## 1 Introduction
It is this long memory property and global nature that the fractional order calculus more aptly describes; the damping properties of viscoelastic materials [22; 23; 24; 25; 26]. [PERSON] et al. [22] considered nonlinear Duffing oscillators in the presence of fractional damping and showed that oscillations exist in the fractional order derivatives and that their amplitude and asymptotic time change abruptly with small changes in the fractional parameter. [PERSON] et al. [23] proposed a model of oblique crack dynamics based on fractional order damping and experimentally verified that the proposed fractional order model compensates for the deficiencies of the integer order model. [PERSON] et al. [24] investigated the distribution of poles and the dynamic response to several excitations for different values of the parameters of the fractional damping model, and showed that the fractional model differs from the conventional viscous damping system by observing specific behaviors that cannot be replicated by the classical behavior. [PERSON] et al. [25] studied the dynamics of fractional-order thin magnetostrictive actuators and showed that, compared to integer order systems, fractional order systems have more complex chaotic properties and extreme multistability. [PERSON] et al. [26] studied a frictionally damped system with fractional order derivative damping under Gaussian white noise excitation, analyzed the stochastic bifurcation phenomenon caused by system parameters such as fractional order and fractional order coefficients, by setting critical conditions, and finally gave the differences between the bifurcation regions of the fractional order model and the integer order model. So far, most of the studies on GMA dynamical systems are based on the mechanical behavior described by integer-order calculus, which has some limitations for studying the stability of the system. The above studies show that it is important to consider fractional-order calculus in the model of dynamical systems and that GMM is a viscoelastic material with memory [25; 27], and for systems with viscoelastic materials, the damping term depends not only on the current time and position but also on the previous state, and the memory properties of the viscoelastic material are characterized by the power-law kernel function associated with the fractional-order derivatives. The use of the fractional order model to describe the damping characteristics of GMA systems is essential to provide theoretical guidance for the stability design of GMA structures, not only to further reveal the intrinsic mechanism during the nonlinear motion of the system, but also to study the stability of GMA systems from a new perspective. In this paper, based on the working mechanism of the GMA system based on previous research, the GMA dynamical system model is extended to fractional order based on fractional order calculus theory, the dynamical equations of nonlinear GMA system containing fractional damping terms are established, and the nonlinear vibration response of the system is analyzed based on the averaging method; the numerical solution of the system is solved using the algorithm defined by Grunwald-Letnikov fractional order derivatives, and the effects of fractional damping coefficient \(\delta\), and different external excitation amplitudes f, and fractional order \(\alpha\), on the main resonant output response of the GMA system are studied through MATLAB numerical simulations. By varying the fractional order \(\alpha\) and the magnitude of fractional damping coefficient \(\delta\) and excitation amplitude \(f\) at different orders, the bifurcation and chaos characteristics of the GMA system are investigated from qualitative and quantitative perspectives using bifurcation diagrams, Lyapunov exponent diagrams, phase trajectory diagrams, amplitude spectrum diagrams, Poincare diagrams and time domain waveform diagrams.