**4. Nonlinear dynamic behavior outline**

The basin of attraction is the set of all starting points (initial values) that converge to the attractor. A qualitative change in the behavior (i.e., attractor) of a dynamic system is associated with a change in the control parameter. It is a qualitative leap that progresses to more complex dynamics [22–26].

Therefore, we analyzed the behavior of the basin of attraction considering the parameters p ∈ [0.0.08] and η ∈ [0.0.1]. And thus determine an initial condition to perform the scan of other parameters of nonlinear dynamics, such as the maximum Lyapunov exponent (MLE) and the bifurcation diagram.

The MLE describes the divergence rate of the trajectories described by Eq. (8). The MLE has some characteristics: (i) with positive values for the Lyapunov exponent, they indicate that the trajectories of the system are divergent, this corresponds to the system having a chaotic behavior, (ii) if the MLE is negative, there is a contraction of the phase space, corresponding to a periodically stable system. For our calculations of the MLE exponent, we use the algorithm proposed by Ref. [27] with the variational calculus and the Jacobian matrix of Eq. (8).

The bifurcation diagram, in this work, is calculated considering the maximum points of the time series resulting from the integration of the system of Eq. (8) with the variation of the control parameter p. With this, we can observe the periods formed with this parametric variation. And so, to establish possible values of intervals that have multiple periods and thus diagnosing a chaotic behavior of the system, together with the MLE. And thus, obtaining the phase maps that describe the behavior of the trajectory for a given value of the parameter *p*.

For the dynamics calculations, we considered the parameters and properties of the cantilever described in **Table 1** for the silicon-silicon system, as seen in Ref. [28].

**Table 2** are the dimensionless parameters used with the values from **Table 1** and dimensionless Eq. (8).

*MEMS-Based Atomic Force Microscope: Nonlinear Dynamics Analysis and Its Control DOI: http://dx.doi.org/10.5772/intechopen.108880*


**Table 2.**

*Parameters used for numerical analysis.*

#### **4.1 Basins of attraction analysis: entropy basins and uncertainty coefficient**

Suppose that we have a dynamic system with *NA* attractors for a choice of parameters in a certain region Ω of the phase space. We discretize Ω through a finite number of boxes so that we cover Ω with a grid of linear size ε. Now we build an application C: Ω ! N that relates each initial condition to its attractor (which will have an associated color). Each box contains, in principle, infinite trajectories, each of which leads to an attractor labeled from 1 to *NA* [22, 23].

We consider the colors in the box to be randomly distributed according to some proportions. We can assign a probability to each color *j* within a box *i,* as *pi,j* is evaluated by calculating statistics about the trajectories inside the box. Considering that the trajectories inside a box are statistically independent, we can define the Gibbs entropy of each box *i* is begin:

$$\mathbf{S}\_{i} = \sum\_{j=1}^{m\_{i}} p\_{ij} \log \left(\frac{\mathbf{1}}{p\_{ij}}\right) \tag{9}$$

Where *mi* ∈ [1, *NA*] is the number of colors (attractors) in box *i* in *pi,j* is the probability that each color *j* is determined by the number of trajectories leading to that color divided by the total number of trajectories in the box. We choose nonoverlapping boxes covering Ω so that the entropy of the entire grid is calculated by adding the entropy associated with each of the *N* boxes given by:

$$\mathbf{S}\_{i} = \sum\_{i=1}^{N} \mathbf{S}\_{i} \tag{10}$$

$$S = \sum\_{i=1}^{N} \sum\_{j=1}^{m\_i} p\_{ij} \log \left(\frac{1}{p\_{\vec{\eta}}}\right) \tag{11}$$

Therefore, we can consider the entropy of the basin of attraction (*Sb*) as follows:

$$\mathcal{S}\_b = \frac{\mathcal{S}}{N} \tag{12}$$

An interpretation of this quantity is associated with the degree of basin uncertainty, ranging from 0 (a single attractor) to *log(NA)* (completely random basins with

equiprobable *NA* attractors). This latter higher value is rarely realized in practice, even for extremely chaotic systems. In some cases, we may only be interested in the uncertainty of boundaries between basins of attraction. We often want to know if the boundary is fractal. For this purpose, we can restrict the calculation of the basin entropy to the boxes that fall within the boundaries of the basin of attraction. We can calculate the entropy only for the *Nb* boxes that contain more than one attractor (color),

$$S\_{bb} = \frac{S}{N\_b} \tag{13}$$

where *S* is defined by Eq. (11). We refer to this *Sbb* number as the basin entropy quantifies the uncertainty regarding the boundaries only. The nature of this *Sbb* quantity is different from the entropy of the Sb basin, since Sb is sensitive to the size of the basins, so it can distinguish between different basins with smooth boundaries, *Sbb* provides a sufficient condition to easily assess that some boundaries are fractals [22–24].

Another way to quantify this uncertainty in the initial conditions for its final state is through the uncertainty coefficient. The uncertainty coefficient is related to the sensitivity of the final state of the trajectories in the phase space. An exponent close to 1 means that the basin has smooth contours, while an exponent close to 0 represents fully fractalized basins, also called sieve basins [26].

A phase portrait with a fractal boundary can cause uncertainty in the final state of the dynamical system for a given initial condition. To determine the uncertainty coefficient, one must probe the basin of attraction with balls of size ε at random. If there are at least two initial conditions that lead to different attractors, a ball is marked "*uncertain."* In this way, we can denominate the fraction of "uncertain balls" (*fε*) for the total number of attempts in the basin. In analogy to the fractal dimension, there is a scaling law between, *f<sup>ε</sup>* � ε α . The number that characterizes this scale is called the uncertainty exponent α [26]. For our analysis, we considered the set of differential equations that describe the interactions of the atomic force microscopy system and the parameters described in **Table 1**.

For this we consider *p* ∈ [0,0.08] and η∈[0,0.1], and for the numerical analysis of *Sb*, *Sbb*, and α we use an interval of initial conditions *x*<sup>0</sup> <sup>1</sup> � *<sup>x</sup>*<sup>0</sup> <sup>2</sup> ¼ �½ �� � 0*:*9,0*:*9 ½ � 0*:*9,0*:*9 with approximately 250000 initial conditions and making a 200 x 200 grid considering ε=0.002 for each attraction basin formed during the numerical analysis. **Figure 3a** shows the behavior of *Sb*, **Figure 3b** shows behavior of *Sbb*, and **Figure 3c** shows behavior of *α* for set parameter *p*∈½ � 0,0*:*08 e *η*∈ ½ � 0,0*:*1 .

We can see in **Figure 3a** and **b** that the red regions show the maximum values for *Sb* and *Sbb*, that is, the regions where the basins of attraction have more attractors and

#### **Figure 3.**

*(a) shows the behavior of Sb, (b) shows behavior of Sbb*, *and (c) shows behavior of α for set parameter p* ∈½ � 0,0*:*08 *e η* ∈½ � 0,0*:*1 *.*

*MEMS-Based Atomic Force Microscope: Nonlinear Dynamics Analysis and Its Control DOI: http://dx.doi.org/10.5772/intechopen.108880*


**Table 3.**

*Summarizes the parameters p and η that provide the maximum. values for* Sb *and* Sbb *and the minimum value of α that produce the basins of attraction.*

their edges are fractalized. However, for the uncertainty coefficient in **Figure 3c** the uncertainty coefficient is close to 1 showing the smooth basins. According to Ref. [20] the difference between *Sb* and *Sbb* and the uncertainty coefficient of the attraction basins is that when *Sb* and *Sbb* are minimum the uncertainty coefficient is maximum, or when *Sb* and *Sbb* are maximum the uncertainty coefficient is minimum. This corroborates the analysis of the behavior of the initial conditions with the parametric variation of p and η. **Table 3** shows the parameters p and η that provide the maximum values for *Sb* and *Sbb* and the minimum value of α that produces the basins of attraction.

**Figure 4a–c** show the behavior of the attraction basins considering the maximum values of *Sb* and *Sbb* and for the minimum value of α.

**Figure 5a–c** represent attractor points referring to the basins of attraction of **Figure 5a–c**.

#### **Figure 4.**

*Basins of attraction. (a) p = 0.0012 and η = 0.0045, (b)* p *= 0.0012 and η = 0.0045, and (c)* p *= 0.0764 and η = 0.0121.*

### **4.2 Numerical dynamics analysis**

Taking into account the analysis of the basins of attraction for the interval of p∈[0,0.08] and η∈[0.0.25], we adopted a larger interval for η, for analysis of the maximum Lyapunov exponent (MLE) and the initial condition [0.1, 0.0], because depending on the values of *p* and *η* the initial condition can participate in different attractors, as we saw in **Figures 4a** and **b** and **5a** and **b**.

Using the Jacobian matrix for the variational calculus and the sweep of *p*∈ [0,0.08] and *η*∈ [0,0.25], we have the behavior of the MLE. **Figure 6** shows the space of MLE parameters in which the region of white to black shows the regions in Eq. (7) has a periodic behavior. However, for the region between yellow and green, it shows the chaotic behavior with the parameter sweep *pη*.

**Figure 7a** shows the behavior of the bifurcation diagram for the following parameters, so we can observe the periodic windows p∈ [0, 0.0777] (black region), the intervals are confirmed by the maximum Lyapunov exponent. However, for the interval p∈ [0.0777, 0.08] there is a chaotic window (red region).

#### **Figure 5.**

*Attractive orbits that form the basins of attraction of the figure. (a) p = 0.0012 and η = 0.0045, (b)* p *= 0.0012 and η = 0.0045, and (c)* p *= 0.0764 and η = 0.0121.*

*MEMS-Based Atomic Force Microscope: Nonlinear Dynamics Analysis and Its Control DOI: http://dx.doi.org/10.5772/intechopen.108880*

**Figure 6.** *Behavior of the maximum Lyapunov exponent (MLE) for the parameters p* ∈½ � 0,0*:*08 *and η* ∈½ � 0,0*:*25 *.*

#### **Figure 7.**

*(a) Bifurcation diagram for displacement x1 in black region shows the periodic behavior, red region shows the chaotic behavior, and (b) Lyapunov exponent.*

Therefore, considering the previous nonlinear dynamic results, we obtain the phase portrait and the Poincare map for p =0.0791 and *η* = 0.2043 (**Figure 8**). **Figure 9a** shows the phase portrait (Gray Line) and Poincare map (Black Dots). **Figure 9b** and **c** show the time series of displacement *x1* and velocity *x2.*
