## 5 Bifurcation and Chaotic Behavior of GMA Fractional Damping Systems
### Chaotic Properties of the System When the Fractional Order a Changes  
In order to further reveal the mechanism and dynamics of the effect of fractional damping on the motion of the nonlinear GMA system, the system parameters in Equation (14) are taken as \(\omega_{0}^{2}=-\) 0.3, \(\omega=1.1\), \(\gamma=0.5\), \(\beta=-\)0.4, \(2\xi\), \(=\) 0.3, \(\delta=0.2\), and f = 0.4. The numerical solution of Equation (14) is solved using the algorithm defined by the Grunwald-Letnikov fractional order derivative, and its numerical solution is simulated by simulation, and the steady-state response part is selected for the analysis of the GMA system. The fractional order \(\alpha\) is varied from 0 to 2, the step size is taken as 0.01, and the initial condition is [0, 0, 0]. Figure 11 shows the bifurcation diagram and the Lyapunov exponent of the GMA system with the fractional order as the control parameter. The bifurcation and chaos occur in the GMA system as the order changes, and the path to chaos in the GMA system when the order \(\alpha\) changes is as follows: period \(3\rightarrow\) chaotic motion \(\rightarrow\) period \(4\rightarrow\) chaotic motion \(\rightarrow\) period \(4\rightarrow\) period \(2\rightarrow\) period \(2\rightarrow\) chaotic motion \(\rightarrow\) period \(2\rightarrow\) period \(4\rightarrow\) chaotic motion. Figure 11: Bifurcation diagram and Lyapunov exponent diagram of the system when \(\alpha\) varies: (a) Bifurcation diagram; (b) Lyapunov exponent diagram. Figure 10: Amplitude and frequency characteristic curves of the system under different \(\delta\): (a) \(\delta~{}<~{}0\); (b) \(\delta~{}>~{}0\). From Figure 11, the GMA system enters into chaotic motion from period 3 when in the \(\alpha\ \in\ (0,\ 0.04)\) interval, at which time the Lyapunov exponent in Figure 11b changes from negative to positive. When in the \(\alpha\ \in\ (0.04,\ 0.22)\) interval, the GMA system degenerates from chaotic motion to period 4. When in the \(\alpha\ \in\ (0.22,\ 0.61)\) interval, the GMA system enters from period 4 to chaotic motion. As shown in Figure 12 (\(\alpha=0.41\)), the time domain waveform of the system shows that the waveform is a non-periodic irregular reciprocating motion; the phase trajectory curve fills the phase diagram space area, and the trajectory curve is an irregular multi-turn coil and cannot be closed for a long time; there are multiple frequency peaks in the amplitude spectral response that cannot be metric; the Poincare diagram shows a \"horseshoe\" attractor, and the GMA system can be judged to be in a state of chaotic motion. When in the \(\alpha\ \in\ (0.61,\ 0.65)\) interval, the GMA system degenerates into a period 4 motion via chaotic motion, as shown in Figure 13 (\(\alpha=0.63\)), and the system waveform diagram behaves as a superposition combination of four waveform signals; the system phase trajectory curve is a closed coil of four multi-turn phase sets; the spectral components in the amplitude spectrogram are mainly concentrated in the four frequency regions of 0.28, 0.57, 1.1 and 1.69; and four discrete attractors appear in the Poincare diagram, when the Lyapunov exponent corresponding to Figure 11b also changes from positive to negative in this local range, so that the system is in a period of 4 motion.