**5. Bi-Stable Oscillator with Isolated Periodic Solution**

The third dynamical system studied in this work is a weakly damped variant of the system proposed in the previous section and sketched in Figure 4. Here, the damping parameters are reduced by a factor of 10 to *d*<sup>x</sup> = *d*<sup>y</sup> = *d*<sup>z</sup> = *d*lin = 0.002. This system configuration has already been studied in [13,14] where the authors found an isolated solution branch resulting from the damping variation. Figure 7a displays the bifurcation diagram for the horizontal stiffness *k*x. The fixed point solution loses stability through a sub-critical Hopf bifurcation at *k*<sup>x</sup> = 32.24 to a limit cycle solution, hereafter denotes

as LC1. Interestingly, a second stable limit cycle solution is born for *k*<sup>x</sup> < 29.9, which is found to be an isolated branch [14], hereafter denoted as LC2. That is, this solution is not connected to any other solution path. As a result, the system may jump from the fixed point solution to the first limit cycle for 32.24 <sup>≤</sup> *<sup>k</sup>*<sup>x</sup> <sup>≤</sup> 33.0, and then from the limit cycle to the isolated branch for *<sup>k</sup>*<sup>x</sup> <sup>&</sup>lt; 29.9. Hence, within a rather narrow parameter range, two jumping phenomena between different solutions may occur. It is, therefore, of great interest to investigate the probability of arriving on either of those solutions for some prescribed set of initial conditions.

**Figure 7.** Bifurcation diagram for the weakly damped friction oscillator exhibiting an isolated solution branch (**a**) and basin stability values (**b**) for all three stable solutions along the horizontal stiffness *k*x. Initial conditions for each solution are given in Appendix C.

Figure 7b displays the basin stability values for both periodic orbits and the fixed point solution. For the reference subset, we arbitrarily chose Q (*x*, *y*, *z*, *x*˙, *y*˙, *z*˙) : [0, 10] × [0, 10] × [0, 10] × [−2, 2] × [−2, 2] × [−2, 2] using *n* = 1000 sampling points. For for the bi-stability range featuring the two periodic solutions LC1 and LC2 (*k*<sup>x</sup> < 29.9) the basin stability analysis reveals that LC1 is the by far most probable solution. A maximum of 21% of the trajectories converge to the isolated solution branch, while the remaining trajectories converge to the first periodic orbit. Particularly interesting is the parameter regime 27.4 ≤ *k*<sup>x</sup> ≤ 29.9. Here, the basin stability indicates that LC1 is globally stable, even though the stable isola still co-exists. However, due to the choice of Q, no initial conditions were drawn for the basin related to LC2. Hence, if the range of initial conditions and perturbations can be quantified or limited for some specific system, the basin stability analysis can also help to rule out jumping phenomena between co-existing solutions.

Another interesting observation is the following: the basin stability values in this specific setting do not follow the qualitative trend of the respective amplitudes reported in Figure 7a. S<sup>B</sup> (LC1) keeps increasing along the stiffness parameter, while the corresponding amplitude of the horizontal vibration amplitude shows a different behavior. Theoretically, it is clear that the vibration amplitudes do not relate to the size of the basins of attraction. However, on the first sight classical bifurcation diagrams may suggest that one solution is *more attractive* if it has a larger vibration amplitude. At this point, the basin stability represents a technique to quantify the attractiveness in a highly consistent manner.

Lastly, we discuss our previous thought on the benefits of having a robust methodical approach to estimating the basin volumes through Monte Carlo sampling irrespective of the dynamical system at hand (so-called *model-agnostic* techniques). Especially for such low-dimensional systems as shown before, one might raise the issue of using computation-heavy sampling methods, even though the basins of attraction are readily available once the bifurcation diagram is known. Figure 8 displays the state space of each DOF at *k*<sup>x</sup> = 27, hence in a configuration where the two periodic orbits co-exists. It becomes clear that even for this 3 DOF oscillator (6 states), the analytical calculation of the basin

volumes can quickly become a challenge. There is no straight-forward way to computing the volumes in the six-dimensional space from the intertwined basins separated by the unstable orbits, especially looking at the *z* coordinate. Therefore, the basin stability analysis is not only relevant for systems featuring larger number of states, but also for rather low-dimensional systems.

**Figure 8.** State space of all DOFs (horizontal direction in (**a**), vertical direction in (**b**) and diagonal direction in (**c**)) at *k*x = 27.0 for the weakly damped oscillator.

## **6. Conclusions**

This work proposed augmenting the classical local stability analysis of friction-excited oscillators by their basin stability. The concept of basin stability allows assigning global stability metrics to multi-stable solutions in a highly automated manner including error estimates. For three different friction-excited systems, we show that the knowledge of global stability with respect to a specific set of initial conditions can provide important insights into the long-term dynamics. Particularly for well-controlled perturbations, this approach allows estimating the probabilities of arriving on either of multiple stable solutions, and even to rule out some steady-state behavior. As a result, we suggest to include the basin stability analysis into the toolbox of techniques that are applied to study the nonlinear dynamics of multi-stable systems, especially when operating conditions are well-known.

**Author Contributions:** Conceptualization, M.S. and N.H.; methodology, M.S.; software, M.S.; validation, M.S.; formal analysis, M.S.; investigation, M.S. and A.P.; resources, N.H.; data curation, M.S.; writing—original draft preparation, M.S.; writing—review and editing, N.H. and A.P.; visualization, M.S.; supervision, N.H. and A.P.; project administration, N.H.; funding acquisition, N.H. All authors have read and agreed to the published version of the manuscript.

**Acknowledgments:** Publishing fees supported by Funding Programme **Open Access Publishing** of Hamburg University of Technology (TUHH).

**Funding:** This research was funded by the German Research Foundation (Deutsche Forschungsgesellschaft DFG) within Priority Program 1897 'calm, smooth, smart', grant number 314996260.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
