is obtained as In the rest of the proof, the dynamical properties of the periodic trajectories of the Thakur [33] iteration method defined by Equation ( 8) for the chaos control of chaotic operators will be determined.For this, first of all, the chaos controlling mechanism of the Thakur [33] iteration method is redefined for periodic trajectories of period-m as follows: where Thereby, T m and T(m) share the same set of real periodic fixed points of period-m.In order to examine the stability condition of the controlling mechanism defined as Equation (32), the composite operators are determined such that The Fréchet derivative of the operator T(m) defined by Equation ( 24) at the periodic fixed point x * ∈ X, it can be computed as Norm of operator T(m) at periodic fixed point x * ∈ X, the following can be obtained Notice that under stability condition , if similar the processes of calculation in Equation ( 28) are applied to (32), then we get the same stability range of Equation (31), that is, Consequently, the stability ranges of the controlling mechanisms of Picard-S [26] and Thakur [33] iteration methods, to stabilize unstable periodic trajectories of chaotic operator T are equivalent.This completes the proof.□ In the iteration methods given by Equations ( 9) and (10) in Table 1, the theorem showing that the control intervals are the same if the operator T is chaotic.Theorem 6.Let (X, ‖.‖) be a Banach space, T, I ∶ X → X be a continuous and an unit operator respectively and x * = Tx * .Let the chaos controlling mechanisms established by iteration methods given by Equations ( 9) and ( 10) be denoted by x n+1 = T (x n ).Then T and T have the same set of fixed points.However, let T be a chaotic operator with unstable (repelling) fixed points x * ∈ X satisfying the Fréchet derivative ‖T ′ (x * )‖ > 1.In this case, the stability ranges of the control mechanisms of S-iteration [32] and Normal-S [34], which transform unstable trajectories into stable (attracting), under the stability condition ‖ ‖ T′ (x * ) ‖ ‖ < 1, are equivalent.Also there are always range of control parameters Proof.Let the fixed point chaos controlling mechanism established by the S-iteration [32] method given in (9) be defined as where ( n ) , ( n ) ⊂ (0, 1) are control parameter sequences.Before calculating the Fréchet derivative, since the transform T (x n ) is in the form of a composition of the operators, if the composite operator is taken as the controlling mechanism defined by Equation (34) turns into the following: Using the Fréchet derivative of the operator T at the unstable fixed point x * ∈ X, we have Since every Gâteaux differentiable operator is Fréchet differentiable, it is obtained from the definition of the operator norm as The operator T must satisfy the stability condition ‖ ‖ T′ (x * ) ‖ ‖ < 1 in order to stabilize the unstable fixed point x * ∈ X.It can be observed that If the Equation ( 1) is applied to Equation (37), we have Making the necessary calculation, the stability range determined by the control parameters of T, which stabilizes the unstable fixed point x * ∈ X of the chaotic operator T, is determined as follows: Applying similar procedures, let us consider the Normal-S [34] iteration method given by (10) and show that the stability intervals are the same with the chaos controlling mechanism composed S-iteration [32] method.If equation x n+1 = T (x n ) is modified for Equation (10), the fixed point controlling mechanism based on a one parameter such that where ( n ) ⊂ (0, 1) is control parameter sequence.It can easily be seen that the control mechanism T defined by Equation (39) and the operator T have the same set of fixed points.On the other hand, in Equation (39) if the composite operator is defined as the controlling mechanism can be written Using Equation (3), the Fréchet derivative of the operator T defined by Equation (40) at the fixed point x * ∈ X is Norm of operator T at fixed point x * ∈ X, it can be found as Under stability condition ‖ ‖ T′ (x * ) ‖ ‖ < 1, similar the processes of calculation in Equation ( 34) are applied to (39), we obtain the stability range Consequently, the stability ranges of the control mechanisms of S -iteration [32] and Normal-S iteration [34] methods, to stabilize a chaotic operator T are equivalent.This completes the proof.□ The theorem determining the control intervals of both iteration mechanisms given above for periodic trajectories, which is an important argument in terms of providing chaos control of transformations, will be given below.Theorem 7. Let (X, ‖.‖) be a Banach space, T, I ∶ X → X be a continuous and an unit operator respectively and x * = Tx * .Suppose x * ∈ X be an periodic fixed point of period-m of an operator T, that is, x * = T m (x * ).Let the chaos controlling mechanisms established by iteration methods given by Equations ( 9) and (10) for periodic trajectories with period-m be denoted by x n+1 = T(m) (x n ).Then operators T m and T(m) have the same set of periodic fixed points.However, let T be a chaotic operator with unstable (repelling) periodic fixed point of period-m satisfying the Fréchet derivative In this case, the stability ranges of the control mechanisms of S-iteration [32] and Normal-S [34], which transform unstable periodic trajectories into stable (attracting), under the stability condition there are always range of control parameters ( n ) , ( n ) ⊂ (0, 1) (n ∈ N) of the mechanism T(m) such that Proof.Assume that T m denotes the mth recurrent process of T. Define the fixed point controlling mechanism established by the S-iteration [32] method given in (9) for periodic trajectories of period-m as where ( n ) , ( n ) ⊂ (0, 1) are control parameter sequences.Let x * ∈ X be an periodic fixed point of period-m of T.
Then, in Equation ( 43), since x * = T m (x * ), we obtain So, operators T m and T(m) share the same set of real periodic fixed points.Now, let x * ∈ X be an unstable periodic fixed point of period-m that provides the condition In order to provide the condition of stability , first the controlling mechanism T(m) will be written as composite operators and then the Fréchet derivative will be calculated.In the controlling mechanism given by (43), when the composite is taken in terms of operators H (x n ) ∶= (1 −  n ) x n +  n T m (x n ), the control mechanism turns into the following form: Using the Fréchet derivative of the operator T(m) (x n ) at the fixed point x * ∈ X in the direction of the h for t ∈ R, it can be deduced The norm of the operator T(m) with the Fré chet derivative is Under the stability condition , if the property of normed spaces Equation ( 1) is applied, we obtain that The stability range determined by the control parameters of the control mechanism T(m) of S-iteration [32] method given in (43) for periodic trajectories of period-m is found as In the continuation of the proof, the dynamical properties of the periodic trajectories of the Normal-S [34] iteration method defined by Equation (10) for the chaos control of chaotic operators will be determined.The chaos controlling mechanism of the Normal-S [34] iteration method is modified for periodic trajectories of period-m as follows: where ( n ) ⊂ (0, 1) is control parameter sequence.If the composite operator is determined H (x n ) ∶= (1 −  n ) x n +  n T m (x n ) controlling mechanism given by Equation ( 46) such that then the Fréchet derivative of the operator T(m) defined by Equation (47) at the periodic fixed point x * ∈ X , it can be computed as Norm of operator T(m) at periodic fixed point x * ∈ X, the following can be obtained: Notice that under stability condition , by making the necessary adjustments, the following control range is obtained for the control parameter: Consequently, the stability ranges of the controlling mechanisms of S -iteration [32] and Normal-S [34] iteration methods, to stabilize unstable periodic trajectories of chaotic operator T are equivalent.This completes the proof.□ Theorem 8. Let (X, ‖.‖) be a Banach space, T, I ∶ X → X be a continuous and an unit operator, respectively, and x * = Tx * .Let the chaos controlling mechanisms established by iteration methods given by Equations ( 11) and ( 12) be denoted by x n+1 = T (x n ).Then T and T have the same set of fixed points.However, let T be a chaotic operator with unstable (repelling) fixed points x * ∈ X satisfying the Fréchet derivative ‖T ′ (x * )‖ > 1.In this case, the stability ranges of the control mechanisms of Karakaya [35] and M-iteration [36], which transform unstable trajectories into stable (attracting), under the stability condition ‖ ‖ T′ (x * ) ‖ ‖ < 1, are equivalent.Also, there is always range of control parameter Proof.Since the proof is similar previous methods, we omit it.□ Theorem 9. Let (X, ‖.‖) be a Banach space, T, I ∶ X → X be a continuous and an unit operator, respectively, and x * = Tx * .Suppose x * ∈ X be an periodic fixed point of period-m of an operator T, v x * = T m (x * ).Let the chaos controlling mechanisms established by iteration methods given by Equations ( 11) and ( 12) for periodic trajectories with period-m be denoted by x n+1 = T(m) (x n ).Then operators T m and T(m) have the same set of periodic fixed points.However, let T be a chaotic operator with unstable (repelling) periodic fixed point of period-m satisfying the Fréchet derivative In this case, the stability ranges of the control mechanisms of Karakaya [35] and M-iteration [36], which transform unstable periodic trajectories into stable (attracting), under the stability condition there is always range of control parameter Proof.Since the proof is similar previous methods, we omit it.□

APPLICATION I: STABILITY INTERVALS OF THE FIXED POINT CONTROLLING MECHANISMS ON LOGISTIC SYSTEM IN BANACH SPACES
Let (X, ‖.‖) be a Banach space, and T, I ∶ [0, 1] → [0, 1] be a continuous and an unit operators, respectively.Additionally, I (x) = x and ‖I‖ = 1.As an one-dimensional chaotic discrete dynamical system, consider the logistic operator given by For x * ∈ [0, 1], if Tx * = x * is solved, T has two fixed points: x * 1 = 0 and x * 2 = 3 4 .To determine whether these fixed points are stable or not, we can give the following procedure.Firstly, the Fréchet derivative of the logistic operator ( 52) is taken Later, the norm of the operator T with the Fréchet derivative is calculated as If the fixed point x * = x * 2 = 3 4 is substituted in the Equation (53), it can be obtained From Definition 2, it can be seen that the fixed point x * 2 = 3 4 is the unstable (repelling) fixed point for the logistic operator T, and T is chaotic at this point.Now, if the chaos controlling mechanisms established by Picard-S [26] and Thakur [33] iteration methods given by Equations ( 18) and ( 24) will be remodified by applying the logistic operator, respectively, we get and Under the hypotheses of Theorem 4, the fixed point x * 2 = 3 4 , which is unstable (repelling) for the logistic operator T, turns into a stable (attracting) fixed point in the stability range determined by the control parameters ( n ) , ( n ) ⊂ (0, 1) of the Picard-S and Thakur controlling mechanisms, which satisfies the stability condition ) ‖ ‖ ‖ ‖ < 1.In this case, the stability ranges of systems Equation (54) and Equation (55) using Equation ( 17), it can be observed ) ) (56) Equation ( 56) is reduced to the following control parameter intervals and After this stage, the analytical results obtained above using the MATLAB program will be illustrated by computer simulations.Let's take the initial value x 0 = 0.7333.Let's choose the control parameter ranges  n = 0.3,  n ∈ (0.8333, 1) from the stability intervals Γ 1  , Γ 1  given by Equation ( 57) and apply them to the control mechanisms Equations ( 54) and ( 55) separately.In Figure 1, it is clearly seen that the fixed point x * 2 = 3 4 , which has a repelling (unstable) trajectory for the logistic operator T, turns into an attracting (stable) trajectory with the help of the Picard-S controlling mechanism, which is represented by T. On the bifurcation diagram given in Figure 2, not only the stable region of the system, but also the behavior of all trajectories, which are convergent, periodic and chaotic, of the control mechanism outside the stability range of Γ Let us apply the chaotic logistic operator T to the S-iteration [32] and Normal-S [34] iteration method.Firstly, if the S-controlling mechanism given by Equation ( 34) is arranged under the T(x) = 4x(1 − x) , we obtain for the control parameters ( n ) , ( n ) ⊂ (0, 1).According to Theorem 6, the stability condition ) ‖ ‖ ‖ ‖ < 1 must be satisfied in order to obtain the interval that will stabilize the unstable (repelling) fixed point x * 2 = 3 4 of the logistic operator T. For this reason, it is sufficient to substitute the derivative of the fixed point x * 2 = 3 4 in the control interval given as Equation (38), since the Equation (3) will be applied to the same process as given in the proof of Theorem 6.In Equation (38 ) taking ) ) These ranges are reduced to the following parameter intervals: Thus, the control parameter ranges determined by the controlling mechanism defined as Equation (3) that will stabilize the chaotic logistic operator given by Equation (52) are obtained.
By similar thought, applying the T(x) = 4x(1 − x) to the Normal-S controlling mechanism given by (39), it can be obtained In the proof of Theorem 6, in the stability range obtained as Equation (42) for the Normal-S controlling mechanism, when the derivative of the fixed point x * 2 = 3 4 is substituted for , the following stability range determined by the control parameter ( n ) ⊂ (0, 1) obtained that Therefore, the Normal-S controlling mechanism given by Equation (60) transforms the unstable fixed point x * 2 = 3 4 the chaotic logistic operator given by Equation (52) into a stable in the control range (61).
Let x 0 = 0.7333 be the initial point and choose the control parameter values  n = 0.25,  n ∈ (0.6666, 1) from the stability interval Γ 1  , Γ 1  given by Equation (59).In Figure 5, it can seen that the S-controlling mechanism converges to the fixed point x * 2 = 3 4 and forms attracting trajectories at the selected parameter values.Also, in Figure 6, the dynamical behavior of the controlling mechanism both in the stability range and out of the range is given with the bifurcation diagram.In addition to these, in Figure 7, the unstable fixed point of the logistic operator T is stable in the  n ∈ (0.1666, 0.5) stability range Γ  given by (61) of the Normal-S controlling mechanism.It is seen that the convergence to the fixed point is realized.Besides, the dynamical behavior of the Normal-S controlling mechanism outside the stability range is shown on the bifurcation diagram given in Figure 8.

FIGURE 6
Bifurcation diagram of the S-controlling mechanism for x 0 = 0.7333,  n ∈ (0, 1). [Colour figure can be viewed at wileyonlinelibrary.com]Likewise, applying the logistic operator T(x) = 4x(1 − x) to the Karakaya controlling mechanism established by iteration (11), it can be observed ) and if applied to the M-controlling mechanism established by iteration (12), we have Under the Theorem 8 hypotheses, the unstable (repelling) fixed point x * 2 = 3 4 of the logistic operator T turns into a stable (attracting) fixed point in the control range determined by the ( n ) ⊂ (0, 1) parameter of the Karakaya and M-controlling  50), the stability range determined by the control parameter ( n ) ⊂ (0, 1) is as follows: In other words, the unstable fixed point x * 2 = 3 4 of the logistic operator T chosen as the chaotic operator becomes stable in the control parameter range of Karakaya and M-controlling mechanisms.
Let x 0 = 0.7333 be the initial value.Let the stability range  n ∈ (0.25, 0.4166) determined by Equation (64) be chosen.The trajectory diagram in Figure 9 shows that the fixed point x * 2 = 3 4 acts as the attracting fixed point in the stability range Γ  of the Karakaya controlling mechanism.In addition, with the bifurcation diagram given in Figure 10, periodic and chaotic regions formed by the controlling mechanism outside the stability range are seen.In the same parameter range, the dynamical trajectories and bifurcation diagram of the M-controlling mechanism in the stability range Γ  are given in Figures 11 and 12 graphic drawings, respectively.

APPLICATION II: LYAPUNOV EXPONENT OF THE CONTROLLING MECHANISMS FOR PERIODIC TRAJECTORIES ON LOGISTIC SYSTEM IN BANACH SPACES
In this section, for convenience controlling the chaotic state of unstable periodic trajectories of period-2 with fixed point iteration methods will be discussed.Let us reconsider the logistic operator as a one-dimensional chaotic discrete dynamical system given by Equation (52).Our aim in this section is to stabilize the unstable periodic fixed points of period-2 of the chaotic logistic operator T with fixed point controlling mechanisms.
Firstly, for x * ∈ [0, 1], if the 4th degree equality T 2 x * = x * is solved, we get two trivial fixed points of period-2 such that x * 1 = 0, x * 2 = 3 4 and a pair of nontrivial fixed points of period-2 such that x * (2) . In order to determine the stability of these periodic fixed points, the Fréchet derivative of the operator T 2 will be calculated as follows: The norm of the operator T with the Fréchet derivative, we get Hence, according to Definition 2, since
Now, let's modified the Picard-S and Thakur controlling mechanisms given in Equations ( 28) and (32), respectively, by taking m = 2, we obtained as where ( n ) , ( n ) ⊂ (0, 1) are control parameter sequences.Instead of resolving the controlling mechanisms given above, the Equation ( 27) proved in Theorem 5 for the stability of periodic trajectories of period-m is determined for the fixed points of period-2, the following control parameter ranges are determined In order to calculate the Lyapunov exponent, first of all, the Fréchet derivatives of the operators at the periodic fixed points

1
) and 2 of the controlling mechanisms are obtained Let us choose the parameter values  n = 0.20 ∈ Γ 1  ,  n = 0.96 ∈ Γ 1  from the interval given in Equation ( 68) and calculate the Lyapunov exponent for orbits of period-2.By using Equations ( 69) and ( 16), we have If we apply the same method to the Thakur controlling mechanism given by Equation (67), we get Lyapunov exponent  = −0.080910< 0 for the parameter values  n = 0.20 ∈ Γ 1  ,  n = 0.96 ∈ Γ 1  .Thus, since  < 0, it is proved that the unstable periodic fixed points of period-2 of the logistic system T are stabilized by the Picard-S and Thakur controlling mechanism defined as Equations ( 66) and (67), respectively.
Similarly, let us control the unstable periodic trajectories of the logistic operator with the S and Normal-S controlling mechanisms defined as Equations ( 43) and ( 46), if these controlling mechanisms are arranged for orbits of period-2, respectively, we get where ( n ) , ( n ) ⊂ (0, 1) is control parameter sequences.Using the proof of Theorem 7, in the stability range given by Equation (45) of the S-iteration [32], if

1
) and 2 ) are calculated by substituting the derivatives of unstable periodic fixed points, the control parameter ranges that provide stability are obtained Fréchet derivatives of the operators at the periodic fixed points

1
) and 2 ) of the S-controlling mechanism are obtained as Let us choose the parameter values  n = 0.2 ∈ Γ 1  ,  n = 0.9 ∈ Γ 1  from the interval given in Equation ( 72) and calculate the Lyapunov exponent for orbits of period-2.By using Equations ( 73) and ( 16), we get Therefore, as  < 0, the unstable periodic fixed points of period-2 of the logistic operator T are stabilized by the S-controlling mechanism defined as Equation (70).
Besides, if it is arranged for the stability range given by Equation (49) of the Normal-S [34] iteration, we have Fréchet derivatives of the operators at the fixed points of period-2 of the Normal-S controlling mechanism is calculated as Let us choose the parameter value  n = 0.2 ∈ Γ  from the interval given in Equation (74) and calculate the Lyapunov exponent for orbits of period-2.By using Equations (75) and ( 16), we get Hence, since  < 0, the unstable periodic fixed points of period-2 of the logistic operator T are stabilized by the Normal-S controlling mechanism defined as Equation (71).
Finally, let us consider the stability ranges defined by Equation (51) given for controlling the periodic trajectories of the Karakaya and M-controlling mechanisms.By taking m = 2 in Equation (51), the control parameter range for period-2 orbits is found as Fréchet derivative of the Karakaya controlling mechanisms at the periodic fixed points of period-2 of the logistic operator, it can be evaluated as (77) Choosing the parameter value  n = 0.19 ∈ Γ  from the interval given in Equation ( 76) and calculate the Lyapunov exponent for orbits of period-2.By using Equations ( 77) and ( 16), we have If the similar method is applied to the M-controlling mechanism for  n = 0.19 ∈ Γ  , we calculated Lyapunov exponent  = −0.0809104< 0. Consequently, as  < 0, it is proved that the unstable periodic fixed points of period-2 of the logistic system T are stabilized by the Karakaya and M-controlling mechanisms.
Remark 1.Consider the mth iteration of the logistic system T(x) = rx (1 − x).It is obvious that this original system is chaotic for r = 4.However, considering the control intervals created by the Picard-S [26], Thakur [33], S-iteration [32], Normal-S [34], Karakaya [35] and M-iteration [36] methods given in the text, the chaotic behavior original system T(x) turns from unstable to stable in the control intervals formed by the parameters ( n ) , ( n ) of the iteration classes.The control ranges created by the parameters ( n ) , ( n ) of the iteration classes form a suitable control range depending on the iteration parameters in the state of r > 4 the original system, in other words, a dynamic region.It is possible to see that r ≥ 4 in inter-orbit periodic behaviors obtained by the Lyapunov exponential technique.It is clear that all four fixed point control mechanisms that we discussed in the article control the chaotic original system with coefficient r = 4 by appropriate selections of parameters ( n ) , ( n ) of iteration classes.In other words, the unstable fixed point of the original system turns into a stable fixed point with the help of the parameters of the iteration classes, and the system transforms from a chaotic behavior to a dynamical system.Control intervals obtained by Lyapunov exponential technique for Picard-S [26], Thakur [33], S-iteration [32], Normal-S [34], Karakaya [35] and M-iteration [36] methods are shown in Figures 14, 15, and 16, respectively.Figure 13 shows the graphical representation of positive Lyapunov exponent  for r ∈ [1,4] of the logistic system T(x).That is, the logistic system behaves chaotic.Further, from Figures 14, 15, and 16, it is observed that the Lyapunov exponent approaches to negative Lyapunov exponents  which means, for each initial point x 0 ∈ [0, 1] the orbit of the map converges to stable attractor.

REAL LIFE APPLICATION: CHAOS CONTROL OF LOGISTIC POPULATION GROWTH FOR MAYFLIES
Simple mathematical models are used to predict the long-term behavior of populations given observable and experimentally determined parameters.Mathematical models used to predict the population behavior of a particular species as it grows or shrinks over generations are called exponential growth models.The logistic growth model takes its name from the differential equation used by P. F. Verhulst [37,38] to model population growth in a limited environment.This equation describes how any type of population changes from one generation to the next.In 1976, biologist Robert L. May [1] published an article examining the basic properties of nonlinear deterministic systems.He realized that the equations he used to calculate increases and decreases in a population were much more complex than he had anticipated.He observed that unexpected changes occurred in the system when he gave different values to the parameters affecting the equation.For example, he observed that as the parameter values of the logistic population equation grew, the system bifurcated and the population oscillated between two different values.
Consider the logistic population model representing a population of mayflies whose individuals are born and die in the same season as follows: In the one-dimensional and nonlinear discrete logistic growth model given above, it is assumed that there is a maximum population initially supported by environmental conditions, and x n corresponds to the maximum population living in the nth generation.If overpopulation occurs in the population, food shortages and species extinction will occur.Therefore, it is taken as 0 ≤  n ≤ 1 to keep the population balanced.While r corresponds to the parameter that affects the system, it is also the factor that makes the system the most complex dynamical system.For this reason, the parameter r is the value that limits the population and is kept in the range of 0 < r ≤ 4. Changing values of the parameter r lead to the onset of chaotic behavior.Under this model, May obtained the following findings as a result of analysis.
• In range 0 ≤  n ≤ 1, r < 1 means that as the number of seasons increases, the population will not be able to reproduce itself in the next generation and mayflies will become extinct.• When the growth parameter reaches the critical value r = 3, the trajectories suddenly change and show interesting behavior.The population size begins to fluctuate between two different constant values in successive generations.
Considering the mayfly population, it can be said that the population has now moved from annual cycles to biennial cycles.That is, the population rate is high 1 year, low the next, then high again, then low again, and so on.Since the population rate returns to the same value every 2 years, it corresponds to period-2 behavior.• At the parameter value r = 3.45, the period doubles again and undergoes a new bifurcation, and an attractor containing four points as 0.852, 0.433, 0.847, 0.447 is formed in the system, that is, the system moves to period-4.This time the population moves from 2-year cycles to 4-year cycles.• When r = 3.56, each attractor is divided into two and the population oscillates between eight different levels.These bifurcations gradually accelerate to 4, 8, 16,32, … .As r = 3.569 approaches, an infinite number of bifurcations occur and the number of attractors in the population reaches infinity.Thus, a region of chaos occurs where orbits never repeat and there are no stable periods.That is, the population begins to oscillate between an infinite number of different values.
Inspired by May's work, the chaos of the logistic population growth model with dynamic features can be controlled with chaos control mechanisms established by fixed point iteration methods.When viewed with the arguments of the fixed point theory, the growth model given by (78) corresponds to the Picard iteration method.
Consider the logistic population growth model given by (78) as an operator with chaotic properties.The new population growth control mechanism established with the Picard-S (18) and Thakur (24) iteration methods are defined as follows, respectively, and where were obtained with the help of the Fréchet derivative depending on the stable or unstable fixed points of the chaotic behavior of the iteration on this operator.

1 𝛼 , Γ 1
are shown.It is also shown on the trajectory diagram in Figure 3 that the Thakur controlling mechanism transforms the unstable trajectories of the fixed point x * 2 = 3 4 of T into stable or convergent trajectories at the same starting point and selected parameter ranges.Moreover, on the bifurcation diagram Figure 4, all the dynamic regions formed by the trajectories of the Thakur controlling mechanism outside the stability range are given.

TABLE 1
Some fixed point iteration classes.
Lyapunov exponent diagram of logistic system T for r ∈ [1, 4].[Colour figure can be viewed at wileyonlinelibrary.com]