## 5 Bifurcation and Chaotic Behavior of GMA Fractional Damping Systems
### Chaotic Properties of the System When the Fractional Order a Changes
When in the \(\delta\ \in\ (\ -0.2,\ -0.12)\) interval, the system degenerates from paroxysmal chaotic motion into period 2 motion, at which time the Lyapunov exponent corresponding to Figure 16b changes repeatedly from positive to negative up and down to negative values, and the side-response system transforms from chaotic motion to periodic motion. As shown in Figure 17 (\(\delta=-0.18\)), the waveform of the system shows a reciprocal motion with neither regularity nor periodicity; its phase trajectory curve is a long-term unclosable, irregular multi-turned coil; a continuous wider amplitude spectrum appears in the amplitude spectrum; the Poincare mapping is neither a finite point nor a closed curve, and the Lyapunov index corresponding to Figure 16b is positive at this time. All the above indicators show that the GMA system is in a typical chaotic motion at this time. When in the interval \(\delta\ \in\ (-0.12,\ -0.01)\), the system performs period 2 motion, and the Lyapunov exponent in Figure 16b is less than 0. When in the interval \(\delta\ \in\ (\ -0.01,\ 0.055)\), the system enters from the period 2 motion into the paroxysmal chaotic motion. When in the interval \(\delta\ \in\ (\ 0.055,\ 0.085)\), the system degenerates from paroxysmal chaotic motion to periodic motion, and the Lyapunov exponent in Figure 16b changes from positive to negative at this time. At \(\delta=0.09\), after jumping motion times, the period bifurcation changes into the period 2 motion. When in the \(\delta\ \in\ (0.09,\ 0.2)\) interval, the system degenerates from the period 2 motion inverted times period bifurcation to the periodic motion, and then the GMA system enters into the periodic motion. Figure 16: Bifurcation diagram and Lyapunov exponent diagram of the system with \(\delta\) at \(\alpha=1\): (**a**) Bifurcation diagram; (**b**) Lyapunov exponent diagram. To facilitate the distinction between the details of the bifurcation diagrams of the GMA system caused by the fractional damping coefficient \(\delta\) at different fractional orders \(\alpha\). Figure 18 shows the local enlarged plots in the interval of damping coefficient \(\delta\in(-0.3,\ -0.15)\) when the order \(\alpha\) is taken as 0.55, 0.75, 1, 1.25 and 1.45, respectively. It is easy to see from the plots that the GMA system has similar dynamical behaviors at different orders \(\alpha\) but the regions where chaotic motions are generated are different. At \(0<\ \alpha<\ 1\), the area of the chaotic region in the system decreases significantly with the increase of order \(\alpha\) and enters the chaotic region later; at \(1<\ \alpha<\ 2\), the area of the chaotic region in the system increases significantly with the increase of order \(\alpha\) and enters the chaotic region earlier. Chaotic Characteristics of the GMA System When the Excitation Amplitude f Varies at Different Fractional Orders \(\alpha\)  
Similarly, the external excitation amplitude f is taken as the control parameter, and different orders \(\alpha\) are chosen to study the dynamic behavior of the GMA system, respectively. To facilitate the observation of the relationship between the global motion form of the system and the amplitude f, the conventional GMA system at \(\alpha=1\) is still chosen first, and the external excitation amplitude f \(\in(0,\ 1.5)\) is chosen with a step size of 0.01 and initial conditions of [0, 0, 0], and the bifurcation diagram and Lyapunov exponent diagram of the GMA system are obtained, as shown in Figure 19.  
Figure 10:0 Nonlinear response of the GMA system at \(\delta=-0.18\): (**a**) Time domain waveform diagram; (**b**) Phase trajectory curve; (**c**) Amplitude spectrum; (**d**) Poincaré diagram.