## 5 Bifurcation and Chaotic Behavior of GMA Fractional Damping Systems
### Chaotic Properties of the System When the Fractional Order a Changes
When in the \(\alpha\ \in\ (0.65,\ 0.94)\) interval, the GMA system bifurcates into the period 2 motion via the period 4 inverse times period, as shown in Figure 14 (\(\alpha=0.83\)), the time domain waveform of the GMA system is an iterative combination of two period waveform signals; the closed-loop smooth curve in the phase trajectory diagram forms two mutually nested ellipses; the spectral components in the amplitude spectrum diagram are relatively single, mainly concentrated in the frequency region of 0.57 and 1.1; the Poincare diagram is presented as two mutually isolated attractors, at which time the Lyapunov exponent  
Figure 12: Nonlinear response of the GMA system at \(\alpha=0.41\): (**a**) Time domain waveform diagram; (**b**) Phase trajectory curve; (**c**) Amplitude spectrum; (**d**) Poincaré diagram. corresponding to Figure 11b is less than 0, and thus, it can be determined that the GMA system is in period 2 motion at this point. When in the \(\alpha\in(0.94,1.7)\) interval, the GMA system degenerates into stable periodic motion by a period 2 motion inverse times period bifurcation, as shown in Figure 15 (\(\alpha=0.97\)), the time domain waveform of the GMA system shows a single and regular reciprocal motion; the closed-loop smooth curve in the phase trajectory forms an ellipse; the frequency component in the amplitude spectrum is relatively single, mainly concentrated in the frequency region of 1.1; the Poincare diagram shows a single isolated attractor, which corresponds to the Lyapunov index greater than 0 in Figure 11b, and thus, it can be concluded that the GMA system has a typical stable periodic motion at this point. Figure 14: _Cont._  
Figure 13: Nonlinear response of the GMA system at \(\alpha=0.63\): (**a**) Time domain waveform diagram; (**b**) Phase trajectory curve; (**c**) Amplitude spectrum; (**d**) Poincaré diagram. When in the \(\alpha\in(1.7,\,1.87)\) interval, the GMA system enters period 2 motion through the period motion times period bifurcation. When in the \(\alpha\in(1.87,\,2)\) interval, the GMA system enters the chaotic motion after bifurcating into the period 4 motion through the period 2 motion times period, which corresponds to the change of Lyapunov exponent from negative to positive in Figure 11b. Figure 14: Nonlinear response of the GMA system at \(\alpha=0.83\): **(a)** Time domain waveform diagram; **(b)** Phase trajectory curve; **(c)** Amplitude spectrum; **(d)** Poincaré diagram. Figure 15: Nonlinear response of the GMA system at \(\alpha=0.97\): **(a)** Time domain waveform diagram; **(b)** Phase trajectory curve; **(c)** Amplitude spectrum; **(d)** Poincaré diagram. Chaotic Characteristics of the GMA System When the Damping Coefficient \(\delta\) Varies at Different Fractional Order \(\alpha\)  
To further study the influence law of fractional order \(\alpha\) on the GMA system, the fractional damping coefficient \(\delta\) is taken as the control parameter, and different orders \(\alpha\), respectively, are selected to study the dynamic behavior of the GMA system. When \(\alpha=1\) is the traditional integer order GMA system, in order to facilitate the observation of the relationship between the global motion form of the system and the damping coefficient \(\delta\ \in\ (-0.2,\ 0.2)\), the damping coefficient \(\delta\) is selected, the step size 0.005 is chosen, and the initial condition is \([0,0,0]\), and the bifurcation diagram and Lyapunov exponent diagram of the GMA system are obtained, as shown in Figure 16.  
From Figure 16, it can be seen that the GMA system appears as bifurcation and chaos with the change of fractional damping coefficient \(\delta\). The path of the GMA system to chaos when the damping coefficient \(\delta\) changes is: paroxysmal chaotic motion \(\rightarrow\) period \(2\rightarrow\) paroxysmal chaotic motion \(\rightarrow\) period motion \(\rightarrow\) jump motion \(\rightarrow\) period \(2\rightarrow\) period motion.