**Appendix B. Mode-Coupling Instability Oscillator**

The equations of motion are given by

$$\mathbf{M}\ddot{\mathbf{x}} + \left(\mathbf{D} + \mathbf{G}\right)\dot{\mathbf{x}} + \left(\mathbf{K} + \mathbf{N}\right)\mathbf{x} + \mathbf{f}\_{\text{nl}} = \mathbf{0}, \quad \mathbf{x} = \left[\mathbf{x}, y, z\right]^\top, \mathbf{y}$$

$$\begin{aligned} \mathbf{M} &= \begin{bmatrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & m\_1 \end{bmatrix}, \quad \mathbf{D} = \begin{bmatrix} d\_\mathbf{x} & 0 & 0 \\ 0 & d\_\mathbf{y} & 0 \\ 0 & 0 & d\_\mathbf{x} \end{bmatrix}, \quad \mathbf{G} = \mathbf{0}, \quad \mathbf{K} = \begin{bmatrix} k\_\mathbf{x} & -0.5k\_\mathbf{y}\mu & 0 \\ -0.5k\_\mathbf{y}\mu & k\_\mathbf{y} & 0 \\ 0 & 0 & k\_\mathbf{z} \end{bmatrix}, \\\ \mathbf{N} &= \begin{bmatrix} 0 & -0.5k\_\mathbf{y}\mu & 0 \\ 0.5k\_\mathbf{y}\mu & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad F\_{\text{nl}} = uk\_{\text{lin}} + u^3k\_{\text{nl}} + i d\_{\text{lin}}, \quad \mathbf{f}\_{\text{nl}} = \begin{bmatrix} -\frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} \\ 1 \end{bmatrix} F\_{\text{nl}} \end{aligned} \tag{A2}$$

where *u* is the relative displacement in the joint between the main mass and the secondary mass, given by *u* = − √ 2 2 *x* − √ 2 2 *y* + *z*. The parameter values for the reference configuration are given by *m* = *m*<sup>1</sup> = 1, *k*<sup>x</sup> = 32.5, *k*<sup>y</sup> = 20, *k*<sup>z</sup> = 100, *k*lin = 10, *k*nl = 5, *d*<sup>x</sup> = *d*<sup>y</sup> = *d*<sup>z</sup> = *d*lin = 0.02, *µ* = 0.65.
