In this paper, we investigated the chaotic behavior of a Jerk system proposed by Sambas et al. (2024), which features symmetrical attractors arising from the interplay of sinusoidal, hyperbolic, and absolute nonlinearities. The system's complex dynamics were analyzed using established numerical methods such as phase portraits, stability analysis, bifurcation diagrams, and Lyapunov exponents. Furthermore, through amplitude modulation, we showed that the control parameter δ can enhance or attenuate signal amplitudes without disrupting the system's stability or chaotic nature. The theoretical findings were further validated through Multisim circuit simulations, with experimental attractors closely matching the numerical results. In addition, a Radial Basis Function Neural Network (RBFNN) was implemented to approximate the chaotic trajectories of the system. The model was trained using simulated data and optimized via the least squares method. Network performance was evaluated using Root Mean Square Error (RMSE) and relative error. The results showed that the RBFNN accurately predicts the system's state variables, achieving MSE values on the order of 10^-10–10^-9 and relative error below 1.1 × 10^-8.

This paper introduces a matrix-free variant of the Davidon-Fletcher-Powell (DFP) method for unconstrained optimization problems with applications in compressive sensing and image restoration. The main contribution lies in the new search direction incorporating a scaling parameter that ensures the satisfaction of the sufficient descent condition, independent of the line search conditions. A rigorous convergence analysis guarantees the boundedness and theoretical validity of the proposed method. Comprehensive numerical experiments on benchmark unconstrained optimization test problems and compressive sensing problems demonstrate the efficiency and robustness of the algorithm. Specifically, in image restoration tasks, our method outperforms CG-DESCENT, MDL, and NSMA, achieving a 100% success rate compared to 95.8%, 84.5%, and 53.5%, respectively. Additionally, results on computational time, relative error, and PSNR confirm the superior performance of the proposed approach. These findings establish the proposed method as a competitive alternative for large-scale optimization problems.