An Integrated Numerical and Machine Learning Feedforward ANN-LM for Solving Variable Fractional Chaotic Attractor Systems with Power Law, Exponential, and Mittag-Leffler Operators

This paper presents an innovative numerical scheme designed to simulate variable-order fractional differential operators using power-law, exponential-law, and Mittag-Leffler kernels. Our approach combines the fundamental theorem of fractional calculus with Lagrange polynomial interpolation to improve both accuracy and efficiency. Here we examine a chaotic system of fractional order, characterized by the Caputo derivative, Caputo Fabrizio, and Atangana Baleanu Caputo with a focus on the implications of constant and variable fractional-order behavior. Furthermore, the paper presents a schematic diagram of the fractional-order chaotic system and details simulations conducted in Multisim. For the suggested Artificial Neural Network based Levenberg Marquardt (ANN-LM) model, data is generated by numerical computations that include various chaotic dynamics, periodic orbit, and fixed-point conditions. The ANN-LM techniques are used to further train the outcomes that the numerical methods produced. Dataset is to examine the chaotic behavior of equations that have been modelled, 15% is used for testing, 70% is used for validation, and 15% is used for training. In these sections we also compare numerical solutions with the outcomes that Levenberg Marquardt Neural Networks (LM-NNs) are expected to produce. Error histograms, performance assessments, regression analysis, correlation indices, and fitness curves based on the Mean Squared Error (MSE) for the chaotic dynamics, periodic orbits, and fixed points are used to evaluate the ANN accuracy and efficacy. Tables and graphs representing the chaotic dynamics, periodic orbits, and fixed points are used to further validate the outcomes by ANN.