Semi-analytical solution and nonlinear characterization analysis of fractional-order nonlinear systems based on the time-domain minimum residual method

This study presents a novel application of the time-domain minimum residual method (TMRM) to fractional-order nonlinear systems, with particular emphasis on the fractional-order trigonometric series expansion of operators. The primary contributions of this work include the derivation of high-precision periodic solutions for fractional-order nonlinear systems, a comprehensive decay analysis of residuals, the proof of S-asymptotically T-periodic behavior in these systems, and the development of a complete methodology for applying the TMRM to obtain semi-analytical solutions for such systems. The key steps of this study are as follows: Firstly, derive the trigonometric series expansions of various types of fractional-order differential operators. Then, a thorough decay analysis of the residuals reveals that these systems exhibit S-asymptotically T-periodic behavior, rather than strictly periodic solutions. Next, numerical examples are presented, including a single-degree-of-freedom system and a coupled van der Pol-Duffing system to validate the proposed method, which demonstrates complex nonlinear phenomena such as periodic-doubling bifurcation and chaos. The results underscore the effectiveness of TMRM in obtaining high-precision solutions for fractional-order systems, offering a robust foundation for the analysis and control of nonlinear dynamics in practical applications.