## 6 Results Obtained from Simulating the Circuit  
In case of numerical results, we mainly observed the dynamical behaviors of the considered system under varying parameter, \(\omega_{\rm nd}\). In the case of electronic circuit implementation, we have observed the phase portraits for different \(\omega_{\rm nd}\). From the Equation (15), we can say that the linear frequency \(f\) of the applied sine wave is dependent on the non-dimensional angular frequency \(\omega_{\rm nd}\) of the considered mechanical system. So, to validate the numerical predictions using the electronic circuit set-up, we have varied \(f\) to obtain different \(\omega_{\rm nd}\). When, \(\omega_{\rm nd}=3.89\), from the Equation (15), the expression of the linear frequency \(f\) is 12.382 kHz, where \(C=1\) nF, and \(R=100\) k\(\Omega\). Figure 18 shows the time-series waveforms of the four state-variables of the analog electronic circuit of the considered coupled mechanical system. The waveforms are straight lines parallel to the \(x\)-axis, implying that the circuit has a zero-fixed point solution. We observed the same time-series waveforms in the case of numerical simulation as shown in the Figure 6a. The zero fixed point exists for any initial conditions. The corresponding amplitudes of the applied ac sine waves of the first and second oscillators equivalent circuits are 0.6497 V and 0.7756 V, respectively. The voltages have been calculated using the Equations (14) and (16). When \(\omega_{\rm nd}=4.00\), the Figure 19 shows the corresponding phase space diagrams of the circuit shown in the Figure 17. From the Equation (15), \(f=12.732\) kHz. The single loop in the phase space confirms that the orbit is periodic of an order 1. The same thing we predicted in the numerical result in the Figure 7a. Using the Equations (14) and (16), we can calculate the values of the amplitudes of the applied ac sine waves. The amplitudes of \(V_{\rm in}\) and \(V_{\rm in1}\) for the first and second oscillators are 0.6319 V and 0.7543 V, respectively. Please note that we have chosen the initial condition as \((0.333,0,0,0)\). Suppose we choose the initial condition as \((0,0,0,0)\). In that case, we obtain the zero fixed point trajectories for all the state variables, which confirm that the system has a co-existence of two attractors, one is a zero fixed point attractor, and another one is a period-1 orbit. The same agreement is shown in the numerically obtained bifurcation diagram in the Figure 10, where one can see that the two attractors coexist in the bifurcation diagrams. The red color is the fixed point attractor for the initial conditions \((0,0,0,0)\), and the green color is the different  
Figure 17: The analog circuit diagram of the considered mechanical system. The resistance values are: \(R=R_{1}=R_{2}=R_{3}=R_{4}=R_{5}=R_{6}=R_{16}=R_{22}=R_{23}=R_{24}=R_{25}=R_{6 \delta 8}=R_{69}=R_{71}=R_{72}=R_{74}=R_{81}=R_{82}=R_{91}=R_{92}=R_{93}=100\) k\(\Omega\), \(R_{30}=576.04\) k\(\Omega\), \(R_{73}=482.62\) k\(\Omega\), \(R_{14}=2.50\) k\(\Omega\), \(R_{15}=18.26\) k\(\Omega\), \(R_{75}=2.1\) k\(\Omega\), \(R_{76}=14.91\) k\(\Omega\). The capacitor values: \(C=C_{1}=C_{3}=C_{8}=1\) nF. ‘OA’—series are the op-amps with different operations, \(M_{13}\)–\(M_{16}\) are the multipliers. The initial capacitor voltage of \(C_{1}\) is chosen at 0.333 V, and the remaining capacitor initial voltages are kept at zero voltage. (Color Online). periodic orbits depending upon the different \(\omega_{\rm nd}\) values. For the electronic circuit, the initial condition values have been chosen as the initial capacitor voltages in the circuit, as shown in the Figure 17.  
When \(\omega_{\rm nd}\) is 4.05, the corresponding value of the linear frequency \(f\) is calculated using Equation (15) as 12.892 kHz. The phase space diagram of the equivalent circuit is shown in Figure 20. The amplitudes of the two sine waves \(V_{\rm in}\) and \(V_{\rm in1}\), used at the circuit diagrams Figures 15 and 16 are 0.6241 V and 0.7450 V, respectively.