# Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter
## 4 Parametric Uncertainty Analysis
### Influence of the Electromechanical Coupling Coefficient  
The electromechanical coupling coefficient \(\alpha\) plays a key role in the behavior of TEH, so we consider \(\alpha\) as the bifurcation parameter. Let \(\lambda=-4.5\), \(\tilde{\lambda}=-4.5\), \(\mu=2.5\), \(\zeta\)\(=0.2\), \(A=1.2\), \(\omega=0.9\), \(\beta=0.5\) and \(\sigma=0.01\), the multivalued bifurcation diagrams and corresponding TLE of DR and EMR are shown in Figure 9(a1,a2), respectively. When \(\alpha\approx 0.77\), DR keeps the periodic motion, but EMR enters into transient chaotic motion on account of the influence of the uncertain parameter. After \(\alpha\) decreases to the critical value about \(\alpha\approx 0.52\), DR undergoes several period doubling bifurcation cascades and finally becomes chaotic in Figure 9(a1), where the corresponding TLE of DR gradually changes from negative to positive. Instead, the crisis from periodic to chaotic motion can be clearly observed for EMR and the TLE of EMR suddenly changes from negative to positive within \(0.46\leq\alpha\leq 0.52\). Besides, the harvester under the initial condition IC-3 leaves the chaotic region firstly, which can be seen more clearly in Figure 9(b2).  
Especially, when \(\alpha=0.51\), DR keeps 2T-period motion for different initial conditions in Figure 10a. As we take the stochastic intensity \(\sigma\)\(=0.01\) in Figure 10b, there is almost no distinction between DR and EMR with the initial condition IC-4, but EMR with the initial condition IC-3 becomes chaotic. This indicates that the dynamic behavior of TEH is sensitive to the initial conditions around the given parameters. Furthermore, as \(\sigma\) is increased to  
Figure 8: The TLE with the variation of damping coefficient \(\zeta\) under different intensities; (**a**) is obtained under the initial condition IC-1; (**b**) is obtained under the initial condition IC-2.  
Figure 9: Bifurcation diagrams and the TLE with the variation of electromechanical coupling coefficient \(\alpha\). (**a1,a2**) Multivalued bifurcation diagram of DR and EMR (\(\sigma=0.01\)); (**b1,b2**) Multivalued TLE of DR and EMR (\(\sigma=0.01\)). The blue dots and red dots are obtained under the initial conditions \([x_{0},x_{1},x_{2},x_{3},x_{4}]=[-1.0,0,0,0,0]\), \([x_{0},x_{1},x_{2},x_{3},\)\(x_{4}]=[-1.0,0,0,0,0]\), \([x_{0},x_{1},x_{2},x_{3},\  
0.1 in Figure 10c, the system responses of harvester enter into the chaotic motion under two different initial conditions, which make the output voltages unstable. In addition, the RMS voltages of DR and EMR are calculated in Figure 10d, where EMR(\(\sigma=0.01\)) under the initial condition IC-3 generates 11.6% smaller RMS voltage and EMR(\(\sigma=0.1\)) generates 12.4% smaller RMS voltage.  
From the above analysis, we find that the stochastic dynamics phenomenon of the TEH can be affected more by the bigger intensity of uncertain parameter. For this reason, and then taking the values of intensity \(\sigma\) as 0.0, 0.01, 0.05 and 0.1, the bifurcation diagrams and RMS voltage under the initial condition IC-4 within \(0.5<\alpha<0.58\) are investigated. It can be observed that the critical interval of stochastic period-doubling bifurcation cascade shifts ahead with the increasing of \(\sigma\) in Figure 11a. Moreover, the RMS voltage within \(\alpha<0.52\) is smaller than that in the case of deterministic form in Figure 11b. In addition, when \(0.52<\alpha<0.58\), these RMS values under the stochastic intensities \(\sigma=0.01,0.05\) and 0.1 do not change considerably.  
Figure 11: EMR for \(0.5<\alpha<0.58\) under the stochastic intensities \(\sigma=0.0\), \(\sigma=0.01\), \(\sigma=0.05\) and \(\sigma=0.1\). (**a**) Bifurcation diagrams; (**b**) RMS voltage.  
Figure 10: DR and EMR for \(\alpha\)=0.51. (**a**) Phase portraits of DR; (**b**) Phase portraits of EMR (\(\sigma=0.01\)); (**c**) Phase portraits of EMR (\(\sigma=0.1\)); (**d**) RMS voltage of DR and EMR. The blue dots and red dots are obtained under the initial conditions (IC-3) and (IC-4), respectively.