## 4 Numerical Results
### Time-Series and Phase-Space Plots of the Considered System
In each of the figure, \(x\)-axis is the non-dimensional time and the \(y\)-axis is the non-dimensional displacements, \(y_{1}\) and \(y_{2}\), and the non-dimensional velocities, \(\dot{y}_{1}\), \(\dot{y}_{2}\). The left and the right sides of the upper trace of each figure are the state-variables \(y_{1}\) and \(\dot{y}_{1}\) respectively. The left and the right sides of the lower trace of each figure are the state-variables \(y_{2}\) and \(\dot{y}_{2}\) respectively. (**a**) \(\omega_{\rm nd}=3.89\), \(y_{1}=y_{2}=\dot{y}_{1}=\dot{y}_{2}=0\), i.e., fixed-point solutions, (**b**) \(\omega_{\rm nd}=4.00\), Period-1 orbit, (**c**) \(\omega_{\rm nd}=4.05\), quasi-periodic orbit, (**d**) \(\omega_{\rm nd}=4.093\), quasi-periodic orbit, (**e**) \(\omega_{\rm nd}=4.098\), chaotic orbit. The parameters are: \(\zeta_{1}=0.0868\), \(\zeta_{2}=0.1036\), \(t_{1}=0.2413\), \(t_{2}=0.288\), \(b_{1}=45.7143\), \(b_{2}=54.5737\), \(a_{1}=5.8548\times 10^{-3}\), \(D_{1}=1.5\), \(\beta=11.11\), \(\mu=1.1938\), \(\delta_{1}=1\). The initial condition is chosen to \((0.333,0,0,0)\) to obtain the time-series waveforms. (Color Online). It seems that the Figure 7c,d look like the same orbits. To distinguish between the quasi-periodicity and chaotic orbit, we have computed the poincare sections, which are shown in the Figure 8.  
The poincare sections for different values of \(\omega_{\rm nd}\) are shown in the Figure 8. As the system is a non-autonomous dynamical system, we need to observe the evolution of the state variables with the synchronism of the external periodic signal's frequency. As the value of the frequency is \(2f\), we shall observe the periodicity of the orbits in the interval of \(\frac{T}{2}\). When the parameter value \(\omega_{\rm nd}\) is \(4.00\), the orbit is periodic. The dots in the Figure 8a confirm the period-1 of the coupled system. Please note that the two dots in the sampled state-space confirm the system's symmetric nature. If we replace \(y_{1}\) and \(y_{2}\) by \(-y_{1}\) and  
Figure 8: Poincaré sections of the considered system for different values of \(\omega_{\rm nd}\). \(x\)-axis is the non-dimensional displacements \(y_{1}\), \(y_{2}\) and the \(y\)-axis is the non-dimensional velocities \(\dot{y}_{1}\), \(\dot{y}_{2}\), respectively. The left (blue color) and the right (red color) sides of each figure are the poincaré sections of each oscillator. (**a**) \(\omega_{\rm nd}=4.00\), Period-1 orbit, (**b**) \(\omega_{\rm nd}=4.05\), quasi-periodic orbit, (**c**) \(\omega_{\rm nd}=4.093\), quasi-periodic orbit, (**d**) \(\omega_{\rm nd}=4.098\), chaotic orbit. The parameters are: \(\zeta_{1}=0.0868\), \(\zeta_{2}=0.1036\), \(t_{1}=0.2413\), \(t_{2}=0.288\), \(b_{1}=45.7143\), \(b_{2}=54.5737\), \(a_{1}=5.8548\times 10^{-3}\), \(D_{1}=1.5\), \(\beta=11.11\), \(\mu=1.1938\), \(\delta_{1}=1\). The initial condition is chosen to \((0.333,0,0,0)\). (Color Online). Figure 7: The phase portraits of the considered system for different values of \(\omega_{\rm nd}\). \(x\)-axis is the non-dimensional displacements \(y_{1}\), \(y_{2}\) and the \(y\)-axis is the non-dimensional velocities \(\dot{y}_{1}\), \(\dot{y}_{2}\), respectively. The left and the right sides of each figure are the phase-portraits of each oscillator. (**a**) \(\omega_{\rm nd}=4.00\), Period-1 orbit, (**b**) \(\omega_{\rm nd}=4.05\), quasi-periodic orbit, (**c**) \(\omega_{\rm nd}=4.093\), quasi-periodic orbit, (**d**) \(\omega_{\rm nd}=4.098\), chaotic orbit. The parameters are: \(\zeta_{1}=0.0868\), \(\zeta_{2}=0.1036\), \(t_{1}=0.2413\), \(t_{2}=0.288\), \(b_{1}=45.7143\), \(b_{2}=54.5737\), \(a_{1}=5.8548\times 10^{-3}\), \(D_{1}=1.5\), \(\beta=11.11\), \(\mu=1.1938\), \(\delta_{1}=1\). The initial condition has been chosen at (0.333,0,0,0). (Color Online). in the Equations (4) and (5), we obtain the same expressions of the differential equations. It confirms the symmetricity of the equations of the coupled system. When \(\omega_{\rm nd}\) is 4.05, the system shows the quasi-periodic orbits.