**3.2 Features**

It can be mentioned that for considered coupled MBOs the movement along the trajectory in the direction opposite to the original one can be provided by changing the signs of the derivatives. This property can be called as reversibility of trajectories. The property is valid for stable trajectories as well as for any unstable trajectories before its transition to stable ones. Such a feature may be foundation for the management of coupled MBOs.

In order to get from the original fixed trajectory (for example A) onto given trajectory (for example D), it is enough to choose the intersection point of the predetermined path with the threshold line (*R*<sup>2</sup> ¼ *Rm*) and then to construct the trajectory of leaving it until the inevitable intersection with the original trajectory using change in derivative sign. The control signal with short duration and sufficient amplitude moves the image point to the specified trajectory.

The process of transition to stationary trajectory can be represented using the mapping function of the value *RS* over the period: *RS*ð Þ¼ *n* þ 1 *P R*ð Þ *<sup>S</sup>*ð Þ *n* (**Figure 7**). The value of *RS* for the *n*-th period is given in **Figure 7** at the abscissa axis and similar value as a result of Poincare mapping for *n+1* period is shown at the ordinate axis. The area of stable states belonging to the diagonal D (**Figure 7**) satisfies the condition: �*r*< *Rs* <*r*, that corresponds to the area of stationary trajectories.

Until j j *RS* > 3*r* the return of *RS* to the region of stable states is performed with stepsize equal to 4*r*. If *RS* located in the interval *r*<*RS* <3*r* then the return occurs in one step equal to 2ðj j *RS* � 1).

As follows from this analysis, the speed of the transition process from the excited state to the stationary state depends on the coupling strength or in other words on the coefficient *r* . The width of the stability area also depends on coupling strength. The return to the stationary trajectory after external excitation can be relatively long at low values of factor *r*. It can be expected that the return time is

**Figure 7.** *The function of mapping the difference in the states of the coupled MBOs for period.*

proportional to the amplitude of the external signal in a certain range of amplitude varying.

The situation changes significantly when the excitation has a long duration, comparable to or exceeding the duration of the period *TS*. In this case new values of the threshold resistances for *R*<sup>1</sup> are set during the action of the input signal. Then the value *R*<sup>1</sup> will change within this interval of variation. The duration of the transition to the perturbed state will also depend on the coupling strength. The output signal will appear on the phase detector in this case. The return to the stationary trajectory will repeat again after completion of the input signal and signals at the output of the phase detector will appear again.

The behavior of self-oscillating coupled MBO is described by piecewise-constant differential equations. As a result, the complete analytical solution can be obtained. In practice, it reduces to solving the problem of elastic reflection of a point inside a rectangle with edges positioned depending on the sign of the point's speed.

### **3.3 Simulation examples**

Below the results of simulation of the coupled MBOs are given. The simulation examples demonstrate the opportunity to control the state of the coupling MBOs and illustrate also waveforms of generation of the pulse trains at the input excitation.

The time is defined as dimensionless variable. Also, the dimensionless values of the circuit parameters and variables were used during simulation. Among them: *RM* ¼ 0*:*8, *Rm* ¼ 0*:*4, *r* ¼ 0*:*1.

### *3.3.1 Example 1: managing the state of coupling MBOs*

The considered circuit example has a set of stable and unstable steady- state trajectories and provides complex transformation of input signal. The simulation example illustrates the presence of three stable steady-state periodic solutions (**Figure 8**) correspond to *Rs* ¼ *R*<sup>1</sup> *max* � *R*<sup>2</sup> *max* >0, *Rs* ¼ *R*<sup>1</sup> *max* � *R*<sup>2</sup> *max* <0, *Rs* ¼ *R*<sup>1</sup> *max* � *R*<sup>2</sup> *max* ¼ 0.

Input signals lead to switching of stable trajectories and provides various modes in application.

Let the starting points for the variable resistances be the same al for MBO1 and MBO2 (*R*1ð Þ¼ 0 *R*2ð Þ¼ 0 0*:*5). When the first control pulse with amplitude *VC* ¼ 0*:*4 and duration *T* ¼ 0*:*5 is applied to MBO1 the process is generated in which the amplitude of the resistance oscillations of MBO1 is greater than the similar amplitude for MBO2. The state corresponds to inequality *RS* >0 (**Figure 8**). The second

**Figure 8.** *The computed waveforms in the coupled MBOs. V<sup>С</sup> - the solid line,* R*1(t) - dashed line,* R*2(t) - dotted line.* *Functional Capabilities of Coupled Memristor-Based Reactance-Less Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97808*

#### **Figure 9.**

*Generation of train of output pulses (dotted line) under the action of the input signal of short duration (a) and long duration (b).*

control pulse leads to change of trajectory and generates the steady state corresponding to inequality *RS* <0 (**Figure 8**).

By such a way this simulation example confirms the predicted change of memristor states in the considered circuit under control pulse excitation.
