## 2 Description of the Models
### Vertically Excited Parametric Pendulum with a Torque  
In the next case, we present the vertically excited parametric pendulum with a torque that has a suspension point subjected to a vertical periodic kinematic excitation. This kind  
Figure 2: Numerical plots of the periodically driven pendulum with a torque, which equation is \(\bar{\phi}+\alpha\phi+\sin\phi=v+f\sin\omega t\). We chose the parameter values \(\alpha=0.15\), \(\omega=1\), and different values of \(v\). (**a**,**b**) Trajectories; (**a**’,**b**’) stroboscopic maps. The effect of the torque is observed in (**a**,**a**’), where \(v=0\), and in (**b**,**b**’), where \(v=0.55\). Here, X and Y represent angular displacement and velocity, respectively. of pendulum is also a classical example of chaotic system induced by the motion of the suspension point [9; 23]; therefore, its dynamics is very rich and complex. Figure 3 is given by the equation of motion [24]:  
\[ml^{2}\ddot{\phi}+k\dot{\phi}+m(gl-l\dot{x})\sin\phi=N, \tag{3}\]  
where \(m\) and \(l\) denote the mass of the bob and the length of the pendulum, \(k\) is the damping coefficient, \(g\) is the gravity constant, and \(N\) is the extra constant torque. Additionally, \(x(t)=a\sin\omega t\) represents the vertical periodic forcing of the suspension point of amplitude \(a\) and frequency \(\omega\). The dimensionless form of this equation obtained dividing both sides of Equation (3) by \(ml^{2}\), is the following:  
\[\ddot{\phi}+\alpha\dot{\phi}+(1+\gamma\omega^{2}\sin\omega t)\sin\phi=\
u, \tag{4}\]  
where \(\omega_{0}=\sqrt{g/l}\) and as in the first model, its value is taken as \(\omega_{0}=1\), \(\alpha=k/(ml^{2})\), \(\gamma=a/l\) and \(\
u=N/(ml^{2})\). In the case of the absence of the excitation terms (in Equations (2) and (4)) the shape of the potential can be expressed as:  
\[V(\phi)=1-\cos\phi+\
u\phi. \tag{5}\]  
This potential function is plotted in Figure 4 for different values of the torque \(\
u=0.0,0.2\), and \(0.4\). Term \(\
u\) breaks mirror symmetry \(V(\phi)=V(-\phi)\). Figure 5a,b,a',b' represent the numerical trajectories and the stroboscopic map of the vertically excited parametric pendulum with a torque. Again, on the basis of corresponding phase portraits, we observed some differences in dynamical responses depending on the parameter values. However, the Poincare maps are represented by singular points informing about the agreement between system input and output periodicity (similarly as in Figure 2). We fixed the parameter values as \(\alpha=0.15\), \(\gamma=1\), \(\omega=1\) for all cases, while \(\
u=0\) in Figure 5a,a', and \(\
u=0.6\) in Figure 5b,b'. Both pendula are often used as paradigmatic models of chaotic systems. This means that a tiny change in the initial parameters can cause huge changes in the dynamics of the system after a short period of time, as we show later. Then, after introducing the models of the pendula, we start the study of the excited pendulum with a torque in the limit of fast vertical kinematic oscillations to show how this kind of oscillations can affect the dynamics of the system and the potential shape. Figure 3: A schematic of the vertically excited parametric pendulum. Vibration of the Parametrically Excited Pendulum with a Torque in the Limit of Very Fast Excitation  
Here, we examine the limit of fast kinematic vertical excitations of the vertically excited parametric pendulum with a torque (Equation (4)). Namely, the small amplitude \(\gamma\ll 1\) and large frequency \(\omega/\omega_{0}\gg 1\) enable to perform the system averaging. Using the method  
Figure 4: Plot of the potential function \(V(\phi)=1-\cos\phi+\
u\phi\) for \(\
u=0.0\), \(0.2\) and \(0.5\). Note that \(\
u>0\) (denoted with the dashed line) makes the potential assymetric with respect to the axis \(\phi=0\). Figure 5: Numerical plots of the vertically excited parametric pendulum with a torque (Equation (4)) with parameter values \(\alpha=0.15\), \(\gamma=1\), \(\omega=1\), respectively. (**a**,**b**) Trajectories, (**a**’,**b**’) stroboscopic maps.