Optimal Parameter Estimation Techniques for Complex Nonlinear Systems

Accurate parameter estimation and state identification within nonlinear systems are fundamental challenges addressed by optimization techniques. This paper fills a critical gap in previous research by investigating tailored optimization methods for parameter estimation in nonlinear system modeling, with a particular emphasis on chaotic dynamical systems. We introduce and compare three optimization methods: a gradient-based iterative algorithm, the Levenberg-Marquardt algorithm, and the Nelder-Mead simplex method. These methods are strategically employed to simplify complex nonlinear optimization problems, rendering them more manageable. Through a comprehensive exploration of the performance of these methods in determining parameters across diverse systems, including the van der Pol oscillator, the Rossler system, and pharmacokinetic modeling, our study revealed that the accuracy and reliability of the Nelder-Mead simplex method were consistent. The Nelder-Mead simplex algorithm emerged as a powerful tool, that consistently outperforms alternative methods in terms of root mean squared error (RMSE) and convergence reliability. Visualizations of trajectory comparisons and parameter convergence under various noise levels further emphasize the algorithm's robustness. These studies suggest that the Nelder-Mead simplex method has potential as a valuable tool for parameter estimation in chaotic dynamical systems. Our study's implications extend beyond theoretical considerations, offering promising insights for parameter estimation techniques in diverse scientific fields reliant on nonlinear system modeling.