**3. Bi-Stable Oscillator with Falling Friction Slope**

As a first system, we study the dynamics and the stability of a single-degree-of-freedom oscillator *mx* ¨ + *cx* ˙ + *kx* = *F*, see Figure 2a, with velocity-dependent friction as proposed by Papangelo et al. [12]. Specifically, the friction characteristic *µ* (*v*rel) is a velocity-dependent weakening function

$$\begin{aligned} v\_{\text{rel}} \neq 0: \quad & F = -N\mu \left( v\_{\text{rel}} \right) \text{sign} \left( v\_{\text{rel}} \right), \quad v\_{\text{rel}} = \dot{\text{x}} - v\_{\text{d}}\\ v\_{\text{rel}} = 0: \quad & |F| < \mu\_{\text{st}} N \end{aligned} \tag{5}$$

$$\mu(v\_{\text{rel}}) = \mu\_{\text{d}} + (\mu\_{\text{st}} - \mu\_{\text{d}}) \exp \left( -\frac{|v\_{\text{rel}}|}{v\_{0}} \right)$$

featuring the static friction coefficient *µ* (0) = *µ*st , the dynamic friction coefficient *<sup>µ</sup>* (*v*rel → +∞) = *µ*d, the reference velocity *v*<sup>0</sup> and the contact normal load *N*. The non-dimensional form of the equations of motion is obtained through normalization (˜ ·) of the quantities accordingly to the work of Papangelo et al. [12]. The velocity-dependence introduces a dynamic instability that gives rise to friction-induced vibrations (FIV) for 0 ≤ *v* ˜<sup>d</sup> ≤ 1.84. Moreover, the friction nonlinearity enables the system to exhibit a bi-stable behavior, such that a stable steady sliding state and a stable stick-slip cycle co-exist for a range of belt velocities 1.11 ≤ *v* ˜<sup>d</sup> ≤ 1.84, see Figure 2b. At *v* ˜<sup>d</sup> = 1.15, the steady sliding state loses stability at a subcritical Hopf bifurcation point. In the bi-stability regime, and depending on the initial condition or instantaneous perturbations, the system will either end up in the low-energy steady sliding state, or on the high-intensity stick-slip cycle. Both solutions are locally stable and attractive, i.e., robust against small perturbations.

**Figure 2.** (**a**) single-degree-of-freedom frictional oscillator, (**b**) bifurcation diagram for the non-dimensional belt velocity *v* ˜d, and (**c**) phase plane for *v* ˜<sup>d</sup> = 1.5. Stable (unstable) solutions are indicated by solid (dashed) lines. The stable steady sliding state (blue spiral trajectory) co-exists with the unstable periodic orbit (black dashed line) and the stable stick-slip limit cycle (red trajectory). The non-dimensional system (˜ ·) is evaluated for *µ*<sup>d</sup> = 0.5, *µ*st = 1.0, *ξ* = 0.005, *N* = 1.0 and *v* ˜<sup>0</sup> = 0.5.

For this minimal system, the basin boundaries are directly accessible through the unstable periodic orbit (UPO). However, if this knowledge was not available, the probability of arriving on one of the two steady states would be unknown. Figure 1 displays a sampling with *n* = 100 points uniform at random from Q (*x*, *x* ˙) : [−3, 3] × [−2, 2] at *v* ˜<sup>d</sup> = 1.5, and the resulting basin stability values S<sup>B</sup> (FP) = 0.37 and S<sup>B</sup> (LC) = 0.63. Hence, for this *ρ* (**x**), it is more likely to arrive on the high-amplitude limit cycle solution than on the steady sliding fixed point.

To complement the bifurcation diagram and the complex eigenvalue analysis, the basin stability of the fixed point and limit cycle solution is derived along the normalized belt velocity parameter. In particular, at each velocity value *n* = 1000 initial conditions are drawn from a uniform random distribution in Q (*x*, *x* ˙) : [0.5, 2.5] × [−2, 0], i.e., positive initial displacement and negative initial velocity. Figure 3 depicts the eigenvalue's real part and the basin stability. As *v* ˜ <sup>d</sup> decreases, the real part grows until it crosses into the positive plane at *v* ˜<sup>d</sup> = 1.15. This rather smooth behavior nicely indicates the transition into linear instability of the fixed point solution. However, the eigenvalues at

the exemplary points *v*˜<sup>d</sup> = 1.3 and *v*˜<sup>d</sup> = 1.7 would not allow a statement about the system's probability to converge to this solution instead of converging to the periodic orbit. Additionally, the eigenvalue obviously does not indicate the existence of the competing stable periodic solution in this parameter range. At this point the basin stability analysis comes into play: Below the Hopf bifurcation point, all trajectories converge to the periodic orbit, hence S<sup>B</sup> (LC) = 1.0, and above the bi-stability regime all trajectories converge to the globally stable fixed point, i.e., <sup>S</sup><sup>B</sup> (FP) <sup>=</sup> 1.0 for *<sup>v</sup>*˜<sup>d</sup> <sup>&</sup>gt; 1.84. For the chosen subset Q, the periodic orbit is the dominating behavior in the lower parameter range of the bi-stable regime. For increasing relative velocity the probabilities, i.e., the basin stability values, are more balanced for arriving either on the LC or the FP. For *v*˜<sup>d</sup> > 1.6 the fixed point is the more probable solution to arrive at. Therefore, the basin stability values add an important insight and complement the binary stability statements given by the eigenvalues. Using the basin stability, it is now possible to state *how* stable a solution is against arbitrary and possibly *non-small* perturbations. For more realistic systems, this statement may be of even larger value than the binary stability statement given by local metrics.

**Figure 3.** Bifurcation diagram (**top**), real eigenvalue (**middle**) and basin stability (**bottom**) of the single-DOF friction oscillator along the relative sliding velocity.

#### **4. Bi-Stable Oscillator with Mode-Coupling**

As a second system, we study a frictional oscillator [8,11], which (in contrast to the first system) experiences FIV through a mode-coupling instability. The system features a main oscillating mass that is in dry Coulomb-type frictional contact with a conveyer belt. A second mass is connected to the main mass through a nonlinear joint element in diagonal direction, thereby geometrically coupling the vertical and horizontal movement of the main mass. The relative sliding velocity is assumed to always be positive, such that no stick-slip cycles can arise. For the nonlinear joint element, a cubic stiffness nonlinearity *k*nl is chosen [11]. The equations of motion and parameter values are given in Appendix B and the model is displayed in Figure 4.

**Figure 4.** (**a**) Frictional oscillator with nonlinear joint and mode-coupling instability [11]. (**b**) Trajectories obtained in the reference configuration (see Appendix B) for two different initial conditions of the horizontal displacement *x* (all other states were kept at 0).

Previous studies have revealed the complicated bifurcation behavior of this system, including super- and sub-critical Hopf bifurcations as well as isolated solution branches [11,13,14]. In this study, a variation of the horizontal stiffness *k*x is performed. A sub-critical Hopf bifurcation point is found at *k*<sup>x</sup> = 32.3, see Figure 5a. Below, a stable limit cycle and the unstable fixed point exist. Above this value there is a bi-stable range up to *k*<sup>x</sup> = 33.0 with a co-existing stable limit cycle and the stable fixed point. The eigenvalues' real parts in Figure 5b exhibit the classical forking behavior that is related to the mode-coupling instability mechanism in this system. At the point of instability, one eigenvalue crosses into the positive plane. The basin stability S<sup>B</sup> of both stable solutions is computed for *n* = 500 random initial conditions drawn from Q (*x*, *x* ˙) : [0, 0.5] × [0, 0.25] (all other initial conditions are fixed to 0). Figure 5c depicts the basin stability as a function of the horizontal stiffness. In the bi-stability range 32.3 ≤ *k*<sup>x</sup> ≤ 33.0 the basin stability values indicate that the limit cycle solution is the dominating one for lower stiffness values. For larger stiffness values the fixed point solution is the most probable for our choice of Q. Hence, within this rather short bi-stability range, a minor variation of the horizontal stiffness value would crucially affect the probability of arriving either on the low-energy steady-sliding state, or on the high-energy limit cycle, which may cause increased wear, audible vibrations and other effects in realistic systems. Such kind of statement about the global stability regarding non-small perturbations would not have been easily accessible through the bifurcation diagram or the local stability analysis.

**Figure 5.** (**a**) bifurcation diagram for the horizontal stiffness parameter, (**b**) eigenvalues' real parts and (**c**) basin stability of the fixed point and limit cycle solution. *x* ˆ denotes the maximum amplitude of *x*(*t*) along one vibration period. Solid and dashed lines indicate stable and unstable solutions, respectively.

These results are clearly related to the shape of the unstable periodic orbit, i.e., the separatrix of both basins of attraction. While the qualitative basin stability values for a variation in the initial

conditions for *x* would have been readable from the bifurcation diagram in Figure 5a, this task quickly becomes complex once more degrees-of-freedom (DOFs) are considered. For example, let us consider a reference subset Q that captures certain variations for multiple DOFs, instead of variations for a single DOF as shown before. Figure 6 displays the basin stability values for three different choices of Q, i.e., different variations of the range of possible initial conditions:

$$\begin{aligned} \mathcal{Q}\_1 \left( \mathbf{x}, \boldsymbol{y} \right) &: \left[ \mathbf{0}, \mathbf{0.25} \right] \times \left[ \mathbf{0}, \mathbf{0.5} \right] \\ \mathcal{Q}\_2 \left( \mathbf{x}, \boldsymbol{y}, \boldsymbol{\xi}, \boldsymbol{y} \right) &: \left[ \mathbf{0}, \mathbf{0.25} \right] \times \left[ \mathbf{0}, \mathbf{0.25} \right] \times \left[ -\mathbf{0}.1, \mathbf{0.1} \right] \times \left[ -\mathbf{0}.2, \mathbf{0.2} \right] \\ \mathcal{Q}\_3 \left( \mathbf{x}, \boldsymbol{y}, \boldsymbol{z}, \boldsymbol{\xi}, \boldsymbol{y}, \boldsymbol{z} \right) &: \left[ \mathbf{0}, \mathbf{0.25} \right] \times \left[ \mathbf{0}, \mathbf{0.25} \right] \times \left[ -\mathbf{0.1}, \mathbf{0.1} \right] \times \left[ -\mathbf{0.1}, \mathbf{0.1} \right] \times \left[ -\mathbf{0.1}, \mathbf{0.1} \right] . \end{aligned} \tag{6}$$

In the first case, some initial variations in the horizontal position and large variations in the vertical displacement of the main mass are allowed. In the second case, variations in the initial velocities are studied, and in the third case also variations in the secondary mass' initial conditions are considered. Such scenarios would quickly become impractical for studying permissible perturbations, i.e., the global stability of each solution, using bifurcation diagrams and subdividing the state space by the unstable solutions. The concept of basin stability automates this process through the Monte Carlo sampling, allowing for a easy-scaling and consistent estimation of the relevant basin volumes.

**Figure 6.** Basin stability values in the bi-stability range for the reference sets of initial conditions Q<sup>1</sup> (**a**), Q<sup>2</sup> (**b**), and Q<sup>3</sup> (**c**) defined in Equation (6).

In fact, even though the three reference sets are very different in their value ranges, the resulting basin stability analysis displayed in Figure 6 does not change qualitatively. The *turning point*, i.e., the point after which the FP solution dominates over the LC solution for increasing values of *k*x, changes only slightly: For Q<sup>1</sup> this point is found at *k*<sup>x</sup> = 32.9, while it is *k*<sup>x</sup> = 32.7 and *k*<sup>x</sup> = 32.55 for Q<sup>2</sup> and Q3, respectively. Hence, the basin stability is not very sensitive to the choice of Q for this system. In a situation in which the overall qualitative behavior of the basin stability values may have seem obvious, the quantitative evaluation would have become difficult to obtain from the bifurcation diagrams. Especially for higher-dimensional systems and specific subset choices the basin stability analysis represents a highly robust approach to estimate the probability of arriving on either of the competing solutions, which we will illustrate in the next section.
