# Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter
## 4 Parametric Uncertainty Analysis
### Influence of the Damping Coefficient  
In system (3), the parameters are chosen as \(\lambda=-4.5\), \(\mu=2.5\), \(\alpha\)\(=0.9\), \(A=2.8\), \(\omega=0.8\) and \(\beta=0.5\). When the damping coefficient \(\zeta\) varies from 1.5 to 0.0, the multivalued bifurcation diagram and the top Lyapunov exponent (TLE) of DR are obtained by selecting different initial conditions, as shown in Figure 4(a1,b1). When \(\zeta=1.5\), it can be seen that there exist two 1T-period attractors. After \(\zeta\) decreases to the critical value about \(\zeta=1.03\), the system responses go to chaotic motion through the period doubling bifurcation cascade. Moreover, as \(\zeta\) decreases from 1.03 to 0.04, the system undergoes several period doubling bifurcation cascades, crises and periodic motions. Furthermore, we find a typical crisis near \(\zeta=0.04\), where the chaotic attractor is suddenly destroyed and replaced by a newborn regular 1T-period attractor. Based on the above deterministic analysis, we take \(\bar{\lambda}=4.5\), \(\sigma=0.01\) in the system (19) to investigate the influence of uncertain parameter. The multivalued bifurcation diagram and corresponding TLE of EMR are plotted in Figure 4(a2,b2), from which we can see that the global stochastic dynamics of system (19) is similar to the deterministic case in Figure 4(a1,b1). However, there are still some different stochastic properties in details, which are presented as follows. Now let \(\zeta\)\(=1.5\), the DR and EMR with \(\sigma\)\(=0.01\) are shown in Figure 5. It can be observed that DR and EMR are 1T-period under two different initial conditions and the output voltages are stable. The root mean square (RMS) voltages of DR and EMR are 0.732. For \(\zeta=0.9\), there are two chaotic attractors for DR in Figure 6. Because of the occurrence of the merging crisis, two chaotic attractors turn into a greater chaotic attractor for EMR with \(\sigma=0.01\) and the RMS voltages reduce from 0.750 to 0.729. When \(\zeta=0.02\), DR is periodic and discrete spectrums can be viewed in Figure 7(b1). However, EMR becomes chaotic due to the influence of uncertain parameter, which can be illustrated from the continuous spectrum of the output voltage over a wide range of frequency in Figure 7(b2). Compared with DR, the amplitudes of frequency components of EMR decrease, especially in the frequencies \(3\omega\) and \(5\omega\). Figure 4: Bifurcation diagrams and the TLE with the variation of damping coefficient \(\zeta\). (**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 \([x_{0},x_{1},x_{2},x_{3},x_{4}]=[-0.5,0,0,0,0]\), \([x_{0},x_{1},x_{2},x_{3},\)\(x_{4}]=[1.0,0,0,0,0]\), \([v_{0},v_{1},v_{2},v_{3},v_{4}]=[1.0,0,0,0,0]\)(IC-1) and \([x_{0},x_{1},x_{2},x_{3},x_{4}]=[1.0,0,0,0,0]\), \([x_{0},x_{1},x_{2},x_{3},\)\(x_{4}]=[-0.5,0,0,0,0]\), \([v_{0},v_{1},v_{2},v_{3},v_{4}]=[1.0,0,0,0,0]\)(IC-2), respectively. The stochastic behavior of TEH with \(\sigma=0.01\) has been investigated above. Then, different values of \(\sigma\) are employed to observe how is about the influence under different  
Figure 5: DR and EMR (\(\sigma=0.01\)) for \(\zeta\)=1.5. **(a)** Phase portraits; **(b)** Time history of the output voltage. The blue dots and red dots are obtained under the initial conditions (IC-1) and (IC-2), respectively. Figure 6: DR and EMR (\(\sigma=0.01\)) for \(\zeta\)=0.9. **(a)** Phase portraits of DR; **(b)** Phase portraits of EMR (\(\sigma=0.01\)). **(c)** RMS voltage of DR and EMR (\(\sigma=0.01\)). The blue dots and red dots are obtained under the initial conditions (IC-1) and (IC-2), respectively. Figure 7: DR and EMR (\(\sigma=0.01\)) for \(\zeta\)=0.01. **(a1,a2)** Time history of the output voltage of DR and EMR (\(\sigma=0.01\)); **(b1,b2)** Spectrum of the output voltage of DR and EMR (\(\sigma=0.01\)). intensities.