On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system

Abstract: The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension q and the fractal dimension k. We will look at the wellposedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of q and k.


Introduction
The complexities of physical phenomena in nature have forced researchers into developing mathematical models that can be used to describe and capture the behavior of these natural occurrences. Traditional calculus may seem to be enough in solving problems that arise from science and engineering. However, many physical phenomena may better be described by fractional calculus because it is a well-suited tool to analyze problems of fractal dimension, with long term "memory" and chaotic behavior [1]. Some of the advantages of using fractional calculus over classical calculus

Preliminaries
In this section, we will give a brief discussion of some important definitions and properties from fractal-fractional calculus that are useful for this paper.
The following definitions are discussed in detail in [39].
Definition 2.1. [39] Suppose that u(t) is a continuous function and fractal differentiable on an open interval (a, b) with order k then, a q order fractal-fractional derivative of the function u(t) in a Caputo sense with a power-law type kernel is given by: where θ ≤ 1.
Definition 2.2. [39] Suppose that u(t) is a continuous function and fractal differentiable on an open interval (a, b) with order k then, a q order fractal-fractional derivative of the function u(t) in a Caputo sense with an exponential decay type kernel is given by: where 0 < q, k ≤ n ∈ N and M(0) = M(1) = 1. A generalized version of the above equation is defined as follows: where 0 < q, k, θ ≤ 1.
Definition 2.3. [39] Suppose that u(t) is a continuous function and fractal differentiable on an open interval (a, b) with order k then, a q order fractal-fractional derivative of the function u(t) in a Caputo sense with the generalized Mittag-Leffler kernel is given by:

5)
A generalized version of the above equation is defined as follows: where 0 < α, β ≤ 1 and AB(α) Definition 2.4. [39] Suppose that u(t) is a continuous function and fractal differentiable on an open interval (a, b) with order k then, a q order fractal-fractional derivative of the function u(t) in a Caputo sense with exponential decay kernel is given by: where 0 < q, k ≤ n and M(0) = M(1) = 1. A generalized version of the above equation is defined as follows: where 0 < q, k, θ ≤ 1.
Definition 2.5. [39] Assuming that u(t) is a continuous function on (a, b) , then a q order fractalfractional integral of the function u(t) with power law type kernel is given by: (2.9) Definition 2.6. [39] Assuming that u(t) is a continuous function on (a, b) , then a q order fractalfractional integral of the function u(t) with an exponential decaying type kernel is given by: (2.10) Definition 2.7. [39] Assuming that u(t) is a continuous function on (a, b), then a q order fractalfractional integral of the function u(t) with a generalized Mittag-Leffler type kernel is given by: (2.11)

The Lorenz chaotic system under the fractal fractional Caputo-Fabrizio derivative
In this section, we introduce the Lorenz chaotic system under the definition of fractal fractional Caputo-Fabrizio derivative.
Consider the following three dimensional nonlinear chaotic system called the Lorenz chaotic system [24,46] x (t) = γ(y − x), where x = x(t), y = y(t), and z = z(t) are the dynamical variable of the system and γ, ρ, and δ are the related real constants parameters. Using the definition of the fractal-fractional derivative under the Riemann-Liouville sense with exponential decay kernel for each classical derivative equations in (3.1), we obtain and Φ 3 (x, y, z, t) = xy − δz. The above system of equations can be written as follows Since the fractional integral is differentiable. We can rewrite the Eq (3.3) as Therefore, system (3.4) can be expressed as follows We now replace the Riemann-Liouville derivative with the Caputo-Fabrizio derivative to make use of the integer-order initial conditions. Thus from system (3.5) we get In this study, we will investigate the above system of equations.

The well-posedness of the fractal fractional Caputo-Frabrizio derivative of the Lorenz system
In this section, we use the Pichard Lindelof method [46][47][48] to show the existence and uniqueness of the solution of the following nonlinear system of equations subjected to the following initial conditions To show the existence and uniqueness of the solution we define the following operators.
Now, let We now want to show that f 1 , f 2 and f 3 satisfy the Lipschipitz conditions with respect to x, y, z respectively. This means that for any two given functions For f 1 we have Hence, f 1 is satisfies the Lipschipitz conditions. For f 2 we have Thus, f 2 is satisfies the Lipschipitz conditions. For f 3 we have Therefore, f 3 satisfies the Lipschipitz conditions. Now, let We now continue to apply the Banach fixed point theorem using the metric on spaces of continuous We now define the next operator between the two functional spaces of continuous functions, Picard's operator, as follows (4.10) Defined as follows Where the matrix X is given as From (4.5)-(4.7) we can conclude that F(t, X(t)) satisfies Lipschipitz conditions with respect to the system state variable X(t). Now, we must show that this operator maps a complete nonempty space into itself. We first show that, given a certain restriction on a, D takes values in B 1 , B 2 , B 3 in the space of continuous functions with uniform norm. To obtain good results, we assume that the problem under consideration satisfies M . Using the maximum's metric (4.13) We want to show that the operator is a contraction mapping. So, we have with p < 1. Since F is Lipschitz continuous, we have that the operator D is a contraction for arp < 1.
Hence, this shows that the system under consideration has a unique set of solution.

Numerical scheme
In this section, we present the numerical scheme for the Caputo Fabrizio fractal fractional derivative of the Lorenz chaotic system. The chaotic model (3.1) can be converted to When we apply the Caputo-Fabrizio integral to (5.1), we get For a positive integer n,the solution of the system of Eq (5.1) at t = t n+1 becomes, and at t = t n , we obtain Taking the difference between (5.4) and (5.3), we get , y(t n−1 ), z(t n−1 ), t n−1 ) We now approximate the functions Λ k−1 Φ 1 (x, y, z, Λ), Λ k−1 Φ 2 (x, y, z, Λ) and Λ k−1 Φ 3 (x, y, z, Λ) on the finite interval [t n , t n+1 ] using the piece-wise Lagrangian interpolation such as Substituting (5.6) into (5.5) and integrating, we obtain , y(t n−1 ), z(t n−1 ), t n−1 ), , y(t n−1 ), z(t n−1 ), t n−1 ), (5.7) , y(t n−1 ), z(t n−1 ), t n−1 ), Therefore, we have completed the derivation of the numerical scheme used in this study.

Numerical simulation
In the previous section, we presented a numerical scheme under the Caputo-Fabrizio fractalfractional derivative operator. In this section, we aim to use the numerical scheme presented to approximate the graphical solution for the Lorenz chaotic systems under the Caputo-Fabrizio fractalfractional derivative operator for different values of the fractional dimension q and the fractal dimension k. We now look at the following examples: Example 1. Consider the following system of equations where Φ 1 (x, y, z, t) = γ(y − x), Φ 2 (x, y, z, t) = ρx − y − xz, and Φ 2 (x, y, z, t) = xy − δz. With parameter values γ = 10, ρ = 28 and δ = 8 3 . In this example we solve the system (6.1) using the initial conditions x(0) = y(0) = z(0) = −1. The graphical numerical simulations for different values of q and k for this example are presented in Figures 1-4.
Example 2. For the the second example, we consider the system of equation in example 1 with different initial conditions. Consider the following system of equations In this example, we solve the system (6.2) using the initial conditions x(0) = 0, y(0) = 2 and z(0) = 20. The graphical numerical simulations for different values of q and k for this example are presented in Figures 5-8.

Discussion
In Example 1 and 2, we modeled the Lorenz chaotic system with initial conditions x(0) = y(0) = z(0) = −1 and x(0) = 0, y(0) = 2 and z(0) = 20 respectively, using the Caputo-Fabrizio fractalfractional derivative operator. We then solved the obtained nonlinear systems of equations using a numerical scheme for different fractal dimensions k and fractional order q. Figures 1-4 represent the graphical simulation, for example, 1, and Figures 5-8 represent the graphical simulation, for example 2.
In both examples, we noticed that the fractal power k = 1 and the fractional power q = 1 recover the classical two-step Adams-Bashforth method and the classical differential and integral operators. Figures 1(a) and Figures 5(a) represents the graphical solutions for the case where k = 1 and q = 1 for example 1 and example 2, respectively. While solving the systems of equations, we noticed that for some values of k and q the solution blows up.
For both examples, we noticed that if we keep q = 1 and vary the value of k we can obtain graphical solutions for as far as k = 0.10, see From Table 1, we can see that the solution for Example 1 for some values of k and q blows up faster than those in Example 2. This means that the choice of the initial conditions of the nonlinear system of equations also contributes to when the solution blows up for the nonlinear system under investigation.

Error estimate
In this section, we use numerical results obtained from the numerical scheme to estimate the rate of convergence using the following

19.980149246718597
The results in Tables 2-7 depict order one and two rate of convergence. However, this approach might not be entirely reliable because we deal with chaotic systems that are extremely sensitive to initial conditions. Some other methods are proposed in [49,50].

Conclusions
In this study, we used the newly proposed Caputo Fabrizio fractal fractional operator with different fractal dimension k and fractional order q, to capture and analyze the dynamical behavior of the Lorenz chaotic system. We present the numerical scheme used to solve the system of nonlinear equations and obtained graphical numerical simulations for different values of q and k. We noticed that for the fractal dimension k = 1 and fractional order q = 1 we obtain a two-step Adams Bashforth method and the classical differential and integral operators. We also noticed that for some values of q and k the solutions blow up. Taking into consideration that the chaotic systems are sensitive to initial conditions we compared two examples of the same nonlinear system of equations with different initial conditions, we noticed that the choice of the initial conditions also affect when some solutions blow up. For future work, we want to find out what other factors contribute to the blowing up of the solutions for some values of q and k.