**5. Chaos with hydrodynamic damping in TM-AFM**

The elastic constant of the cantilever *<sup>c</sup> k* must be less than the effective elastic constant of the interatomic coupling *at k* of the sample. Thus the elastic constant of the spring must be *K Kat* , with <sup>2</sup> *K wm at at at* . Typical atomic vibration frequencies are <sup>13</sup> <sup>10</sup> *at Hz* and atomic masses are of order <sup>25</sup> <sup>10</sup> kg and *<sup>K</sup>* 10 [N / m]. Considering the case of *K Kat* and rewriting the equation into state space results:

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_2\\ \dot{\mathbf{x}}\_2 &= -r\mathbf{x}\_2 - b\mathbf{x}\_1 - c\mathbf{x}\_1^3 - \frac{d}{\left(a + \mathbf{x}\_1\right)^2} + \frac{e}{\left(a + \mathbf{x}\_1\right)^8} + g\cos\Omega\mathbf{x}\_3 - \frac{p}{\left(a + \mathbf{x}\_1\right)^3}\mathbf{x}\_2 \end{aligned} \tag{17}$$

where 1 *x y* , 2 *x y* . The phase diagram can be observed in Figure 11a. For the parameters values: 1; r 0.1 ; b 0.05 ; c 0.35 ; d 4/27 ; e 0.0001 ; g 0.2 ; p 0.005 and a 1.6 . The Lyapunov exponents ( <sup>1</sup> 0.23 , 2 0; <sup>3</sup> 0.1 ), can be seen in Figure 11b, indicating that the system has a chaotic attractor.

**Figure 11.** (a): Phase diagram (b): Lyapunov exponents
