# A Passive but Local Active Memristor and Its Complex Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. A Six-Lobe Locally Active Memristor with a Continuous and Derivable State Equation

^{2}− 4ac < 0, where we set the parameters: a = 1, b = 0, and c = 0.1. We adjust the order of magnitude of the memristance with the parameter k so that it matches the actual memristor, where we take k = 10

^{−3}, while parameter k

_{2}is used to control the change rate of state variable x. If k

_{2}= 10

^{3}, the variable rate agrees with the actual memristors. As there is no solution if α(x) = 0 when calculating stable equilibrium points of the memristor on its DC V–I curve, we set α(x) ≠ 0, i.e., a

_{3}≠ 0 and a

_{2}

^{2}− 4a

_{1}a

_{3}< 0, where we set a

_{1}= 3, a

_{2}= 0.2, and a

_{3}= 0.1.

_{3}for balancing the effect of internal force and external force, where k

_{3}= 0.05. In addition, γ(x) can be written as a continuous function γ(x) = −k

_{3}((48x

^{5}− 480x

^{3}+ 240x)/(1 + x

^{2})

^{5}).

#### 2.1. Dynamic Route Map of the Memristor

#### 2.2. Hysteresis Characteristics of Memristor

^{−4}, and the memristor is equivalent to a linear resistor with a resistance of 10

^{4}Ohms. Therefore, the model is consistent with the characteristics of a memristor [13].

#### 2.3. Local Activity

## 3. Small-Signal Analysis for Locally Active Region

#### 3.1. Zero-Pole Analysis of Memristor

_{11}(Q) = ∂i/∂x = 2xv/1000 and a

_{12}(Q) = ∂i/∂v = (x

^{2}+ 0.1)/1000. Furthermore, the operating point Q(V, I) on the DC V–I curve must meet the condition dx/dt = g(x,v). Through differential expansion of the state equation dx/dt = g(x,v) in Equation (1), we obtain

_{11}(Q)b

_{12}(Q)), Rx = b

_{11}(Q)/(a

_{11}(Q)b

_{12}(Q)), and Ry = a

_{12}(Q).

_{x}< 0 and R

_{y}> 0 in the locally active regions, and L

_{x}< 0 for the stable locally active points, while L

_{x}> 0 for the unstable locally active points.

_{12}(Q), Z = (a

_{12}(Q)b

_{11}(Q) − a

_{11}(Q)b

_{12}(Q))/a

_{12}(Q), and P = b

_{11}(Q).

#### 3.2. Frequency Response of the Memristor

_{0}, the real part and imaginary part of the memductance function are ReY(iω,V) = K(ω

^{2}+ pz)/(ω

^{2}+ p

^{2}) and ImY(iω,V) = iKω(z − p)/(ω

^{2}+ p

^{2}), respectively. To make the locally active system generate oscillation, there should be a pair of complex conjugate poles (Hopf bifurcation points) on the imaginary axis for the admittance function Y(s, Q) of the locally active memristor. In other words, it is necessary to add an energy storage element (capacitance or inductance) in series or in parallel with the memristor to form a locally active system. The selection of capacitance or inductance depends on the frequency response of the memristor. Figure 6 shows the frequency response of the memristor where the voltage at both ends of the memristor is 0.6000 V. In order to construct an oscillation system, if Lx < 0 (Lx > 0) in the equivalent circuit of the memristor shown in Figure 4, an inductance L* = 1/(ωImY(iω,V)) (capacitor C* = ImY(iω,V)/ω) in series (parallel) with the memristor is required.

^{−4}S at ω* = 2902 rad/s, indicating that the memristor is inductive and therefore a positive capacitance C* in parallel with the memristor is needed to compensate the ImY(iω*,V), as well as to make the total impedance of the C*-augmented memristive circuit equal to zero at operating point V = 0.6000 V. The compensated capacitance can be obtained using the following formula:

^{−4}S at ω* = 200.7 rad/s, indicating that the memristor is capacitive and therefore a positive inductance L* in series with the memristor is needed to compensate the ImY(iω,V), as well as to make the total impedance of the L*-augmented memristive circuit equal to zero at operating point V = 0.1031 V. The compensated inductance can be obtained using the following formula:

## 4. Second-Order Periodic Circuit of Memristor

_{x}< 0 and L

_{x}> 0. For the two cases, we design two second-order memristive circuits, as shown in Figure 8.

#### 4.1. Properties of the Memristive Circuit in the Unstable Locally Active Region of the Memristor

_{2}, as shown in Figure 9.

_{C}(s, Q) of the circuit satisfies that 1/Y

_{C}(s, Q) = 1/(Y(s, Q) + Y

_{C*}) + R, where Y

_{C*}= sC*. Therefore, Y

_{C}(s, Q) can been written as follows:

_{C}(s, Q) of the circuits are obtained from Equation (9) as follows:

_{1}= (kR + 1 − pRC)/RC, a

_{2}= −(p + kzR)/RC, b

_{1}= (k − pC)/RC, and b

_{2}= −kz/RC; p, z, and k are the pole, the zero, and the coefficient of admittance function Y(s, Q) of the memristor in Equation (7), respectively.

_{1}and P

_{2}) of Y

_{C}(s, Q) with respect to the capacitance C, in which V = 0.6 V and the state variable x = 0.3322. Observe from Figure 10 that Y

_{C}(s, Q) has a pair of complex conjugate poles on the imaginary axis at C = 97.13 nF and Im p

_{1,2}= ±1556, which are the Hopf bifurcation parameters. When C < 97.13 nF, such as C = 65.84 nF, the real parts of the complex conjugate poles are less than zero, and the circuit gradually stabilizes to an equilibrium point, as shown in Figure 10a. However, when C = 97.13 nF, the periodic oscillation shown in Figure 10b,c occurs in the circuit. Moreover, the oscillation amplitude increases with the initial value, as described in Figure 10d. If C > 97.13 nF, the system enters the unstable right half plane of the complex plane and may oscillate.

_{1}and p

_{2}) of Y

_{C}(s, Q) with respect to the voltage V, where C = 97.13 nF. Observe that Y

_{C}(s, Q) has a pair of complex conjugate poles on the imaginary axis at the Hopf bifurcation parameters: V = 0.6 V and Im p

_{1,2}= ±1556.

#### 4.2. Properties of the Memristive Circuit in the Stable Locally Active Region of the Memristor

_{C}(s, Q) of the circuit satisfies

_{L*}= 1/sL*. Therefore, Y

_{C}(s, Q) can be written as follows:

_{3}= (1 − KZL)/KL and a

_{4}= −P/KL.

_{1}and P

_{2}) of the admittance Y

_{L}(s, Q) versus the inductance L, where V = 0.1031 V and x = −1.022. Observe that L = 1.972 H is the Hopf bifurcation point of the memristive circuit where Im p

_{1,2}= ±200.7. If L > 1.972 H, the system enters the right half plane of the complex plane and may oscillate; for example, for L = 2.021 H, a periodic oscillation appears as shown in Figure 14.

_{1}and p

_{2}) of the admittance Y

_{L}(s, Q) versus the voltage V, where L = 1.972 H.

## 5. Memristor-Based Third-Order Chaotic Circuit

_{x}< 0, as shown in Figure 16.

_{C}is the voltage across the memristor, i

_{L}is the current through the inductance, x is the state variable of the memristor, and v is the supply voltage.

#### 5.1. System Equilibrium Points

_{c}/dt = 0, and di

_{L}/dt = 0 in Equation (13); the following four equilibrium points of the system can be obtained: E

_{1}(−1.022, 0.1031 V, 0.179 mA), E

_{2}(−0.98, 0.1031 V, 0.109 mA), E

_{3}(0.026, 0.1031 V, 0.104 mA), E

_{4}(0.633, 0.1031 V, 0.516 mA). The four equilibrium points of the system (13) happens to be the equilibrium points of the memristor. Equilibrium E1 is located in the locally active region.

_{3}).

#### 5.2. Influence of Parameters L and C on System Dynamics

_{1}(−1.022, 0.1031 V, 0.179 mA); the variation in inductance L and capacitance C can cause the system to bifurcate. Figure 18a shows the variation in the system Lyapunov exponent spectrum [16] with the capacitance C within the interval of 10 μF–30 μF, where inductance L = 0.96 H. Figure 18b shows the bifurcation of the state variable x with the capacitance C within the interval of 21 μF–26.2 μF.

_{3}(0.026, 0.1031 V, 0.104 mA).

_{1}(−1.022, 0.1031 V, 0.179 mA), and the capacitance C = 26 μF. Observe from Figure 21 that for 0.925 H ≤ L ≤ 0.962 H, the system generates chaotic oscillation. With the increase in inductance L, the system bifurcates from period doubling to chaos by period-doubling bifurcation. Obviously, a Period 3 window can be observed from Figure 22b. If L > 0.962 H, the system gradually stabilizes to the system equilibrium point E3.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 7.**(

**a**) Small-signal frequency response of memristor and (

**b**) Nyquist diagram of memristor when it operates under V = 0.1031 V, x = −1.022, L* = 1.972 H.

**Figure 8.**(

**a**) Parallel capacitance and (

**b**) series resistance of locally active memristor oscillation circuit.

**Figure 10.**Simulation of memristor oscillation circuit: (

**a**–

**c**) system initial value V = 0.6000 V and x = 0.3322 and (

**d**) increasing oscillation amplitude, if the initial value is far from the equilibrium point.

**Figure 11.**Variation in the real and imaginary parts of the poles of the admittance function in the oscillation circuit with the DC voltage (if C = 97.13 nF).

**Figure 12.**Variation in the real and imaginary parts of the poles of the admittance function in the oscillation circuit with the external capacitance (if V = 0.6000 V).

**Figure 13.**Variation in the real and imaginary parts of the poles of the admittance function in the oscillation circuit with the external capacitance (if V = 0.1031 V).

**Figure 15.**Variation in the real and imaginary parts of the poles of the admittance function in the series inductance oscillation circuit with the DC voltage.

**Figure 18.**(

**a**) Lyapunov exponent of the system, and (

**b**) bifurcation of variable x with the capacitance C (if L = 0.96 H).

**Figure 19.**Variation in system attractor with capacitance C, where V = 0.1031 V, L = 0.96 H. (

**a**) C = 21.5 μF, (

**b**) C = 23 μF, (

**c**) C = 23.8 μF, (

**d**) C = 26 μF.

**Figure 21.**(

**a**) Lyapunov exponent spectrum and (

**b**) bifurcation of system with the inductance L (if C = 26 μF).

**Figure 22.**Variation in system attractor with inductance L, v = 0.1031 V, C = 26 μF. (

**a**) L = 0.88 H, L = 0.91 H, (

**b**) L = 0.925 H, (

**c**) L = 0.95 H, (

**d**) L = 0.963 H.

**Figure 24.**Typical phase diagrams. (

**a**) C = 21.5 μF, L = 0.9 H; (

**b**) C = 24 μF, L = 0.94 H; (

**c**) C = 23.5 μF, L = 0.96 H; (

**d**) C = 24 μF, L = 0.98 H; (

**e**) C = 26 μF, L = 0.98 H.

Range of x | Lobe | Corresponding Voltage (V) | Corresponding Current I (mA) | Stability |
---|---|---|---|---|

(−3.979, −3.667) | 6 | (−1.816 × 10^{−4}, −1.989 × 10^{−4}) | (−2.894 × 10^{−6}, −2.694 × 10^{−6}) | Unstable |

(−1.169, −1.000) | 2 | (0.08855, 0.1034) | (1.298 × 10^{−4}, 1.138 × 10^{−4}) | Stable |

(−0.3963, −0.2702) | 3 | (−0.5395, −0.6600) | (−1.387 × 10^{−4}, −1.140 × 10^{−4}) | Unstable |

(0.2649, 0.3909) | 1 | (0.6369, 0.5206) | (1.084 × 10^{−4}, 1.316 × 10^{−4}) | Unstable |

(0.9921, 1.152) | 4 | (−0.09095, −0.07837) | (−9.862 × 10^{−5}, −1.119 × 10^{−4}) | Stable |

(3.653, 3.954) | 5 | (1.414 × 10^{−4}, 1.295 × 10^{−4}) | (1.902 × 10^{−6}, 2.038 × 10^{−6}) | Unstable |

Lobe | Equivalent Circuit Lx | P | Z | Energy Storage Element |
---|---|---|---|---|

1 | > 0 | > 0 | < 0 | Parallel capacitance |

2 | < 0 | < 0 | > 0 | Series inductance |

3 | > 0 | > 0 | < 0 | Parallel capacitance |

4 | < 0 | < 0 | > 0 | Series inductance |

5 | > 0 | > 0 | < 0 | Parallel capacitance |

6 | > 0 | > 0 | < 0 | Parallel capacitance |

Equilibrium Point | Characteristic Value | Equilibrium Points Types | ||
---|---|---|---|---|

λ_{1} | λ_{2} | λ_{3} | ||

E_{1} | −157.01 | 9.55 + 156.13i | 9.55–156.13i | Saddle focus |

E_{2} | 144.14 | −45.92 + 159.2i | −45.92–159.2i | Saddle focus |

E_{3} | −11810 | −2 + 204i | −2–204i | Stable focus |

E_{4} | 4273.2 | −8.1 + 204.2i | −8.1–204.2i | Saddle focus |

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**MDPI and ACS Style**

Li, F.; Liu, J.; Zhou, W.; Dong, Y.; Jin, P.; Ying, J.; Wang, G.
A Passive but Local Active Memristor and Its Complex Dynamics. *Electronics* **2022**, *11*, 1843.
https://doi.org/10.3390/electronics11121843

**AMA Style**

Li F, Liu J, Zhou W, Dong Y, Jin P, Ying J, Wang G.
A Passive but Local Active Memristor and Its Complex Dynamics. *Electronics*. 2022; 11(12):1843.
https://doi.org/10.3390/electronics11121843

**Chicago/Turabian Style**

Li, Fupeng, Jingbiao Liu, Wei Zhou, Yujiao Dong, Peipei Jin, Jiajie Ying, and Guangyi Wang.
2022. "A Passive but Local Active Memristor and Its Complex Dynamics" *Electronics* 11, no. 12: 1843.
https://doi.org/10.3390/electronics11121843