## 1 Introduction  
The Duffing system, commonly utilized as a mathematical model for nonlinear oscillations, finds its applications in mechanics, circuitry, and communications [1; 2; 3]. Due to its sensitivity to initial conditions, coefficients, and inputs, the Duffing system serves as a useful instrument in comprehending intricate systems [4]. By conducting simulations and analyses on this system, we can grasp its oscillatory traits and enhance system performance. From a numerical analysis standpoint, numerous research outcomes exist concerning the Duffing model. [PERSON] et al. focus on a novel nonlinear Duffing system incorporating sequential fractional derivatives [5]. Their research delves into the complexities and dynamics of this system, highlighting its unique characteristics and behavior. By exploring the implications of such a system, they contribute valuable insights to the field of nonlinear dynamics and differential equations. [PERSON] et al. investigate the development of localized states within the context of weakly coupled Duffing oscillators in a nonlinear regime [6]. Through their study, they shed light on the emergence and properties of these localized states, offering a deeper understanding of the behavior of coupled oscillators and nonlinear systems. [PERSON] et al. present a numerical analysis of synchronization in Rayleigh-Duffing oscillators. Their study explores the dynamics of synchronization within these oscillators, illustrating the potential for coherent behavior and collective phenomena in coupled systems [7]. [PERSON] et al. explore spatiotemporal chaos in Duffing oscillators. Their research involves numerical simulations that demonstrate quasi-periodic chaotic responses within this system [8]. In this paper, we investigate the typical Duffing oscillator,focusing on its superior parameter and excitation regulation response. We also consider its ease of implementation in field-programmable gate arrays (FPGAs) circuits and its potential for fault diagnosis in rotating machinery circuits. This study serves as a foundation for future research. The work mentioned comprises a thorough investigation of the Duffing oscillator, yet most studies primarily focused on modeling aspects rather than precise solution methods. Conventional numerical methods can directly solve nonlinear models. However, errors accumulate with the increase in iteration cycles [9]. Inaccurate computational outcomes undermine research reliability. To enhance computational accuracy for nonlinear systems, [PERSON] et al. [10; 11] introduced a semi-analytical method to analyze analytical solutions of periodic motion in nonlinear systems. This method offers a comprehensive examination of nonlinear systems undergoing periodic motion. [PERSON] et al. [12; 13] extended this approach by utilizing discrete mapping techniques to analyze and predict complex periodic motion in damped, hardening Duffing oscillators subjected to periodic forcing. The combination of discrete mapping and analytical solutions offers a more detailed understanding of the dynamics of such systems. [PERSON] et al. [14] further expanded on these methods by applying a semi-analytical approach to predict bifurcation trees of a time-delay, hardening Duffing oscillator. This innovative technique provides insights into the stability and behavior of systems with time delays, enhancing our ability to analyze complex dynamics. Subsequently, the discrete implicit mapping method has been increasingly utilized to analyze even more intricate systems [15; 16; 17], leading to significant advancements in the field of nonlinear dynamics. By integrating various mapping methods, researchers are able to explore and understand the dynamics of nonlinear systems in greater depth, opening up new possibilities for research and application. Compared to analog circuits, FPGAs provide superior computational performance and parallel processing capabilities, enabling efficient computation of intricate mathematical models [18].