## 1 Introduction  
Giant magnetostrictive material (GMM) is a strategic rare earth functional material [1]. The giant magnetostrictive actuator (GMA) made from GMM rods as the core component has the advantages of a high magneto-mechanical coupling coefficient, good low frequency response, high magnetostrictive coefficient, and a high output force and wide vibration response band. It has been widely used in the fields of micro-displacement drive, sensor, precision positioning control and vibration control [2; 3; 4; 5; 6; 7]. However, due to the influence of multiple coupled nonlinear factors during the operation of the GMA system [8; 9], the nonlinearity of GMA structural materials, i.e., the stress-strain relationship does not satisfy [PERSON]'s law, and the nonlinearity of the system's construction, is reflected in the nonlinearity of deformation and external forces, and the recovery force and damping are nonlinear functions of structural vibration displacement and velocity, respectively. In addition, the GMM rod also has strong hysteresis characteristics, frequency doubling characteristics, and eddy current effect nonlinear characteristics; it is easy to fall into nonlinear instability, resulting in large output error and low control accuracy of the system. At present, studies on enhancing and improving the stability of hysteresis nonlinearities in GMA systems have mainly focused on the hysteresis compensation control of the system [10; 11], and the mechanism of system dynamics instability and dynamics characteristics have not been studied in depth. The existence of nonlinear instability and even chaotic motion in GMA systems makes it difficult to accurately predict and control their output response, which seriously hinders the application of GMA in precision positioning control and active vibration isolation, etc. [PERSON] et al. [12] established a high-power GMA nonlinear mathematical model and showed that increasing the system damping coefficient helps to improve output stability, and chaotic phenomena occur when the system's stiffness is low. [PERSON] et al. [13] established a mathematical model of GMA electromagnetic coupling dynamics and showed that when the system stiffness coefficient and damping coefficient are low, it leads to system instability. [PERSON] et al. [14] established a mathematical model of GMA hysteresis nonlinearity and showed that the system has a complex motion pattern under different structural parameters. [PERSON] et al. [15] established a mathematical model containing a five times nonlinear thin magnetostrictive actuator mathematical model, and it was shown that the system exhibits rich and complex dynamical properties, such as multistability and anti-monotonicity. In order to study the structural vibrations [16; 17; 18; 19] in a system, it is essential to establish an accurate theoretical model. [PERSON] et al. [20] established a new model of beam, this new beam model is free of shear-locking for both thick and thin beams, is easy to apply in computation, and has efficiency in simulating the variable thickness beams. [PERSON] et al. [21] developed a suitable theoretical model to generate nonlinear electrostatic forces acting on the MEMS ring structure and examined the nonlinear dynamic response in the drive and sensing directions by time response, phase diagram and Poincare mapping when both input angular motion and nonlinear electrostatic forces were considered. Although some progress has been made in the study of chaos in GMA systems, there is a lack of more accurate dynamical model descriptions for GMA systems generating chaotic phenomena. By studying the relationship between fractional order calculus and integer order calculus, and the properties of fractional order calculus, it can be found that the fractional order calculus integrates the effects of historical as well as nonlocal distributions and does not reflect only local and some point properties.