Modelling chaotic dynamical attractor with fractal-fractional differential operators

Abstract: Differential operators based on convolution have been recognized as powerful mathematical operators able to depict and capture chaotic behaviors, especially those that are not able to be depicted using classical differential and integral operators. While these differential operators have being applied with great success in many fields of science, especially in the case of dynamical system, we have to confess that they were not able depict some chaotic behaviors, especially those with additionally similar patterns. To solve this issue new class of differential and integral operators were proposed and applied in few problems. In this paper, we aim to depict chaotic behavior using the newly defined differential and integral operators with fractional order and fractal dimension. Additionally we introduced a new chaotic operators with strange attractors. Several simulations have been conducted and illustrations of the results are provided to show the efficiency of the models.


Introduction
Very recently the concept of fractal differentiation and fractional differentiation have been combined to produce new differentiation operators [1][2][3][4]. The new operators were constructed using three different kernels namely, power law, exponential decay and the generalised Mittag-Leffler function. The new operators have two parameters, the first is considered as fractional order and the second as fractal dimension.
The operators were tested to model some real world problems [9][10][11] surprisingly, the new differential and integral operators were found to be powerful mathematical tools able to capture even hidden complexities of nature [5,6,16]. Very detailed attractors could be captured when modelling with such differential and integral operators [8]. One of the great advantage of these new operators is Definition 2.2. [3] The Caputo-Fabrizio fractal-fractional derivative of f (t) with order − κ in the Riemann-Liouville sense is defined as follows: (2.2) where κ > 0, ≥ m ∈ N and M(0) = M(1) = 1.

Numerical approximation of fractal-fractional calculus
In this section, we have given three numerical schemes for Caputo-fractal-fractional, Caputo-Fabrizio-fractal fractional and the Atangana-Baleanu fractal-fractional derivative operators [7].

Caputo-fractal-fractional derivative
Consider the following differential equations in the fractal-fractional Liouville-Caputo sense can be converted to the Volterra case and the numerical scheme of this system using a Caputo-fractal-fractional approach at t n+1 is given by We can approximate the above integral to With in the finite interval [t j , t j+1 ], we approximate the function s −1 , f (t, u, v) using the Lagrangian piecewise interpolation such that So we obtain Solving the integral of the right hand side, we obtain the following numerical scheme

Caputo-Fabrizio-Caputo fractal-fractional derivative
Consider the following differential equations in the Caputo-Fabrizio-fractal-fractional derivative Applying the Caputo-Fabrizio integral, we obtain Here we present the detailed derivation of the numerical scheme. Thus, at t n+1 we have Taking the difference between the consecutive terms, we obtain (3.10) Now using the Lagrange polynomial piece-wise interpolation and integrating, we obtain (3.11)

Atangana-Baleanu-Caputo fractal-fractional derivative
Consider the following differential equations in the Atangana-Baleanu fractal-fractional derivative in the Liouville-Caputo sense Applying the Atangana-Baleanu integral, we have (3.13) At t n+1 , we have the following (3.14) the above system can be expressed as Approximating s −1 f (s,u,v) in [t j , t j+1 ], we have the following numerical scheme

Numerical analysis for Duffing attractor
The Duffing Pendulum is a kind of a forced oscillator with damping, the mathematical model represented in state variable as when we apply Caputo-fractal-fractional derivative is given by where initial conditions are u 1 (0) = 0.01, u 2 (0) = 0.5 and the constant are κ = 0.25, b = 0.3, ω = 1, h = 0.01, and t = 100. The numerical scheme is given by    Involving the fractal-fractional derivative in the Caputo-Fabrizio-Caputo, we have The numerical scheme is given by (4.6) Involving the fractal-fractional derivative in the Atangana-Baleanu-Caputo sense, we have The differential and integral operators used in the case of Caputo and Caputo-Fabrizio have kernel with no crossover in waiting time distribution. Therefore to include the effect of crossover in waiting time distribution, we make use of the differential operator whose kernel has crossover behavior, thus the Atangana-Baleanu operators are used here. Also the numerical simulation is presented in      Additionally adapting the previously used numerical scheme, we obtain the following:

Numerical analysis for EL-Nino Southern Oscillation
The EL-Nino Southern Oscillation known as ENSO is an unconventional repetitive variation in sea area temperatures over the tropical eastern Pacific Ocean and winds that affect the climate change of much of the tropics and sub-tropics The natural behavior of this dynamical system can be classified in two phases. The first phase is the warming phase of the sea temperature, this phase was named EL-Nino. The second phase known as EL-Nino is the cooling phase. The results of some data collection showed that, the Southern Oscillation is following atmospheric component, more importantly the phenomena is coupled with the sea temperature change. On the other hand, the EL-Nino is followed by high air surface pressure in the side of tropical western Pacific and La Nina is accompanied with low pressure. In this section, we consider the mathematical model able to replicate such natural occurrence, nevertheless, here we consider the model with different non-local differential and integral operators.
The behavior of this phenomena is described by the following equation as in Caputo-fractal-fractional derivative sense where κ = 100 and b = 1.
The numerical scheme is given by The above numerical solution is depicted in Figures 9-13, for different value of fractional order and fractal dimension. In this exercise, we chose to simulate the case with Caputo derivative. The numerical simulation depend on the fractional order and the fractal dimension. More precisely for the values 0.8 and 0.5, we observed a very strange behavior where the oscillation are vanishing.     Involving the fractal-fractional derivative in the Caputo-Fabrizio-Caputo, we have The numerical scheme is given by Involving the fractal-fractional derivative in the Atangana-Baleanu-Caputo sense, we have The numerical scheme is given by

Numerical analysis for Ikeda system
Consider the ikeda delay system in Caputo-fractal-fractional derivative sense In [12], the author proposed a second delay parameter. The below given equation is resultant solution here we consider a=24, b=3 and c=1 and the delays τ 1 = 0.01 and τ 2 = 0.1. The numerical scheme is given by The numerical scheme is given by ) .

(6.5)
Involving the fractal-fractional derivative in the Atangana-Baleanu-Caputo sense, we have The numerical scheme is given by (6.7)

Numerical analysis for Dadras attractor
Consider the Dadras system described [13] by the following equation as in Caputo-fractal-fractional derivative sense where A=3, B=2.7, C=1.7, D=2 and E=9.
The numerical scheme is given by Involving the fractal-fractional derivative in the Caputo-Fabrizio-Caputo, we have The numerical scheme is given by Involving the fractal-fractional derivative in the Atangana-Baleanu-Caputo sense, we have The numerical scheme is given by Numerical simulation are depicted in Figures 14-17 for the value of κ = 0.8 and value of = 0.9. The used numerical model is that with the fractal-fractional differential operator. The model with this new differential operator display some important behavior.

Numerical analysis for Aizawa attractor
A very strange attractor have been studied in the last past years, although such system of equations have not being attracting attention of many researchers, but the attractor is very strange as the system is able to display very interesting attractor in form of sphere. The model under investigation is called Aizawa attractor, this system when applied iteratively on three-dimensional coordinates, it is important to point out that, evolving in such a way as to have the consequential synchronizes map out a three dimensional shape, in this case a sphere with a tube-like structure powerful one of its axis. In this section, we consider the model using the fractal-fractional with power law, exponential decay law and the generalized Mittag-Leffler function. Consider the Aizawa system described by the following equation as in Caputo-fractal-fractional derivative sense where A = 0.95, B = 0.7, C = 0.6, D = 3.5, E = 0.25, and F = 0.1. The numerical scheme is given by Involving the fractal-fractional derivative in the Caputo-Fabrizio-Caputo, we have The numerical scheme is given by ×(1 + Ew(t n−1 )) + Fw(t n−1 )u 3 (t n−1 ) .

(8.12)
Involving the fractal-fractional derivative in the Atangana-Baleanu-Caputo sense, we have The numerical scheme is given by Numerical simulation are depicted in Figures 18-21 for the value of κ = 0.8 and value of = 0.9. The used numerical model is that with the fractal-fractional differential operator. The model with this new differential operator display some important behavior.     The numerical scheme is given by The numerical scheme is given by n (−Aw(t n ) + sin u(t n )) − ∆t 2 t −1 n−1 −Aw(t n−1 ) + sin u(t n−1 ) .

(9.12)
Involving the fractal-fractional derivative in the Atangana-Baleanu-Caputo sense, we have where A = 0.19. The numerical scheme is given by

Numerical analysis for 4 Wings attractor
Consider the 4 Wings system proposed in [15] and modifying in [14] is described by the following equation as in Caputo-fractal-fractional derivative sense where A = 4, B = 6, C = 10, D = 5, and E = 1. The numerical scheme is given by The numerical scheme is given by Numerical simulation are depicted in Figures 26-29 for the value of κ = 0.9 and value of = 0.8. The used numerical model is that with the fractal-fractional differential operator. The model with this new differential operator display some important behavior.

Conclusions
Fractal-fractional differential operators have been introduced very recently; however, the new concept has not yet attracted attention of many scholars. In fact, few works have being done where such differential and integral operators are used. The advantages if this work is that it considers operators capturing new complexities of nature with great success. Such differential and integral operators are sophisticated tools to model complex real world problems. Moreover, in order to evaluate the efficiency and the capabilities of the new differential and integral operators, we investigate the behavior of some well-known chaotic attractors and see if one will capture more complexities compared to existing differential and integral operators. Additionally, we introduced a new chaotic model with alternative attractors and we showed by numerical simulation that these new differential and integral operators are powerful mathematical operators able to capture heterogeneity. Since any new numerical method should be validated in terms of convergence, stability and consistency of solutions, these are important research directions left to future work.