# Response Statistics of a Shape Memory Alloy Oscillator with Random Excitation
## 4 Numerical Verification  
To verify the effectiveness of the obtained analytical results, we compared the differences between the numerical results and the approximately analytical ones through numerical simulations at different temperatures \(\theta\) in the next step. In Figure 3, we consider the deterministic case, namely, \(\overline{h}=0\); here, we select the temperature \(\theta=1.9\) and \(\theta=3.5\) for the sake of verification. Clearly, the approximately analytical results are in good agreement with the numerical simulation results, and multi-value response area can be observed. In this area, the steady-state responses of the SMA oscillator system have two possible states (i.e., bistable behavior), which strongly depend on the initial conditions of the system. At the same time, we found that there is a jump phenomenon in the SMA oscillation system. In Figure 3a, when both martensite and austenite exist, as indicated by the arrow, when the frequency reached a certain critical value from Region 2 to Region 3 with the increase of the external excitation frequency \(\Omega\), a sudden jump occurred in the frequency-amplitude response. If the external excitation frequency \(\Omega\) gradually decreased, when the frequency value \(\Omega\) varied from Region 2 to Region 1, a similarly sudden jump can be observed. This phenomenon can be called a bifurcation, essentially a change in the nature of the solution. The same jump phenomenon occurred when the austenite was stable, that is, \(\theta=3.5\). For the deterministic case, we also considered the influences of the temperature \(\theta\) and the amplitude \(f\) of the external excitation on the amplitude of the SMA system (11). In Figure 4, we find that as the temperature \(\theta\) increased, the multi-value response area shifted to the right and the area became smaller. In contrast, the amplitude of the system response and multiple-value response area will become larger with the increase of \(f\). same. For the temperature parameter \(\theta\), as the frequency \(\Omega\) increased and reached a certain threshold, the amplitude of the system gradually increased, and the solution of the system changed at the same time, which indicates that the frequency \(\Omega\) can be regarded as a bifurcation parameter of the system. In addition to the above, we can see from Figure 6a, as the temperature parameter \(\theta\) increased, we found that the frequency of bifurcation point gradually increased, namely, the lower the temperature, the earlier bifurcation will occur. This phenomenon is the same in Figure 6b. In fact, a reversed phenomenon can be found from Figures 3-6. As is shown in Figure 5, we can see that there was only one stable solution in Region 1, then two stable solutions and one unstable solution came later in Region 2, and finally it again came to one stable solution in Region 3. Thus, the switch phenomenon will occur when the frequency \(\Omega\) changes from Region 2 to Region 1 and from Region 2 to Region 3. This phenomenon can be regarded as bifurcation, that is to say, in this bistable area, the noise may cause stochastic switching in the SMA system, and then we consider the influence of random noise in the next. Here, the stochastic switch behaviors of the SMA oscillator system (11) subject to narrow-band random excitation were investigated to observe the effects of the random noise. Monte Carlo simulation was used to demonstrate the steady-state probability density functions (PDFs) and the responses. From here, we consider the effects of the noise intensity \(h\) on the random responses of the SMA oscillator system (13). Figure 7 presents the effects of the noise intensity \(h\) on the steady-state PDF \(P(a)\) of the amplitude \(a(\tau)=\sqrt{x^{2}(\tau)+\left[\dot{x}(\tau)/\Omega\right]^{2}}\), with the fixed system parameters \(f=0.6\), \(\Omega=2.0\). As shown in Figure 7a-d, when adding a small noise, we can see that the noise will cause the appearance of a peak change in the steady-state PDF.