## 3 Periodic Motions to Chaos in Oscillator  
### Periodic Motions of Duffing Oscillator with Excitation Frequency  
Setting system parameters \(\delta=0.3\), \(Q_{0}=2.8\), the bifurcation analysis can be obtained via the discrete mapping method. The global bifurcation diagram of the Duffing oscillator is shown in Figure 1. The excitation frequency is calculated from 0.55 to 5. \"SN\" and \"PD\", respectively, represent saddle-node and period-doubling bifurcations. The stable motion is depicted by the black solid line, whereas the red dashed line illustrates unstable solutions that are unattainable through conventional numerical methods. Through the implicit mapping algorithm, a comprehensive overview of both stable and unstable periods of the Duffing oscillator is presented. In implicit mapping method, instability refers to higher-order period existence. And the eigenvalues provide the bifurcation characteristics in Duffing oscillator motion. From Figure 1, the complete bifurcation tree from period-1 to chaos could be predicted by analyzing the periodic motions of the Duffing oscillator. Period-1, 2, 4, and period-3, 6 are labeled P1, P2, P4, and P3, P6, respectively. To clearly describe the periodic motion, the bifurcation trees and points of periods 1, 2, and 4 are depicted in Figure 2. From Figure 2, it can be observed that period-1 motion exists in the range of \(\Omega\in(0.55,5.00)\). The stable period-1 motion is \(\Omega\in(0.55,0.5577)\), \((1.07,2.76)\), \((1.54,1.79)\), \((3.16,5.00)\), while the unstable period-1 motion takes place in the range of \(\Omega\in(0.5577,1.07)\), \((1.54,2.76)\), \((1.71,5.00)\). There are many interlaced periodic lines indicating different periodic behaviors of the system. The corresponding saddle-node bifurcations in the multiple solution ranges occur at \(\Omega=0.5577\), \(1.55\), \(1.71\), \(1.79\), \(2.76\), where there exists a jumping phenomenon. Additionally, at \(\Omega=3.16\), a period-doubling bifurcation leads to a transition from period-1 to period-2 motion with excitation frequency decrease. The periodic line of period-2 motion has two limit circles. The stable motion range of period-2 is \(\Omega\in(1.79,1.80)\), \((2.89,3.16)\), while the unstable motion range is \(\Omega\in(1.80,2.89)\). The system has period-doubling bifurcations at \(\Omega=1.80\) and \(\Omega=2.89\), respectively. Through the two period-doubling bifurcation points, period-2 undergoes a transition from stable to unstable and then back to stable. At \(\Omega=2.89\), period-doubling bifurcation appears and then the Duffing enters period-4 motion from period-2 motions with excitation frequency decrease. Period-4 motion is characterized by the presence of four limit circles within the range of \(\Omega\in(1.80,2.89)\), with period-doubling bifurcations occurring at \(\Omega=2.85\) and \(\Omega=2.89\). The system exhibits a transition in stability as it moves through these bifurcation points, showcasing the complexity and dynamics of the periodic motion in the system. The bifurcation trees and points of period-3 to period-6 motion are depicted in Figure 3. Period-3 motion can be observed in \(\Omega\in(1.647,3.596)\). The stability of period-3 motion is categorized into two areas: \(\Omega\in(1.647,1.65)\) and \((2.79,3.596)\) for the stable motion; \((1.65,2.79)\) and \((1.647,3.596)\) for the unstable motion. In period-3 motion, there are mainly three mutually influencing periodic lines. The occurrence of four saddle-node bifurcations in  
Figure 1: A global view of bifurcation trees varying with excitation frequency \(\Omega\): **(a)** node displacement \(x_{i}\) (**b**) node velocity \(y\). Figure 2: A zoomed view of bifurcation trees (period-1 motions to period-4 to chaos): **(a)** node displacement \(x_{i}\) (**b**) node velocity \(y\). period-3 motion happens at \(\Omega\) = 1.647, 1.65, 2.92, 3.595. Additionally, at \(\Omega\) = 2.92, it is still period-3 motion despite there being six periodic orbits.