## 2 Description of the Models
### Periodically Driven Pendulum with a Torque  
By simply applying [PERSON]'s second law, a periodically driven pendulum with a torque (Figure 1) is given by the equation of motion [8].  
\[ml^{2}\ddot{\varphi}+k\dot{\varphi}+mgl\sin\phi=N+F\sin\omega t, \tag{1}\]  
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 [21]. The external periodic forcing is \(F\sin\omega t\), which from now on we call harmonic torque component with frequency \(\omega\). The term \(mgl\sin\phi\) is the \(y-\)vertical component of the gravity. For simulation convenience, we consider the dimensionless version of this equation by using the following transformation: \(\alpha=\frac{k}{ml^{2}}\), \(f=\frac{f}{ml^{2}}\), \(\
u=\frac{N}{ml^{2}}\) (see [PERSON] et al. [22]), \(\omega_{0}=\sqrt{g/l}\) where \(\omega_{0}\) is the natural frequency, that we fix as \(\omega_{0}=1\). Therefore, the dimensionless equation of the pendulum reads as follows:  
\[\ddot{\phi}+\alpha\dot{\phi}+\sin\phi=\
u+f\sin\omega t. \tag{2}\]  
In order to provide information on the dynamics of our pendulum, we plot some pictures of the trajectories and the stroboscopic map [8].  
Our stroboscopic map corresponds to the Poincare map with periodic excitation. This can be defined as the intersection of a periodic orbit in the space of a continuous dynamical  
Figure 1: A schematic of a periodically driven pendulum.  
system with a certain lower-dimensional subspace, called the Poincare section, transversal to the flow of the system. A Poincare map can be interpreted as a discrete dynamical system with a state space that is one dimension smaller than the original continuous dynamical system. This map qualitatively shows if the system is chaotic or not depending on the state points distribution. To define a stroboscopic map, we can consider a periodic orbit, with initial conditions within a section of the physical space. The periodic orbit leaves that section afterwards and we record the point at which it returns after one period. We repeat this procedure \(n\) times where \(n\rightarrow\infty\) and therefore we complete the stroboscopic map. If the whole intersection is a set of \(r\) discrete points, where \(r\) is an integer and less than \(n\), the trajectory is periodic of period equal to \(r\). Otherwise, the trajectory is chaotic. A stroboscopic map is indeed a special case of a Poincare map for periodic systems as in the case of pendulum oscillations. The distinguishing feature is that a given phase of the driver period is used for mapping (instead of some other marker event like a local maximum or a zero crossing). In our case, as the driving term is a \(\sin(\omega t)\) term, we build the stroboscopic map by mapping the state of the system in intervals of \(2\pi\) time units.  
Figure 2a,b,a',b' represent the numerical trajectories and the stroboscopic maps of the periodically driven pendulum with a torque. We observed different dynamical behaviors depending on the parameter values showing in phase portrait size and \(x\) mirror symmetry. We fixed the parameter values as \(\alpha=0.15\), \(f=0.25\), \(\omega=1\) for both cases, while \(v=0\) in Figure 2a,a' and \(v=0.55\) in Figure 2b,b'. Figure 2b,b' clearly notes the external torque influence where periodic motions are present.