**3. Behavior of coupled memristor based oscillators with positive couplings**

### **3.1 Operating principles**

The analysis of behavior of two coupled identical MBO with positive connection is presented below.

This circuit is shown in **Figure 5**. It contains MBO1, MBO2, an adder at the input and a phase detector at the output [50]. To provide an external control the excitation signal *VC* is transmitted using an adder at the input. The phase detector at the output is used to identify the synchronization mode of coupled oscillators. If there is no synchronization between the oscillator stages MBO1 and MBO2 then output signal *Vs* ¼ 1 and *Vs* ¼ 0 if there is synchronization.

The coupling strengths between the MBOs specified by coefficient *k* impact on the behavior of this system significantly.

The rates of change of memristor resistances *R*<sup>1</sup> and *R*<sup>2</sup> are equal in modulus for identical MBOs. But these rates may differ in signs. By such a way the variables *R*<sup>1</sup> and *R*<sup>2</sup> and the signs of derivatives *dR*1*=dt* and *dR*2*=dt* can be considered as system states and may specify the behavior of system of two coupled oscillators.

The phase plane with axes *R*<sup>1</sup> and *R*<sup>2</sup> (**Figure 6**) can be exploited for analysis of different behavior versions of such a system. The analysis is based on model Eq. (8). In this case, the trajectories of moving the image points are straight lines. They pass at angles of � π/4 on phase plane. Four trajectories can pass through each point of phase plane. The sign of *dR/dt* defines one from them.

The boundaries of the area of trajectories movement are specified by the threshold resistances. When the trajectory reaches the boundary the sign of the derivative *dR=dt* changes and trajectory is mirrored from the boundary. The boundaries can shift themselves at this time point.

If the external excitations are absent then the threshold of each MBO depends on positive pulse from the neighboring MBO. In particular the lower limit of the resistance of each MBO is reduced to *Rm* � *r* . As a result the area of the allowable system states on the phase plane in self-oscillating mode is determined by the square with vertices ð Þ *RM*, *RM* and ð Þ *Rm* � *r*, *Rm* � *r* (**Figure 6**). The area of

**Figure 5.** *Schematic of coupled memristor based oscillators (MBOs).*

#### **Figure 6.**

*The boundaries and trajectories at phase plane of changing the variables R*<sup>1</sup> *and R*<sup>2</sup> *for coupled MBOs: solid lines – boundaries for case dR*1*=dt* ¼ *dR*2*=dt, dash-dotted lines - boundaries for case dR*1*=dt* 6¼ *dR*2*=dt, solid lines with arrows - the trajectories of R*<sup>1</sup> *and R*<sup>2</sup> *with different initial conditions. The areas of stable trajectories are limited by dashed lines.*

stationary trajectories is located insight this square. This area is limited by the dashed lines in **Figure 6**.

For the existence of a stationary trajectory, the following necessary and sufficient conditions must be met: the image points must be located in the area indicated above, and the signs of the derivatives must be identical.

If the variables are located at the main diagonal in this area and the specified conditions are met, then the variables reach the threshold simultaneously (dotted line A in **Figure 6**). Their moving directions also change simultaneously. They continue to move along the main diagonal. When the threshold line is reached by one variable on the other lines parallel to the main diagonal in this area, the sign of its derivative changes. This is followed by the threshold change for another variable with a corresponding change in the sign of its derivative. The trajectory is saved, but the movement along it occurs in the opposite direction. Note that the phases of the oscillations of the resistors are the same (*Vs* ¼ 0Þ for stable trajectories.

If the starting points of trajectories are located outside area of stationary trajectories (**Figure 6**) then such trajectories are reflected after reaching the boundaries. If in this case the signs of the derivatives are the same then the segments of the trajectories tend to the stability region. The reflection character is defined by the boundaries with different signs of derivatives *dR*1*=dt* 6¼ *dR*2*=dt* (dashed lines in **Figure 6**). Any trajectory ends in the region of stable trajectories in result. Such behavior is illustrated in **Figure 6** by examples of the trajectories B and C. It can be seen that the trajectory B falls into the stability region after two reflections and the trajectory C - after four reflections.

The considered circuit with two coupled identical oscillator elements (**Figure 5**) has a set of stable and unstable steady state trajectories. The difference between the maximal values of the variables *Rs* ¼ *R*<sup>1</sup> *max* � *R*<sup>2</sup> *max* . can be exploited as characteristic of stable steady state trajectories. It can be mentioned that zero value *Rs*ð*Rs* ¼ 0) corresponds to the main diagonal on phase plan (**Figure 6**). This characteristic reaches the value*Rs* ¼ �*r* at the boundaries of the stable region. The each stationary trajectory (each value of *Rs*) corresponds to a certain period of triangular oscillations which equals to

$$T\_S = 2\frac{R\_M - R\_m + r - R\_S}{\chi I}.\tag{10}$$

## *Functional Capabilities of Coupled Memristor-Based Reactance-Less Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97808*

Let duration of the additional external control signal *VC* be shorter than period *TS*. This signal *VC* can change the boundary and the trajectory of movement on the phase plane respectively. **Figure 6** shows the boundary *U* created by an external signal. The trajectory *D* in **Figure 6** illustrates the transition to new stable trajectory under the influence of an external signal. The starting point of trajectory *D* is located at the main diagonal. The trajectory D moves away from the main diagonal under the external excitation. After three reflections (**Figure 6**) the transition of image point to new stable trajectory is carried out.
