## 4 Numerical Results
### Time-Series and Phase-Space Plots of the Considered System  
Figure 6 shows the time-series waveforms of the mathematical Equations (4) and (5). These equations characterize the dynamics of the considered mechanical system. Here, we have varied the non-dimensional angular frequency, \(\omega_{\text{nd}}\), keeping the remaining parameter values fixed at \(\xi_{1}=0.0868\), \(\zeta_{2}=0.1036\), \(t_{1}=0.2413\), \(t_{2}=0.288\), \(b_{1}=45.7143\), \(b_{2}=54.5737\), \(a_{1}=5.8548\times 10^{-3}\), \(D_{1}=1.5\), \(\beta=11.11\), \(\mu=1.1938\), \(\delta_{1}=1\). We have varied the \(\omega_{\text{nd}}\) in the range between 3.89 to 4.1. When the parameter \(\omega_{\text{nd}}\) is increased up to 3.89 from the low value, the solutions of the state-variables, i.e., \(y_{1}\), \(y_{2}\), \(y_{1}\), and \(\dot{y}_{2}\) give only the zero-fixed point solutions for any initial conditions. So, we get a straight line parallel to the time axis for each of the state-variables. Figure 6a shows the time series waveforms of the four state variables which signify the stable fixed point solutions. When \(\omega_{\text{nd}}\) is chosen to 4.00, the period-1 time series waveforms come to an exist for all state-variables, which is shown in Figure 6b. When \(\omega_{\text{nd}}\) is at the value of 4.05, the state-variables show the quasi-periodic nature in their time-series waveforms. Figure 6c shows the quasi-periodic  
Figure 4: Plots of the magnetic spring force terms \(f_{\text{S1}}(y_{1})\) and \(f_{\text{S2}}(y_{2})\) with the variation of the (**a**) state-variable \(y_{1}\) and (**b**) state-variable \(y_{2}\), respectively. In each of the figures, the \(x\)-axis is the non-dimensional displacements (i.e., state-variables \(y_{1}\) and \(y_{2}\)) and the \(y\)-axis is the non-dimensional magnetic spring force terms. The parameter values are: \(D_{1}=1.5\), \(\delta_{1}=1\), (**a**) \(b_{1}=45.7143\), (**b**) \(b_{2}=54.5737\). (Color Online). Figure 5: Comparison the magnetic spring force terms \(f_{\text{S1}}(y_{1})\) and \(f_{\text{S2}}(y_{2})\) obtained from the different expressions with the variation of the (**a**) state-variable \(y_{1}\) and (**b**) state-variable \(y_{2}\). The \(x\)-axes are the non-dimensional displacements (i.e., state-variables \(y_{1}\) and \(y_{2}\)) and the \(y\)-axes are the non-dimensional magnetic spring force terms. The parameter values are: \(D_{1}=1.5\), \(\delta_{1}=1\), (**a**) \(b_{1}=45.7143\), (**b**) \(b_{2}=54.5737\). (Color Online). time-series waveforms for four state-variables. The quasi-periodicity of the state-variables remains when \(\omega_{\rm nd}\) is 4.093. This is shown in Figure 6d. When \(\omega_{\rm nd}\) is 4.098, there is a chaotic attractor comes to an exist in the system. The corresponding time-series waveforms of the state-variables are shown in Figure 6e. We have computed the phase spaces and the poincare sections in the state-space to observe the actual periodicities of the system. In that case, we can distinctly observe different periodicities of the system while the parameter is varied. Figures 7 and 8 show the phase-spaces and the poincare sections of the coupled system for different values of \(\omega_{\rm nd}\) respectively. Figure 7 shows different phase space trajectories of the considered coupled system for different values of \(\omega_{\rm nd}\). Figure 7a shows period-1 orbit in the phase space. The single loop confirms the period-1 in the phase space of the coupled system. When the parameter \(\omega_{\rm nd}\) is increased to 4.05, the system shows the quasi-periodic orbits in the phase-space diagram, which is shown in the Figure 7b. The quasi-periodic orbit with a different shape comes into existence in the parameter value of \(\omega_{\rm nd}=4.093\), which is shown in the Figure 7c. When \(\omega_{\rm nd}=4.098\), the orbit is chaotic. The chaotic attractor is shown in the Figure 7d. Figure 6: Time-Series waveforms of the considered system for different values of non-dimensional frequency, \(\omega_{\rm nd}\).