## 4 Numerical Examples and Illustrations
### Bifurcation and Stability Analysis
#### 4.2.2 Out-of-Plane Excitations  
Provided that the external excitation is transferred from the in-plane to the out-of-plane, the relative dynamical behaviors are investigated. Figure 8 presents the force-response curves when \(\sigma_{1}=-0.1\) and \(\sigma_{2}=0\), and the out-of-plane excitation amplitude \((f_{w})\) is selected as the control parameter. As shown in Figure 8, there is no stable equilibrium solution in the middle of the resonant region. In this region, the suspended cable's responses are expected to be either periodic or chaotic modulation motions. If the out-of-plane excitation amplitude \((f_{w})\) increases from zero, only the out-of-plane motions could be observed. The steady-state solutions are stable until reaching the pitchfork bifurcation PF\({}_{1}\). If \(f_{w}\) increases further, the branch gains stability again at the Hopf bifurcation HB, where the response branches as to the in-plane motion emerge. Along these branches, there is strong coupling between the in-plane and out-of-plane motions. As to some small excitation amplitudes, there are some large response amplitudes due to the one-to-one internal resonances. It is noticed that there are two saddle nodes, and the non-linear internal resonant responses are stable between these two bifurcations while it is unstable in other regions. Within the internal resonant region, the internal excited amplitudes \(a_{v}\) appear to be more sensitive to temperature changes than the directly excited ones \(a_{w}\). Otherwise, when sweeping the out-of-plane excitation amplitude \((f_{w})\) backward from 0.008, the out-of-plane (in-plane) response amplitude \(a_{w}\) (\(a_{v}\)) decreases (increases) and passes through a Hopf bifurcation in the internal resonant region. Then, a stable solution starts to fade away and passes through a pitch-fork bifurcation. As illustrated before, the response amplitude decreases (increases) in the cooling (warming) condition, regardless of the direct or the internal resonant case. Interestingly, in the large excitation amplitude  
Figure 8: Force-response amplitude curves with thermal effects when \(\sigma_{1}=-0.1\) and \(\sigma_{2}=0\): (a) in-plane; (b) out-of-plane. region (e.g., \(f_{w}>0.006\)), the internal excitation amplitude \(a_{v}\) decreases (increases) in the warming (cooling) condition on the contrary. Both the perturbations and numerical solutions have verified this phenomenon. Here, both the static and dynamic bifurcations could be observed. Due to the temperature effects, three bifurcations (PF, SN, and HB) are exhibited in a smaller excitation amplitude \(f_{w}\) in the warming condition. Finally, Figure 9 illustrates the representative frequency response curves when the external excitation is applied on the out-of-plane direction. The corresponding bifurcations are marked, and the stability of the steady-state solutions is described clearly. These curves show the response amplitudes \(a_{v}\) and \(a_{w}\) as a function of the external detuning parameter \(\sigma_{1}\), for fixed values of the out-of-plane excitation amplitude \(f_{w}=0.002\) and the internal detuning parameter \(\sigma_{2}=0\). Specifically, Figure 9 reveals that for \(\sigma_{1}>0\), only the single-mode solution is observed, and the in-plane response amplitude \(a_{v}\) remains at zero. However, for \(\sigma_{1}<-0.2\), a strong interaction between the in-plane and out-of-plane modes is expected, and an internal resonance may occur, leading to a sudden appearance in the response amplitude of the in-plane mode due to energy transfer from the out-of-plane mode. In the internal resonant region, further sweeping of the external detuning parameter causes a decrease in the in-plane and out-of-plane responses until they experience a saddle-node jump to the trivial solutions. It is observed that increasing temperature significantly reduces the response amplitudes.