## 5 Bifurcation and Chaotic Behavior of GMA Fractional Damping Systems
### Chaotic Properties of the System When the Fractional Order a Changes
From Figure 19, it can be seen that the GMA system bifurcates and chaos occurs with the change of external excitation amplitude f. The path to chaos when the amplitude f changes is as follows: periodic motion \(\rightarrow\) period \(2\rightarrow\) paroxysmal chaotic motion \(\rightarrow\) period \(2\rightarrow\) period \(4\rightarrow\) chaotic motion \(\rightarrow\) proposed periodic motion \(\rightarrow\) chaotic motion \(\rightarrow\) period \(3\rightarrow\) chaotic motion \(\rightarrow\) period \(4\rightarrow\) period \(2\rightarrow\) periodic motion \(\rightarrow\) jump motion \(\rightarrow\) periodic motion. When in the f \(\in\ (0,\ 0.4)\) interval, the system changes from a period motion times period bifurcation to period 2 motion, when the Lyapunov exponent in Figure 19b corresponds to less than zero. When in the f \(\in\ (0.4,\ 0.76)\) interval, the system transforms from period 2 motion to paroxysmal chaotic motion, when the Lyapunov exponent in Figure 19b changes from negative to repeatedly changing above and below the positive and negative values, and the side reaction system transforms from periodic motion to paroxysmal chaotic motion. In the interval f \(\in\ (0.76,\ 0.84)\), the system degenerates from chaotic motion to period 2 motion. In the f \(\in\ (0.84,\ 1.07)\) interval, the system changes from period 2 times bifurcation to period 4 motion, and then to chaotic motion, and when f = 0.97, the system is in the proposed periodic motion, at which time the Lyapunov exponent in Figure 19b appears as a spike. At the interval of f \(\in\ (1.07,\ 1.11)\), the system changes from chaotic motion to period 3 motion. As shown in Figure 20 (f = 1.1), the time domain waveform of the GMA system is an iterative combination of three periodic waveform signals; the closed-loop smooth curve in the phase trajectory diagram forms three mutually nested ellipses; the spectral components in the amplitude spectrum diagram are relatively single, and mainly concentrated in three frequency regions; the Poincare mapping diagram is presented as three mutually isolated attractors, at which time the corresponding Figure 19b Lyapunov exponent is negative, thus, it can be determined that the GMA system is in period 3 motion at this point. In the interval f \(\in\ (1.11,\ 1.27)\), the system changes from period 3 to chaotic motion. In the f \(\in\ (1.27,\ 1.43)\) interval, the system degenerates from chaotic motion to period 4, then to period 2 motion through the inverse period bifurcation, and then to period motion. When in the f \(\in\ (1.43,\ 1.5)\) interval, the system goes to the periodic motion after the jumping motion. To facilitate the distinction between the details of the bifurcation diagrams of the GMA system caused by the external excitation amplitude f at different fractional orders \(\alpha\). Figure 21 shows the local enlarged plots in the interval of amplitude f \(\in\ (0,\ 0.7)\) when \(\alpha\) is taken as 0.5, 0.8, 1 and 1.3, respectively. As shown in the Figure, the GMA system produces different regions of chaotic motion at different orders \(\alpha\). As the order \(\alpha\) increases, the area of the chaotic region appears significantly less in the system and enters the chaotic region later. Figure 10:1 Bifurcation diagram and Lyapunov exponent diagram of the system with f at \(\alpha\) = 1: (**a**) Bifurcation diagram; (**b**) Lyapunov exponent diagram. From the above analysis, it can be seen that with the change of external excitation amplitude f, fractional damping coefficient \(\delta\) and fractional order \(\alpha\), the output response of the GMA system will be accompanied by complex forms of motion such as periodic, apropos-periodic and chaotic motion. However, the proposed periodic and chaotic motions are unstable and difficult to predict and control accurately. As a result, uncontrollable and unpredictable chaotic phenomena can occur during the operation of the GMA system due to the changes of the excitation force and the fractional order damping coefficient. Figure 10:2 Cont.