## 5 Computational Simulation Results and Discussion
### The Synchronous Response and the Stability Assessment  
The frequency response curve for the investigated rotor is depicted in Figure 3. The response curve is composed of two discontinuous curves and shows the behaviour of the system with stiffening properties. One can observe that the nonlinear hydraulic forces have the dominant effect in the region of resonance. The resonant peak is shifted and bent in the direction of increasing rotational speed. There are regions of multiple solutions for a single rotational speed, indicating unstable branches of the frequency response curve. Hysteresis behaviour is also present for the increasing/decreasing rotational speed. The three critical points divide the response curve into four regions denoted as A, B, C, and D. In Figure 3, the points are displayed as green circles. According to Floquet theory [38], the bifurcation points are of the saddle-node type, see Figures 4 and 5. Figure 4: The effect of the rotational speed on the maximum of real (**a**) and imaginary (**b**) part of the eigenvalues \(\mu\); the borders of regions are denoted by the red dashed lines and the unstable speed interval is marked by the red line. Figure 3: Frequency response curve (**a**) and detail of the isolated part (**b**); borders of regions are denoted by the red dashed lines and u, m, and l denote the upper, middle, and lower branches. In region A, the response of the rotor exhibits a circular orbit, and the vibration of the rotor is stable, as determined by Floquet theory. In region B, three different circular orbits are possible. Both orbits with the largest and smallest eccentricity are stable, while the branch in between represents an unstable response and is marked by a dashed line. The following region C has only a single stable solution. In the last region, denoted as D, multiple solutions appear. The solution with large eccentricity is isolated and its detailed view is depicted in Figure 3b. The isolated solution arises at the saddle-node point at a certain angular speed value and has two branches. The vibration response is unstable for the lower branch, denoted by the dashed line. The blue curve in Figure 3 was obtained from the solution of the nonlinear algebraic Equation (15) derived with the assumption of a synchronous circular orbit. The steady-state response, marked with the \"+\" symbol, was obtained by the direct time integration and the red \"x\" symbol denotes the solution provided by the HBM combined with the arc length method. The equations of motion (1) were solved using the MATLAB software environment. The time integration was performed using the ode45 routine [43] with the relative and absolute tolerance [43] set to \(1\cdot 10^{-12}\) and \(1\cdot 10^{-15}\), respectively. The calculation was done with an upward and downward sweep of the rotational speed. In Figure 3, the direction of the sweep is indicated by black and green arrows. The initial conditions for the current sweep step were obtained from the previous step. The form of the obtained response curve demonstrates that the direct time integration method is not suitable to compute unstable branches, as depicted in Figure 3.  
The HBM, combined with the use of arc length continuation, was able to identify both stable and unstable regions. The isolated solution was identified by setting the initial conditions near the searched solution. The response was explored using 8 harmonic terms, a 10x oversampling factor, and an arc length parameter of \(1\times 10^{-2}\). The norm of the relative solution error was found to be less than \(7\times 10^{-14}\) across the investigated range of rotational speed. It has been confirmed that all applied methods for identifying the response converge to the same solution. However, the methods differ in terms of computational and time effort, where time integration is the most expensive. Figure 4 displays the dependencies of the real (Figure 4a) and imaginary (Figure 4b) parts of the largest eigenvalue of the transition matrix on the nondimensional speed.