## 5 Computational Simulation Results and Discussion
### The Basins of Attraction  
The solution type of the response is determined by the choice of initial conditions. The response is calculated using the direct time integration method. Unless otherwise specified, the initial conditions for velocities are set to zeros. The initial displacements are selected from the set of points evenly distributed in the damper clearance up to 98% of its size. Figures 12, 13b and 14b are constructed of approximately 8000 points of the initial displacement values. A comparison of a coarser and a finer mesh of points for the construction of basins of attraction can be seen in Figures 12a and 13a. The influence of the initial displacements to characterize the long-time behaviour of solutions is illustrated in Figure 12. The computed basins of attraction are composed of two possible solutions. The set of points coloured red corresponds to a response with a circular orbit of a smaller radius, and the green-coloured points correspond to a response with a larger circular orbit. The investigated nondimensional speeds are selected from within the B region on the frequency response curve. At the nondimensional speed of 1.715, located near the lower bifurcation point, the solution converges to the larger circular orbit in 68% of cases. Figure 11: The frequency response of the synchronous motion and the bifurcation diagram (**a**) and detail of the bifurcation diagram near the origin of the nonsynchronous response (**b**). Figure 12: Illustration of the basins of attraction for the nondimensional speed of 1.715 (**a**), 1.838 (**b**). Conversely, at the nondimensional speed of 1.838, the solution converges to the smaller orbit in 82% of cases. In the case of the nondimensional speed of 1.780, located almost in the middle section of the unstable vibration response, the solution converges to the smaller orbit in 61% of cases, and in 39% of cases to the larger one, as shown in Figure 13a. It was found that the shape of basins of attraction also depends on the choice of initial velocities. Figure 13b shows the basins of attraction for the chosen initial conditions of the velocity with the assumption of a circular orbit. The solutions were attracted to the circular orbit with a smaller radius in 31% of cases and to the larger radius in 69%. Three values of the dimensionless speed (7.1, 8.0, and 8.5) were chosen. The responses were obtained through time integration, assuming zero initial velocities and using the initial displacements from the grid of points shown in Figure 12. The basins of attraction identified a single solution for these three rotational speeds. Figure 14: The basins of attraction for the nondimensional speed of 8.5327 as a result of the zero (**a**) and nonzero (**b**) initial velocities; the red/green points correspond to a smaller/larger orbit radius and the blue points correspond to a quasiperiodic response. Figure 13: The basins of attraction for the nondimensional speed of 1.780 as a result of the zero (**a**), and nonzero (**b**) initial velocities; the red/green points correspond to a smaller/larger orbit radius. The origin of nonsynchronous response occurs at the nondimensional speed of 8.5327 located in the D region, as shown in Figure 11. For this reason, basins of attraction for the initial velocities of zero and non-zero values were studied, and the results are shown in Figure 14. When considering the zero initial velocities, two basins of attraction were identified, see Figure 14a. The response with a smaller orbit radius occurs in 49% of the cases and is marked by the red colour. A quasiperiodic response occurs in 51% and is indicated by the blue colour. The long-time convergence to a given solution was determined for the response after 500 times the period of the excitation force. On the other hand, with nonzero initial velocities, three basins of attraction were identified, depicted in Figure 14b. The non-zero initial velocities were specified with the assumption of a circular motion.