On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives

In this work, we present a design for a Newton-Leipnik system with a fractional Caputo-Fabrizio derivative to explain its chaotic characteristics. This time-varying fractional Caputo-Fabrizio derivative approach is applied to solve the model numerically, and to check the solution’s existence and uniqueness. The existence and uniqueness of results of a fractional-order model under the Caputo-Fabrizio fractional operator have been proved by fixed point theory. As well, we achieved a stable result by applying the Ulam-Hyers concept. Chaos is controlled by linear controllers. Furthermore, the Lyapunov exponent of the system indicates that the chaos control findings are accurate. Based on weighted covariant Lyapunov vectors we construct a background covariance matrix using the Kaplan-Yorke dimension. Using a numerical example, this suggested method is illustrated for its applicability and efficiency.


Stability analysis Equilibrium points
To compute the equilibrum points of the Newton-Leipnikk model (2), consider the following:

Thus
All equilibrium points are saddle points and satisfy the system chaos condition.

Proof
Then, Similarly, one obtains Hence, the proof follows.

Kaplan-Yorke dimension
where σ 1 ≤ σ n are Lyapunov characteristic exponents and j is the largest integer for which Vol.:(0123456789) , where ν is the correlation exponent, σ the information dimension, and D the Hausdorff dimen- sion, then see 19 .
You can now compute the Lyapunov exponent (LE) of (2) using the Danca algorithm: 5,6 and apply the Adams-Bashforth-Moulton numerical scheme.Using Table 2, we can see that the Newton-Leipnik system (2) is dissipative, since the sum of the Lyapunov exponents (LE) in each row is negative.Observe that the Lyapunov exponent depends on x(0) = 0.349 , y(0) = 0 , and z(0) = − 0.160 .Table 1 presents some fractional derivatives with Kaplan-Yorke dimensions: For α = 0.70, For α = 0.90, For α = 0.98, The fact that all of the Kaplan-Yorke dimensions calculated earlier are fractional is another indication that the system is moving in a chaotic direction.Figure 1's simulation results demonstrate the Lyapunov exponential spectrum technique for chaotic fractional-order systems' high accuracy and convergence.
When evaluating the Newton-Leipnik dissipative properties, it turns out that the volume element y 0 shrinks exponentially to y 0 e (−a−0.4+b)t at time t and the asymptotic motion eventually becomes an attractor stabilized.The divergent flow of (2) is dissipative if and only if ∇V < 0, If b − a < 0.4 then the system is dissipative.
The system (2) is symmetric about the z axis since it is invariant under the coordinate transformation (x, y, z) → (−x, −y, −z) .Table 2 displays the Lyapunov exponents for of a fractional Newton-Leipnik system (2).Simulation results in Fig. 1 demonstrate the convergence of the Lyapunov exponential spectrum technique for a fractional Newton-Leipnik system (2).

Equilibrium points Eigenvalues
Nature Index asymptotic motion eventually stabilizes as an attractor.Therefore the divergent flow of ( 5) is dissipative if and , the system is dissipative.By looking at the Lyapunov exponent value, we can see that the fixed point is stable.For α = 0.70, For α = 0.90, For α = 0.98, Table 2. Lyapunov exponents versus α of a fractional Newton-Leipnik system (2).Table 3 displays the Lyapunov exponents for of a controlled fractional Newton-Leipnik system (5).Simulation results in Figure 2 demonstrate the convergence of the Lyapunov exponential spectrum technique for a fractional Newton-Leipnik system (5).

Constant-order numerical schemes in sense of Caputo-Fabrizio
We present the following Cauchy problem with new fractional derivative: From the definition of the Caputo-Fabrizio integral, we can reformulate the above equation as We write Eq. ( 6) at the point  Taking the difference of these equations, we can write the following: Putting its Lagrange polynomial into the above equation, we can get the following: and we get the following: The integrals on the right hand side of the above equation can be calculated as Thus, we have the following numerical scheme:

Variable-order numerical schemes in sense of Caputo-Fabrizio
The following variable-order numerical schemes is the same as 22,28 .A variable order fractional differential equation is shown below.
The fundamental theorem of fractions has been applied, and we have In this way and When ( 7) is substituted for (8), we get where The numerical solution is given by The system (2) is given by where 1 , 2 , 3 are defined as in section "Model analysis".

Discussion of results
A fractional mathematical model based on the Caputo-Fabrizio operator was developed to describe the Newton-Leipnik system (2).Furthermore, the results depicted in Figs. 3, 4,5, 6, 7, 8, 9, 10, 12 and 13, with the decreasing and increasing values of α , can observe how effective infection is in the model's behavior.The equilibrium points of the system (2) and the corresponding eigenvalues of the Jacobian matrix are shown in Table 1.In chaotic 3D chaos, the equilibrium points of the Newton-Leipnik system (2) yield all unstable eigenvalues as illustrated in Table 1.A balance with exactly five unstable eigenvalues, the saddle point or saddle focus with index 2, is responsible for the generation of the rolling attractor.Therefore, the theoretically calculated minimum effective size of the Newton-Leipnik system is 2.82 as illustrated in Sheu et al. 4 , and this finding is further verified in the numerical simulation results in section "Control of Newton-Leipnik systems".The system shows good dynamic behavior. .

Conclusions
A fractional mathematical model based on Caputo-Fabrizio fractional operators is presented.Thus, in this paper, we examined the dynamics of the Newton-Leipnik system under the fractional derivative with non-singular kernels.Via the fixed-point theorems of Sachuder and Banach, we have explained the existence theory of the model under the Caputo-Fabrizio fractional operator.We have used the fixed point approach for the existence of a unique solution under the non-singular operator.An approximate solution to the model has been calculated using the Caputo-Fabrizio fractional scheme.The results show that the system is stable at equilibrium points and each obtained function converges at its equilibrium point.To investigate the effect of derivative order on the model results, the functions obtained from the model are plotted for different fraction degrees.The results show that the general behavior of the functions is the same with small changes in derivative order but the numerical results are different.The model can also be controlled linearly.We have noticed that the nonsingular fractional operator produces excellent results for the model under consideration compared to the Caputo-Fabrizio and fractional operators.Thus, we have concluded that modeling with the nonsingular operator is much better than modeling with the singular operator.Vol.:(0123456789)