The main purpose of the study is to present a numerical approach to investigate the numerical performances of the fractional 4-D chaotic financial system using a stochastic procedure. The stochastic procedures mainly depend on the combination of the artificial neural network (ANNs) along with the Levenberg-Marquardt Backpropagation (LMB) i.e., ANNs-LMB technique. The fractional-order term is defined in the Caputo sense and three cases are solved using the proposed technique for different values of the fractional order α. The values of the fractional order derivatives to solve the fractional 4-D chaotic financial system are used between 0 and 1. The data proportion is applied as 73%, 15%, and 12% for training, testing, and certification to solve the chaotic fractional system. The acquired results are verified through the comparison of the reference solution, which indicates the proposed technique is efficient and robust. The 4-D chaotic model is numerically solved by using the ANNs-LMB technique to reduce the mean square error (MSE). To authenticate the exactness, and consistency of the technique, the obtained performances are plotted in the figures of correlation measures, error histograms, and regressions. From these figures, it can be witnessed that the provided technique is effective for solving such models to give some new insight into the physical behavior of the model.

The characteristics based on the dynamics of the system provides the interest of many scientists with its complex and important features. This behavior depends directly on the parameters, which are used to describe the physical phenomenon as well as the sensitivity of the dynamical models with slight change in the initial conditions [

The concept of fractional calculus has been first introduced by two famous scientists Riemann et al. [

Recently, ANNs have been considered one of the most important techniques that have been used to find the solutions to the complex problems. Due to the wide real-life applications in different areas of science and engineering, scientists have been striving for expanding the applications of these techniques along with different properties. For example, El-Mahelawi et al. [

This work aims to simulate the nonlinear 4-D chaotic financial model represented in the system (1) to gain more insight into the dynamics of this important model. The ANNs-LMB technique consists of merging the regular artificial neural networks (ANNs) along with the Levenberg-Marquardt backpropagation (LMB), resulting as a new technique. The proposed algorithm is a promising one since it can deal with different complex problems. The main concept of this algorithm is based on the samples of training, testing and verification samples to get the obtained solutions by using the accurate and efficient technique. This method proves to be a valuable key player in stimulating real

The novelty of the proposed technique can be summarized in the following few points:

A model of the 4-D chaotic financial model under some Mittag-Leffler nature and have the benefit of a more realistic application.

The proposed technique is used to solve system (1) with the combination of the ANNs-LMB method.

The correction of the proposed scheme is observed through the comparison of the proposed and reference solutions.

The absolute error is provided in good measure, which authenticates the proposed solver is an accurate and reliable to solve the fractional form of the 4-D chaotic financial problem.

The combined features of the ANNs with the LMB enhance the accuracy of the obtained results in terms of error for solving the 4-D chaotic financial problem.

This section is illustrated the main steps for the proposed technique named the ANNs-LMB method. This method shall be used to solve the 4-D chaotic financial model represented by

A numerical method with stochastic features based on the ANNs-LMB technique will be used to solve the model (1).

The effectiveness and robustness of the proposed algorithm will be tested for solving the 4-D chaotic model through the application of the proposed technique.

These main steps are illustrated in

This section is devoted to validating the performance of the proposed technique by using the numerical results. The performance of the method in solving the problem (1) is tested with three different cases of the fractional terms. The mathematical representations of the system (1) are simulated for the setup of parameters in the following form as:

Three cases are investigated for the solution of the system (2) with different values of the fractional order

The acquired results of the proposed 4-D chaotic system are presented in

Case | MSE | Performance | Gradient | Mu | Epoch | Time | ||
---|---|---|---|---|---|---|---|---|

Training | Verification | Testing | ||||||

1 | 8.98 × 10^{−09} |
3.28 × 10^{−00} |
1.09 × 10^{−09} |
8.98 × 10^{−09} |
3.46 × 10^{−07} |
1 × 10^{−09} |
176 | 3 |

2 | 5.12 × 10^{−09} |
2.43 × 10^{−10} |
6.48 × 10^{−10} |
5.12 × 10^{−09} |
9.87 × 10^{−08} |
1 × 10^{−09} |
165 | 3 |

3 | 3.37 × 10^{−09} |
9.63 × 10^{−09} |
1.10 × 10^{−09} |
3.38 × 10^{−09} |
9.94 × 10^{−08} |
1 × 10^{−10} |
122 | 2 |

The current study aims to investigate the numerical simulation of a 4-D chaotic financial system under some Mittag-Leffler laws using a combination of the artificial neural network and the Levenberg-Marquardt backpropagation techniques named the ANNs-LMB technique. The fractional order derivatives have been implemented to perform more realistic solutions of the 4-D chaotic financial system. The computational scheme is applied for three variations of the fractional kinds of models using different fractional values. The statics proportions have been applied as 73%, 15%, and 12% for training, testing, and certification for the 4-D chaotic model. The number of neurons used is 14 to solve the model. The numerical outcomes for the nonlinear fractional chaotic system are obtained by using the ANNs-LMB technique in order to reduce the MSE for the acquired approximate solutions. To ensure the reliability, and effectiveness of the scheme, the numerical measure is plotted and compared to a reference solution. The absolute error is provided in good ranges, which shows the competence of the proposed stochastic solver. The performance of the technique is witnessed to be ideal for the proposed model in terms of precision and accuracy.

_{2}composites using Levenberg–Marquardt backpropagation algorithm

_{2}gas with an efficient method