## 3 Periodic Motions to Chaos in Oscillator
### Periodic Motions of Duffing Oscillator with Excitation Amplitude  
Given system parameters \(\Omega\) = 2.4 and \(\delta\) = 0.3, Figure 5 shows the predicted trends of node displacement \(x\) and velocity \(y\) in relation to changes in excitation amplitude \(Q_{0}\). In Figure 5, the predictions of a bifurcation tree completely transitioning from period-1 to chaos are illustrated. The period-1, 2, 4, and 3 motions are labeled P1, P2, P4, and P3, respectively. The period-2 motion arises from the period-doubling bifurcation of period-1 motion, and period-4 motion results from the period-doubling bifurcation of period-2 motion.  
Period-1 motion is found within the range of \(Q_{0}\in\) (0, 9.0). Stable period-1 occurs when \(Q_{0}\in\) (0, 0.79), (4.23, 7.85), (2.15, 9.0), while unstable period-1 motion takes place within \(Q_{0}\in\) (0.79, 4.23), (2.15, 7.85), (0, 5.10). As shown in Figure 5, there are many interacting periodic orbits in the bifurcation tree. The corresponding saddle-node bifurcations in the multiple solutions are at \(Q_{0}\) = 4.23, 5.10, where there exists a jumping phenomenon.  
Figure 4: Phase trajectories and waveforms: (a) stable periodic motions; period-1 with \(\Omega\) = 1.73; (b) stable periodic motions; period-2 with \(\Omega\) = 2.9; (c) stable periodic motions; period-3 with \(\Omega\) = 3.309; (d) stable periodic motions; period-4 with \(\Omega\) = 2.8606.  
Figure 5: A global view of bifurcation trees varying with excitation excitation amplitude \(Q_{0}\): (a) node displacement \(x\); (b) node velocity \(y\).  
Furthermore, period-doubling bifurcations are observed at \(Q_{0}=4.23\), 5.10. At \(Q_{0}=5.10\), through a saddle-node bifurcation, the stable periodic orbit splits while the excitation amplitude decreases, but the periodic motion is still period-1. The stable period-2 motion takes place within the range of \(Q_{0}\in(0.63,0.83)\), (4.04, 4.23), while the unstable range is within \(Q_{0}\in(0.82,4.04)\). At the bifurcation points \(Q_{0}=0.82,4.04\), a shift from stable to unstable motion occurs and then goes back to stable motion, signifying a transformation in the behavior of the system. Additionally, at \(Q_{0}=4.04\), the system transitions from period-2 motion to period-4 motion, where the phase trajectory encompasses four circles within the range of \(Q_{0}\in(0.82,4.04)\). The stable period-4 motion is observed within \(Q_{0}\in(0.82,0.87)\), \((4.01,4.04)\), while unstable motion falls within the interval of \(Q_{0}\in(0.87,\) 4.01). These findings highlight the complex dynamics and transitions exhibited by the system under varying values of \(Q_{0}\).  
The movements of period-3 and 6 are depicted in Figure 6. Period-3 motion occurs within the interval of \(Q_{0}\in(1.02,\) 6.12). Period-3 exhibits stability within the range of \(Q_{0}\in(1.02,\) 1.64), (5.86, 6.12), while instability is observed in the intervals of \(Q_{0}\in(1.02,\) 6.12), (1.44, 5.86), (1.64, 5.86). There are three limit circles associated with the periodic trajectory of period-3. Five saddle-node bifurcations in period-3 are located at \(Q_{0}=1.02,\) 1.44, 5.8, 5.86, 6.12, respectively. The period-doubling bifurcation points are \(Q_{0}=1.64,\) 1.69. Furthermore, at \(Q_{0}=1.44\), the Duffing model is still in period-3 motion, although the periodic orbit splits into six lines here. The system enters period-6 motion from period-3 motion with an excitation amplitude decrease at \(Q_{0}=2.80\). The stable period-6 motion range is \(Q_{0}\in(1.64,\) 1.69), while the range for unstable motion is \(Q_{0}\in(1.69,2.80)\).