According to Equations (1) and (2), when the parameters are set as
, the color POP of Equation (1) with arrow heads is shown in
Figure 1. It can be observed that there are five intersections with
, namely E
1, E
2, E
3, E
4, and E
5. By adjusting the dynamic route of the five points, we can judge that the equilibrium points E
1, E
3, and E
5 are asymptotically stable, whereas the equilibrium points E
2 and E
4 are unstable. The green lines represent the negative slope ranges, while the blue lines represent the positive slope ranges. The arrow above the x = 0 axis faces right, and the curve arrow below the x = 0 axis faces left.
2.1. Pinched Hysteresis Loops
When the memristor is driven by a periodic signal source with amplitude A and frequency , the curves of the memristor are pinched hysteresis loops passing through the origin. While the parameters of the memristor and the periodic signal source take different values, and the initial values of Equation (1) also take different values, the memristor will exhibit a variety of characteristics. The parameters of Equations (1) and (2) are set as .
Let
, and the parameter
is changed. When
, the dynamic trajectory displays three coexisting pinched hysteresis loops, as shown in
Figure 2b, and the corresponding time-domain waveforms are shown in
Figure 2a.
It can be observed from
Figure 2 that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 2b; if the initial value
, the pinched hysteresis loop is the blue curve of
Figure 2b; if the initial value
, the pinched hysteresis loop is the green curve of
Figure 2b. The red and green curves are symmetrical hysteresis loops about the origin. When
, the dynamic trajectories display double coexisting pinched hysteresis loops, as shown in
Figure 3b, and the corresponding time-domain waveforms are shown in
Figure 3a. It can be observed from
Figure 3 that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 3b; if the initial value
, the pinched hysteresis loop is the green curve of
Figure 3b. The red and blue curves are symmetrical hysteresis loops about the origin, and they have two pinched points. When
, the dynamic trajectory displays only one pinched hysteresis loop, as shown in
Figure 4b, and the corresponding time-domain waveforms are shown in
Figure 4a. It can be observed that the dynamic trajectories have three stable states at the beginning, and as time goes on, the final curves are independent of the initial values and eventually converge to the same curve.
Let the amplitude
, and the parameter
is changed. When
, the dynamic trajectory displays three coexisting pinched hysteresis loops, as shown in
Figure 5b, and the corresponding time-domain waveforms are shown in
Figure 5a. It can be observed from
Figure 5 that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 5b; if the initial value −
, the pinched hysteresis loop is the blue curve of
Figure 5b; if the initial value
, the pinched hysteresis loop is the green curve of
Figure 5b. The red and green curves are symmetrical hysteresis loops about the origin. When
and −
, the dynamic trajectories display double coexisting pinched hysteresis loops, as shown in
Figure 5d, and the corresponding time-domain waveforms are shown in
Figure 5c. It can be observed that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 5b; if the initial value
, the pinched hysteresis loop is the green curve of
Figure 5b. The red and blue curves are symmetrical hysteresis loops about the origin, and they have two pinched points. When
, the dynamic trajectory displays only one pinched hysteresis loop, as shown in
Figure 5f, and the corresponding time-domain waveforms are shown in
Figure 5e. It can be observed that the pinched hysteresis loop has three pinched points and is symmetric about the origin.
Let the amplitude
, and the parameter
is changed. When
, the dynamic trajectory displays three coexisting pinched hysteresis loops, as shown in
Figure 6b, and the corresponding time-domain waveforms are shown in
Figure 6a. It can be seen that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 6b; if the initial value −
, the pinched hysteresis loop is the blue curve of
Figure 6b; if the initial value
, the pinched hysteresis loop is the green curve of
Figure 6b. The red and green curves are symmetrical hysteresis loops about the origin. When
, the dynamic trajectories display double coexisting pinched hysteresis loops, as shown in
Figure 6d, and the corresponding time-domain waveforms are shown in
Figure 6c. It can be observed that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 6d; if the initial value
, the pinched hysteresis loop is the green curve of
Figure 6d. The red and blue curves are symmetrical hysteresis loops about the origin and they have two pinched points.
Let the amplitude
, and the parameter
is changed. When
, the dynamic trajectory displays double coexisting pinched hysteresis loops, as shown in
Figure 7b, and the corresponding time-domain waveforms are shown in
Figure 7a. It can be observed that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 7b; if the initial value
, the pinched hysteresis loop is the blue curve of
Figure 7b. The red and green curves are symmetrical hysteresis loops about the origin. When
, the dynamic trajectories display three coexisting pinched hysteresis loops, as shown in
Figure 7d, and the corresponding time-domain waveforms are shown in
Figure 7c. It can be observed that if the initial value
, the pinched hysteresis loop is the red curve of
Figure 7d; if the initial value −
, the pinched hysteresis loop is the blue curve of
Figure 7d; if the initial value
<
, the pinched hysteresis loop is the green curve of
Figure 7d. The red and blue curves are symmetrical hysteresis loops about the origin. It can be seen from the above analysis that the memristor presented different stable states with changes in parameters and input signals.
2.2. Local Activity
The DC V-I plot reflects Ohm’s law for the memristor, which describes the DC characteristics of the memristor and can analyze the intrinsic locally active features of the memristor. Let
in Equation (1), we can obtain the relationship between voltage
and state variable
:
Then, setting
. Based on Equation (3), we can obtain a function between the state variable
and the DC voltage
:
Substituting Equations (4) and (5) into Equation (2), the DC current
can be calculated, respectively:
Let
, based on Equations (4) and (6), when −
we drew points
and
in the
and
planes, respectively, and obtained the DC
plots shown in
Figure 8a–c. Based on Equations (5) and (7), we drew points
in the
plane and obtained the DC
plot shown in
Figure 8d in the same way. It was observed that the slopes of all of the curves had negative values; therefore, we determined that the memristor was a locally active memristor.
Comparison with the existing memristor model is shown in
Table 1. It was found that the model proposed in this paper had more parameters and equilibrium points. In order to verify the correctness of the analysis, we designed a circuit on the proposed memristor, as shown in
Figure 9. When we placed a sinusoidal excitation at both ends of the memristor, pinched hysteresis loops were observed, as shown in
Figure 10. It could be concluded that the analysis was correct by comparison.