## 5 Equivalent Circuit of the Coupled Mechanical Oscillators  
In order to validate the numerically predicted results, an equivalent circuit diagram of the coupled mechanical system is developed. To achieve this, we split the two-second order differential Equations (4) and (5) by two pairs of the first-order differential equations, which are shown below.  
\[\dot{y}_{1} =z_{1} \tag{9a}\] \[\dot{z}_{1} =-\big{[}f_{\rm{R1}}(z_{1})+f_{\rm{S1}}(y_{1})+\beta_{1}(\tau) \big{]} \tag{9b}\]  
\[\dot{y}_{2} =z_{2} \tag{10a}\] \[\dot{z}_{2} =-\big{[}f_{\rm{R2}}(z_{2})+f_{\rm{S2}}(y_{2})+\beta_{2}(\tau) \big{]} \tag{10b}\]  
where,  
1. \(f_{\rm{R1}}(z_{i})=f_{\rm{LDi}}(z_{i})+f_{\rm{DPi}}(z_{i})=2\zeta_{i}z_{i}+ \frac{t_{i}z_{i}}{\big{(}z_{i}^{2}+a_{i}^{2}\big{)}^{\frac{1}{2}}};i=1,2\). \(f_{\rm{Ri}}(z_{i})\) is the non-dimensional resistance force of the bearings. \(f_{\rm{LDi}}(z_{i})\) is the linear damping force, and \(f_{\rm{DPi}}(z_{i})\) is the force due to the dry-friction. The values of \(\zeta_{1}\), \(\zeta_{2}\), \(t_{1}\), \(t_{2}\) and \(a_{1}\) are 0.0868, 0.1036, 0.2413, 0.2880, and \(5.8548\times 10^{-3}\) respectively.
2. \(f_{\rm{Si}}(y_{i})=b_{i}\Big{[}\frac{1}{(1+D_{1}(\delta_{1}-y_{i}))^{4}}-\frac {1}{(1+D_{1}(\delta_{1}+y_{i}))^{4}}\Big{]}\); \(i=1,2\)., \(f_{\rm{Si}}(y_{i})\) is the non-dimensional force due to the magnetic spring. The values of \(b_{1}\), \(b_{2}\), \(D_{1}\), and \(\delta_{1}\) are \(45.7143,54.5737\), \(1.5\), and \(1\) respectively. To reduce the complexity of the equations, we have chosen the most straightforward expressions of the non-dimensional force due to the magnetic spring, which are in the Equations (7) and (8).
3. The non-dimensional stiffness couplings of the considered system, \(\beta_{1}(\tau)=\Big{[}\Big{(}\frac{1+\beta}{2}\Big{)}+\Big{(}\frac{1-\beta}{ 2}\Big{)}\cos(2\omega_{\rm{nd}}\tau)\Big{]}\big{(}y_{1}-y_{2})\) and \(\beta_{2}(\tau)=\mu\Big{[}\Big{(}\frac{1+\beta}{2}\Big{)}+\Big{(}\frac{1-\beta }{2}\Big{)}\cos(2\omega_{\rm{nd}}\tau)\Big{]}(y_{2}-y_{1})\). The parameter values are \(\beta=11.11\), \(\mu=1.1938\). The non-dimensional angular frequency, \(\omega_{\rm{nd}}\), will have to be varied in order to obtain the bifurcation diagram. In order to draw the circuit diagram, the Equations (9) and (10) can be written as  
\[CR\dot{y}_{1} =-\Big{(}\frac{R}{R}\Big{)}(-z_{1}) \tag{11a}\] \[CR\dot{z}_{1} =-\Big{(}\frac{R}{R}\Big{)}\big{[}f_{\rm{R1}}(z_{1})+f_{\rm{SI}}( y_{1})+\beta_{1}(\tau)\big{]} \tag{11b}\]  
Figure 10: Co-existing attractors in the bifurcation diagram of \(y_{1}\): The \(x\)-axis is the non-dimensional parameter \(\omega_{\rm{nd}}\) and the \(y\)-axis is the sampled value of \(y_{1}\). The parameters are: \(\zeta_{1}=0.0868\), \(\zeta_{2}=0.1036\), \(t_{1}=0.2413\), \(t_{2}=0.288\), \(b_{1}=45.7143\), \(b_{2}=54.5737\), \(a_{1}=5.8548\times 10^{-3}\), \(D_{1}=1.5\), \(\beta=11.11\), \(\mu=1.1938\), \(\delta_{1}=1\). (Color Online).  
\[CRy_{2} =-\Big{(}\frac{R}{R}\Big{)}(-z_{2}) \tag{12a}\] \[CRz_{2} =-\Big{(}\frac{R}{R}\Big{)}\big{[}f_{\rm R2}(z_{2})+f_{\rm S2}(y_{2} )+\beta_{2}(\tau)\big{]} \tag{12b}\]  
First we have drawn the circuit diagrams of each of the components \(f_{\rm Ri}(z_{i})\), \(f_{\rm Si}(y_{i})\), \(\beta_{1}(\tau)\), and \(\beta_{2}(\tau)\).