**3.3 Destabilization analysis after the introduction of coupling**

For the emergence of pattern formation, it is essential that the identical cells lose stability after the resistive coupling is established between them. In the previous section, we have studied the stability of the uncoupled cell, while in this section, we investigate the destabilization process after a resistive array is added to couple the elements of the network. Since a direct stability analysis that requires exploring the complete network itself would be complicated and time-consuming, it would be beneficial to apply a simplified stability analysis. Considering highly accurate

*Pattern Formation in a RD-MCNN with Locally Active Memristors DOI: http://dx.doi.org/10.5772/intechopen.100463*

illustrative examples, the stability analysis of two-cell [37] and three-cell [29] arrays can be found in the literature. In this work, we present a quick inspection method, which is conceptually introduced in [9] and has been implemented through an example in [28], where the idea is to terminate the coupling node of the single cell with a coupling resistor, resulting in the circuit depicted in **Figure 5(a)** for the case study presented here.

To investigate the possible destabilization scenarios of the resistively terminated cell in **Figure 5(a)**, we follow the same circuit theoretical approach as in the previous subsections and examine the corresponding AC equivalent circuit, which is shown in **Figure 5(b)**. Here, the poles of the impedance function *Zc*ð Þ*s* coincide with the eigenvalues of the second-order system equivalent of the circuit in **Figure 5(a)** at a given equilibrium point and should be examined to determine the destabilization conditions. At this point, it should be noted that the AC equivalent circuits depicted in **Figures 3(b)** and **5(b)** are qualitatively equivalent to each other, while *Rc* appears additionally in parallel with *Rb* in **Figure 5(b)**. Therefore, the expression of *Zc*ð Þ*s* , readily given in Eq. (5), is identical to the expression of *Z s*ð Þ, where the term *Rb* is replaced with the parallel equivalent term *Rbc*, that is, *Rbc* ¼ *Rb* ∙ *Rc=*ð Þ *Rb* þ *Rc* .

$$Z\_c(s) = \frac{sL(R\_1 + R\_2)R\_{bc} + R\_1R\_2R\_{bc}}{s^2LC(R\_1 + R\_2)R\_{bc} + s[L(R\_1 + R\_2 + R\_{bc}) + R\_1R\_2R\_{bc}C] + (R\_1 + R\_{bc})R\_2} = \frac{N\_c(s)}{D\_c(s)}\tag{5}$$

To claim instability of *Zc*ð Þ*s* , at least one coefficient of the *Dc*ð Þ*s* polynomial has to be negative, as dictated by Routh-Hurwitz criteria. Here, it is straightforward to see that the coefficient of the *s* <sup>2</sup> term of *Dc*ð Þ*<sup>s</sup>* is readily positive. Similarly, it is possible to show after some algebraic rearrangement that the coefficient of the *s* term remains positive as long as the stability precondition given by Eq. (4), with *Rb* replaced by *Rbc*, is satisfied. Thus, for instability, the constant term of *Dc*ð Þ*s* , namely ð Þ *R*<sup>1</sup> þ *Rbc* ∙ *R*2, has to be negative. Starting with the inequality ð Þ *R*<sup>1</sup> þ *Rbc* ∙ *R*<sup>2</sup> < 0 and replacing *Rbc* with *Rb* ∙ *Rc=*ð Þ *Rb* þ *Rc* directly give us the destabilization condition introduced by Eq. (6).

$$R\_c < -\frac{R\_1 \bullet R\_b}{R\_1 + R\_b} = R\_{c-\max} = -(R\_1 \| R\_b) \tag{6}$$

In **Figure 4(a)**, we plot (in orange) *Rc*� *max* as a function of the equilibrium point of the memristor current *IQ*�*NDR* across the entire NDR region, for two different values of *Rb*. Similar to the characteristics of *Cmax*, the smaller value of *Rb* results in a larger value for *Rc*� *max* , which would relieve the design constraints for an

#### **Figure 5.**

*(a) A simplified scenario for the resistively coupled single cell where Rc stands for the coupling resistor. (b) AC equivalent circuit of the cell in (a). The poles of Zc*ð Þ*s shall be investigated for analyzing the stability of the circuit in (a) at a given equilibrium point.*

hardware realization. On the other hand, Eq. (6) reveals the fact that considering the memristor model-dependent quantities, *Rc*� *max* depends only on the slope of the I-V curve (i.e., *R*<sup>1</sup> �1 ), while *Cmax* depends on other quantities (see Eq. (4)) as well. Besides, as a function of *IQ*�*NDR*, the characteristics of *Cmax* increase (decrements), while the characteristics of *Rc*� *max* decrease (increments). Moreover, it is interesting to mention that the condition derived in Eq. (6) is qualitatively equivalent to the results presented in [29, 37], indicating the fact that the simplification considered in this work is accurate and provides a quick insight into the destabilization scenario of the coupled cell in **Figure 5(a)**.

Considering a typical time constant based design approach, we characterize the term *Rc*� *max* ∙*Cmax*, introducing the exact and an approximate product in Eq. (7), and plot these quantities in **Figure 4(b)** as a function of *IQ*�*NDR*, for two different values of *Rb*. Here, we would like to note that the approximate product term *L=*ð Þ *R*<sup>1</sup> þ *Rb* is obtained under the assumption ð*R*<sup>1</sup> þ *R*<sup>2</sup> þ *Rb*Þ ffi *R*2.

$$R\_{c-\max} \bullet C\_{\max} = -\frac{R\_1 \bullet R\_b}{R\_1 + R\_b} \bullet \frac{L(R\_1 + R\_2 + R\_b)}{-R\_1 R\_2 R\_b} = \frac{L(R\_1 + R\_2 + R\_b)}{R\_2(R\_1 + R\_b)} \cong \frac{L}{(R\_1 + R\_b)}\tag{7}$$

It can be seen from **Figure 4(b)** that the approximated time constant term given in Eq. (7) follows the original expression very closely, especially for the smaller value of *Rb*. Moreover, the approximated value is independent of *R*2, but rather dependent on the dynamic quantity *L*, or in other words, the switching speed of the memristor device. As a final and crucial remark, referring to the instability region of parameters, that is, the sharp-edge-of-chaos domain, it is possible to rigorously define the necessary circuit design variables by taking into account the parametric equations given by Eqs. (4) and (6).
