## 6 Conclusions  
In this paper, according to the working principle of the GMA system, the fractional order calculus is invoked to describe the damping characteristics of the system, the dynamics equations of the GMA system containing fractional damping are established, and the nonlinear vibration response of the GMA system is studied; the numerical solution of the system is solved by using the algorithm defined by Grunwald-Letnikov fractional order derivatives, and the effects of fractional order \(\alpha\) as well as the external excitation amplitude f and fractional damping coefficient \(\delta\) on the bifurcation and chaotic characteristics of the GMA system at different orders are studied from qualitative and quantitative perspectives through MATLAB simulations, and the results are shown as follows:  
1. Compared with the integer-order GMA system, the fractional-order GMA system can more accurately express the physical change process of the system and is closer to the actual situation.
2. When \(0\;<\;\alpha\;<\;1\), with the increase of fractional order \(\alpha\), the resonance frequency and the main resonance peak of the system tend to decrease, and the lag region also decreases; when \(1\;<\;\alpha\;<\;2\), the main resonance peak of the system tends to increase. With the increase of excitation frequency, the system amplitude tends to increase and then decrease, along with the increase of excitation amplitude, the resonance curve shows the phenomenon of outward expansion, the skeleton curve deflects to the right, the hard nonlinearity, the resonance region and the instability region gradually widen, and the hysteresis phenomenon appears. With the increase of absolute value of damping coefficient \(\delta\), the resonance peak of the system decreases accordingly, the resonance curve inwardly shrinks, and the resonance region and the instability region gradually become narrower.
3. The order \(\alpha\) has a more significant effect on the dynamical behavior of the GMA system. When the order \(\alpha\) is less than 0.61 and greater than 1.88, complex dynamical behaviors such as paroxysmal chaos and chaotic motion appear in the GMA system.
4. At different orders \(\alpha\), chaos exists in the system, as the external excitation amplitude f and damping coefficient \(\delta\) vary; at \(0\;<\;\alpha\;<\;1\), the fractional order GMA system enters into chaotic motion earlier than the integer order GMA system, showing richer and different instability conditions of the dynamical system.  
Conceptualization, Q.M.; methodology, Q.M.; software, X.F.; formal analysis, Q.M.; investigation, Q.M.; resources, J.W. and H.Y.; data curation, J.Y.; writing--original draft preparation, Q.M.; writing--review and editing, Q.M.; supervision, H.Y.; project administration, H.Y.; funding acquisition, H.Y. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the Inner Mongolia Natural Science Foundation Project, grant number 2020 LH05023.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Some or all of the data, models, or code generated or used during the study are available from the corresponding author on reasonable request.  
**Acknowledgments:** The authors gratefully acknowledge the support of the Inner Mongolia Natural Science Foundation Project (2020 LH05023).  
**Conflicts of Interest:** The authors declare no conflict of interest.