## 6 Results Obtained from Simulating the Circuit
The phase portraits obtained from the circuit diagram agree with the numerical prediction of the phase space diagrams for \(\omega_{\rm nd}=4.05\) as shown in the Figure 7b. The orbit is quasi-periodic in nature which is confirmed from the Figure 8b. Here also, the initial condition is \((0.333,0,0,0)\). If we change the initial condition \((0,0,0,0)\), we only observe the zero fixed-point solutions. Figure 19: Phase-space diagrams of the circuit when \(f=12.732\) kHz. The corresponding value of \(\omega_{\rm nd}\) is 4.00. (**a**) \(x\)-axis is the voltage of \(y_{1}\) in V and \(y\)-axis is the voltage of \(z_{1}\) in V, (**b**) \(x\)-axis is the voltage of \(y_{2}\) in V and \(y\)-axis is the voltage of \(z_{2}\) in V. The initial condition has been chosen at \((0.333,0,0,0)\) to obtain the phase portraits. (Color Online). Figure 20: Phase-space diagrams of the circuit when \(f=12.892\) kHz. The corresponding \(\omega_{\rm nd}=4.05\). (**a**) \(x\)-axis is the voltage of \(y_{1}\) in V and \(y\)-axis is the voltage of \(z_{1}\), (**b**) \(x\)-axis is the voltage of \(y_{2}\) in V and \(y\)-axis is the voltage of \(z_{2}\) in V. The initial condition has been chosen at (0.333,0,0,0) to obtain the phase portraits. (Color Online). Figure 18: Time-series waveforms of the four state-variables obtained from the circuit simulation for \(\omega_{\rm nd}=3.89\). For each figure, \(x\)-axis is the time in sec and the \(y\)-axis is the values of \(y_{1}\), \(y_{2}\), \(z_{1}\), and \(z_{2}\), respectively in V. (**a**) Time-series waveform of \(z_{1}\), (**b**) Time-series waveform of \(y_{2}\), (**c**) Time-series waveform of \(y_{1}\), and (**d**) Time-series waveform of \(z_{2}\). The frequency of the sine wave generator is 12.382 kHz. The initial condition has been chosen at (0.333,0,0,0) to obtain the time-series waveforms. (Color Online). When \(\omega_{\rm nd}\) is equal to the value 4.098, the corresponding linear frequency \(f\) of the circuit is 13.044 kHz. The phase portraits of the circuit at this frequency are shown in the Figure 21. The phase portraits are chaotic in nature. The numerical phase portraits which we obtained in the Figure 7d are the same as the phase portraits obtained from the circuit diagram. Please note that the amplitudes \(V_{\rm in}\) and \(V_{\rm in1}\) of the ac sine waves for the oscillator 1 and oscillator 2 are 0.6168 V and 0.7363 V, respectively. The coexisting attractors are validated in this case also. So, the circuit in the Figure 17 can be used as an equivalent electronic circuit of the considered mechanical system. The waveforms and the phase portraits of the circuit validate the numerical predictions of the system. If one can notice carefully, the phase portraits obtained from the numerical simulation and obtained from the circuit have different scaling values, although the shapes are the same. These things have happened because the differential equations are non-dimensionalized, and the circuits equations are dimensional. Also, the parameter values of the two cases are nearly equal, not the same. But the periodicities of the phase space trajectories are the same.