## 5 Effect of the Coupled Action of Double Damping Parameters on the Stability of the System  
The analytical process in Sections 3 and 4 weakens the physical properties of damping, but in real working conditions, the equivalent linear damping and pressure film damping are coupled with each other, i.e., the change of one parameter is accompanied by the change of the other parameter, so in this section, we analyze the effect of the coupling effect of the dual damping parameters on the stability of the system. By associating Equations (6), (7) and (15) and introducing dimensionless quantities, the relationship between the dimensionless linear damping coefficient \(r\) and the dimensionless piezoelectric membrane damping coefficient \(p\) is obtained as  
\[p=r.\frac{16\mu_{eff}R^{7}}{l_{b}^{3}}, \tag{16}\]  
Figure 9: Bifurcation diagram and phase trajectories of system (2): (**a**) bifurcation diagram of system (2) for \(M\) = 1000, \(p\) (0, 0.01); (**b**) phase trajectory of system (2) for \(M\) = 500, \(p\) = 0.001; (**c**) phase trajectory of system (2) for \(M\) = 1000, \(p\) = 0.001.  
and from Equation (16), \(r\) is positively correlated with \(p\). Under the premise that the material and structure of the microcantilever beam are fixed, \(R\) and \(l_{b}\) are constant, but when \(r\) is changed, i.e., the medium changes, the effective medium gas-viscosity coefficient \(\mu_{\textit{eff}}\) will likewise change, so \(r\) is nonlinearly related to \(p\). Combining the relevant experimental data in references [21, 22], we introduce here the quadratic nonlinear fitting relationship curves \(L_{1}\) and \(L_{2}\) for \(p\) and \(r\),  
\[L_{1}:p=-0.98r^{2}+0.198r,L_{2}:p=0.79r^{2}+0.022r. \tag{17}\]  
The two-parameter bifurcation diagram in the parameter plane of system (2) \(r\times p\) when \(r\in(0,\,0.1)\), \(p\in(0,\,0.01)\), and the number of iterative cycles \(M=500\) is shown in Figure 10.  
In Figure 10, the black dashed line \(p=0.006\) is equivalent to Figure 4, and the blue dashed line \(r=0.008\) is equivalent to Figure 8. Meanwhile, we add the fitting curves \(L_{1}\) and \(L_{2}\) as auxiliary lines in Figure 10. It is worth noting that the magnitude of the effective medium gas-viscosity coefficient \(\mu_{\textit{eff}}\) is not only related to the medium, but the air pressure and temperature also make it change, and at the same time, due to the coupling relationship between the dimensionless linear damping coefficient \(r\) and the dimensionless piezoelectric film damping coefficient \(p\), in Figure 10, system (2) has real physical significance only in the region wrapped by \(L_{1}\) and \(L_{2}\), while region \(A\) has actual physical significance. To further analyze the coupling relationship between \(r\) and \(p\), and to find the boundary conditions for system (2) to reach a steady state of motion under the condition of a fixed number of iterative loops, Figure 10 is locally enlarged (the part of the black dashed box in Figure 10), i.e., we obtain the two-parameter bifurcation diagram of system (2) in the \(r\times p\) parameter plane at \(r\in(0,\,0.02)\), and \(p\in(0,\,0.002)\), which is shown in Figure 10(a). In Figure 10(a), we can see the bifurcation characteristics of system (2) in the \(r\times p\) parameter plane, and at the same time, we obtain the boundary conditions for the transition of the motion state of system (2) from an unstable to a stable state under the condition of \(M=500\), i.e., the intersection point \(Q\) of the period 2 region in the positive direction and \(L_{2}\), at which time, the coordinate of the point \(Q\) is \((r,p)=(0.01014,\,0.00031)\). In addition, Figure 10(a) is further enlarged to obtain the two-parameter bifurcation diagram of system (2) in the \(r\times p\) parameter plane for \(r\in(0,\,0.006)\) and \(p\in(0,\,0.0001)\) as shown in Figure 10(b), and in Figure 10(b), we can observe that there are obvious hierarchical, branching repetitive, and scaling properties in the two-parameter inverse bifurcation diagram.