**6. Scanner position control**

54 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 9.** Phase diagram

**Figure 10.** (a): FFT, (b): Lyapunov exponent

1 2

a 1.6 . The Lyapunov exponents ( <sup>1</sup>

indicating that the system has a chaotic attractor.

*x x*

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

3

and rewriting the equation into state space results:

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

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*

2 2 11 2 8 3 2 3

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

> 0; <sup>3</sup>

0.23 , 2

*d e <sup>p</sup> x rx bx cx g x <sup>x</sup>*

*ax ax a x*

(17)

1 1 1 cos

0.1 ), can be seen in Figure 11b,

*at Hz* and

*K Kat* , with <sup>2</sup> *K wm at at at* . Typical atomic vibration frequencies are <sup>13</sup> <sup>10</sup>

A laser beam focus on the top of the microcantilever and the reflection is detected by a photodiode. The light is converted into an electrical signal, and stored in the computer as a reference. An oscillation of the microcantilever deflects the laser beam on the photodiode, allowing the system to compute the microcantilever motion. The error signals are then forwarded, and the piezoelectric scanner moves vertically to scan the sample, as shown in Figure 3. The control techniques are diverse: PID or PD, sliding mode, LQR, or other.
