## 4 Numerical Examples and Illustrations
### Bifurcation and Stability Analysis
#### 4.2.1 In-Plane Excitations
As the excitation amplitude \(f_{v}\) increases, the in-plane response \(a_{v}\) also increases until reaching the first pitch-fork bifurcation PF\({}_{1}\). A jump phenomenon is observed, and the jump point comes earlier in the warming condition. Similarly, the response amplitude in the warming condition is larger than the one in the cooling condition. As to the coupled motions, when the forcing amplitude \(f_{v}\) is dropped from 0.004, both the in-plane and out-of-plane response amplitudes are decreased accordingly until reaching the saddle-node bifurcation SN\({}_{1}\). Specifically, both the in-plane and out-of-plane response amplitudes decrease when the temperature drops from 40 \({}^{\circ}\)C to \(-40\)\({}^{\circ}\)C. It is observed that the temperature effects on the saddle-node bifurcation are not apparent. Although there are some differences between the numerical solutions and perturbation ones in the coupled motions, the quantitative changes induced by thermal effects are confirmed. Figure 3: Force-response amplitude curves with thermal effects when \(\sigma_{1}=0.15\) and \(\sigma_{2}=0\). For certain initial conditions, small excitation amplitudes \(f_{v}\) can lead to extremely large vibration amplitudes due to internal resonances. These resonant responses are highly sensitive to temperature variations and resulting in energy transfer from in-plane to out-of-plane modes. Direct numerical simulations and perturbation methods demonstrate good agreement in this case, although some discrepancies are also observed. Figure 5 presents typical frequency-response curves when \(f_{v}=0.0015\) and \(\sigma_{2}=0\), exhibiting the temperature effects on the dynamic behaviors through bifurcation diagrams obtained using the external detuning parameter \(\sigma_{1}\) as a control factor. In the non-resonant region, numerical simulations yield a small steady-state solution, which is used as the starting point for a pseudo-arclength continuation to construct the rest curves. As the excitation detuning parameter \(\sigma_{1}\) decreases from 0.3, only in-plane motions are observed in the small response region, with out-of-plane motions remaining at zero. The frequency-response curves bend to the left side, indicating a softening-type response of the non-linear system. Notably, as the temperature drops from 40 \({}^{\circ}\)C to \(-\)40 \({}^{\circ}\)C, the curves show an even greater leftward bend. Two types of bifurcations, saddle-node bifurcation SN\({}_{1}\) and pitch-fork bifurcation PF\({}_{1}\), are observed in the force-response curves, with the pitch-fork bifurcation occurring at a smaller detuning parameter \(\sigma_{1}\) in the warming condition and the saddle-node bifurcation occurring at a larger detuning parameter \(\sigma_{1}\) in the same condition. As a result, the resonant region between the two bifurcations becomes larger at higher temperatures. These internal resonant phenomena can easily be excited under very small external excitation amplitudes and become more sensitive to temperature variations. From a non-linear dynamics perspective, these observations highlight the importance of considering internal resonances and their sensitivity to temperature in the study of vibrational behavior. Exciting a non-linear system near Hopf bifurcations can cause the system's natural frequencies to change due to thermal effects, resulting in sudden and significant changes in  
Figure 4: Force-response amplitude curves with thermal effects when \(\sigma_{1}=-0.15\) and \(\sigma_{2}=0\): (**a**) in-plane; (**b**) out-of-plane. Figure 5: Frequency-response amplitude curves with thermal effects when \(f_{v}=0.0015\) and \(\sigma_{2}=0\): (**a**) in-plane; (**b**) out-of-plane. and a main frequency could be observed in the frequency spectrum. In the warming condition, the previous periodic motion bifurcated into chaotic motions. The time history curves presented in Figure 6 indicate that the system's response is chaotic, a conclusion supported by the corresponding phase plane, frequency spectra, and Poincar'e sections.