Detecting Boosting Weak Signal via A Meminductive Multistable Chaotic System

In this paper, we rebuild a new meminductive chaotic circuit model based on a Wien-bridge oscillator. Due to the extreme multistability of the meminductive system, it can produce the phenomenon of many infinitely coexisting attractors. Systems that spontaneously produce coexisting oscillations are relatively rare in the study of meminductive circuit systems and are well suited as pseudo-random number generators (PRNG). In this study, a new weak signal detection model is established based on the proposed meminductive system. This detection model can detect boosting weak signals with different amplitudes. The trajectory of objects can be predicted effectively by the boosting line spectrum in the frequency domain. The experimental result shows the feasibility through which the meminductive multistable system is applied in the detection field.

1 Introduction

Memristor, a nonlinear memory resistor, has been used extensively in recent studies. The concept of memristor first appeared in a paper by Chua [1]. Based on the relationship between electrical physical quantities, Chua outlines that there must be a set of relationships between the charge and the flux. In subsequent studies, Chua’s team defined the resistance-like relationship between them as memristor. Until 2008, according to the conceptual model of memristor, HP laboratory manufactured the first nanoscale memristor using TiO2 material [2]. It also confirms the correctness of Chua’s prediction. After further scientific certification, some researchers found that the volt-ampere characteristic curve of the memristor is a hysteresis loop due to its nonlinear characteristic [3–8]. Unlike normal volatile resistors, memristors are efficient at retaining energy, i.e. memristor is a component with a memory function. The memory properties of the memristor are a good match for the resistive components that will be used to build future artificial brains, for which it made headlines in scientific journals [9, 10]. Then, the Ventra team proposed the concept of meminductor [11], which adds better memory features to conventional storage devices.

Research on memristive circuit systems has made great progress, with continuous breakthroughs in memristive hardware [12–15]. The memristor oscillation circuit based on Chua’s circuit model has become a hotspot of nonlinear system research. Due to the more nonlinear parts of the memristor, it can often produce complex dynamic behavior and higher sequence complexity. Thus, the memristive dynamical system is widely used in the related fields of physics [16–19], computer science [20–24], and cryptography [25–28]. Ye et al constructed a memristive circuit model based on Wen’s bridge oscillating circuit. By replacing a memristor of the original 4D Wen’s bridge oscillator, the new circuit equations with a memristor have more nonlinear terms. Based on this, the dynamic behavior of the memristive circuit system becomes more complex, and there is even multi-attractor coexistence. There is even a phenomenon of multi-attractor coexistence [29]. Bao et al proposed an inductor-free memristive circuit using linear coupling of a BPF and found many coexisting infinite attractors by using numerical simulation [30]. There is a large amount of research on memristors in the existing literature. However, there are relatively few studies on meminductor, especially on the meminductive multi-stable system. Thus, it is very important to further explore complex dynamic behavior and the application of a memory-sensing system.

In this article, we focus on a novel meminductive chaotic system of extremely multistability and its application in signal detection. The article is organized as follows: in section 2, the meminductive equivalent circuit model is proposed. In section 3, the meminductive chaotic system model is built. In section 4, the proposed meminductive system is applied to weak signal detection. Finally, the chaotic attractors of the meminductive system are implemented by circuit experiment.

2 Design of Meminductive Equivalent Circuit

Based on the i-φ characteristics of meminductor, we have

$i\left(t\right)=M^{-1}\left(\mu \right)\varphi \left(t\right).(1)$

in the formula above, i(t) and φ(t) are the current and magnetic flux flowing through the meminductor, and M(μ) reflects the inductive resistance of the meminductor. Then, we get

$\mu ={\displaystyle {\int }_{-\infty }^t\phi \left(\tau \right)}d\tau .(2)$

Based on Eqs 1, 2, the mathematical model of meminductor can be designed as

$q\left(\mu \right)=a_{\mu }\mu +b_{\mu }{\mu }^3.(3)$

in which, a_μ and b_μ are the positive real numbers, and then we have

$i\left(t\right)=(a_{\mu }+3b_{\mu }{\mu }^2)\phi \left(t\right).(4)$

Based on the mathematical model of the meminductor above, the equivalent circuit model of the meminductor can be designed as shown in Figure 1. According to the relationships of all components reflected in Figure 1, we can achieve the i-φ characteristic of the meminductor:

$i\left(t\right)=(\frac{R_9}{R_8}+\frac{R_9}{R_7}{\mu }^2)\phi \left(t\right).(5)$

FIGURE 1

FIGURE 1. The equivalent circuit of meminductor.

According to Eqs 4, 5, a_μ = R₉/R₈ and b_μ = R₉/R₇. Set the component parameters in Figure 1 R₁ = R₃ = R₄ = 200kΩ, R₅ = R₆ = R₈ = R₉ = 10kΩ, R₂ = 1MΩ, R₇ = 3kΩ, and C₁ = C₂ = 10nF. By using the circuit simulation software, as shown in Figure 2, we can get the v-i hysteresis loop under different frequencies.

FIGURE 2

FIGURE 2. The v-i hysteresis loop of a meminductive equivalent circuit under different frequencies. (A) f = 50 Hz. (B) f = 70 Hz. (C) f = 180 Hz. (D) waveform of f = 50 Hz.

3 Constructing a Meminductive Circuit

3.1 Constructing a Meminductive Circuit Based on Wien-Bridge Oscillator

By building an RLC oscillation structure based on a Wien-bridge oscillator, a novel meminductive circuit model can be rebuilt. As reflected in Figure 3, the new meminductive circuit model has four resistors R₁∼R₄, three capacitors, one operational amplifier, two diodes, and one meminductor.

FIGURE 3

FIGURE 3. A novel meminductive circuit based on a Wien-bridge oscillator.

3.2 Mathematical Model

Based on the circuit model reflected in Figure 7, we can obtain the circuit equations

$\{\begin{array}{l}{C}_1\frac{d{u}_1}{dt}=\frac{R_4{u}_1}{R_2{R}_3}-\frac{u_1}{R_1}-\frac{u_2}{R_2}-i_d\hfill \\ C_2\frac{d{u}_2}{dt}=\frac{R_4{u}_1}{R_2{R}_3}-\frac{u_2}{R_2}\hfill \\ C_3\frac{d{u}_3}{dt}=i_d-L^{-1}\left(\rho \right)\phi \hfill \\ \frac{d\rho }{dt}=\phi \hfill \\ \frac{d\phi }{dt}=u_3\hfill \end{array},(6)$

in Eq. 6, the equations of the diodes can be expressed as

$i_e=g_e[u_1-u_3+0.5\left(\right|u_1-u_3-U_e|-|u_1-u_3+U_e\left|\right)],(7)$

in which, U_e shows the threshold voltage, g_e is the conductance. Then, let u₁ = U_ex, u₂ = U_ey, u₃ = U_ez, φ = U_eC₁(R₂R₃)^1/2v, ρ = U_thC₁²R₁R₂w, i_e = U_eg_eH, t = C₁(R₁R₂)^1/2τ, c=(R₁/R₂)^1/2, d = R₄/R₃, e=(R₁R₂)^1/2g_e, C₁ = C₂ = C₃, a = U_eC₁R₁R₂a₁, b = U_e³C₁⁵R₁³R₂³b₁, L⁻¹(ρ) = a+3bρ², we have

$H=x-z+0.5\left(\right|x-z-1|-|x-z+1\left|\right).(8)$

Finally, we can get an equivalent mathematical model

$\{\begin{array}{l}\dot{x}=(cd-1/c)x-cy-eH\hfill \\ \dot{y}=cdx-cy\hfill \\ \dot{z}=eH-(a+3b{w}^2)v\hfill \\ \dot{w}=v\hfill \\ \dot{v}=z\hfill \end{array}.(9)$

According to the mathematical equations of the system (9), the dynamical behavior of the proposed system can be analyzed. When the system parameters are a = 40, b = 10, c = 0.6, d = 8, e = 30, the initial condition is (1,1,1,0.1,0.01). We can calculate Lyapunov exponents are Ly₁ = 0.4403, Ly₂ = 0.0127, Ly₃ = 0, Ly₄ = -0.4536, Ly₅ = -35.18. This indicates the proposed system is a hyperchaotic circuit system.

4 The Application of the Meminductive System in Weak Signal Detection

4.1 The Construction of the Detection System Model

Due to the initial value sensitivity of chaotic systems, the system model may generate resonance signals with some weak signals. The detection model of the weak signal can be built based on the system (9). In the process of signal detection, the test weak signal is often mixed with the noise signals. In particular, some noise signals have a close frequency to the test weak signal. The detection model can be expressed as

$\{\begin{array}{l}\dot{x}=(cd-1/c)x-cy-eH\hfill \\ \dot{y}=cdx-cy+K_1\cos \left(w_{1}t\right)+K_2\cos (w_{2}t+\phi )\hfill \\ \dot{z}=eH-(a+3b{w}^2)v\hfill \\ \dot{w}=v\hfill \\ \dot{v}=z\hfill \end{array},(10)$

where K₁ is the amplitude of the test weak signal, w₁ is its angular frequency. K₂ is the amplitude of the noise signal, w₂ is its angular frequency, and is its initial phase. The newly constructed detection model (10) inherits the extremely multiple stability from the meminductive system (9). Let system parameter a = 40, b = 10, c = 0.6, d = 8, e = 30, C₁ = 1, C₂ = 2, w₁ = 1, w₂ = 2, φ = 0, and the initial conditions is (1,1,1,w(0),0.01). As shown in Figure 4, when w(0) = -0.1, 0.2, 0.5, 0.7, and 0.9 respectively, the detection model can produce infinite coexisting attractors. Figure 5 shows the boosting signals with different amplitudes when w(0) = 0.1, 0.2, 0.3, 0.4, 0.5 respectively.

FIGURE 4

FIGURE 4. Infinitely coexisting attractors generated by the system (10). (A) x-z-w plane. (B) v-y-w plane.

FIGURE 5

FIGURE 5. The boosting chaos signals of the time domain. (A) The boost signals on w direction at 0–200 s. (B) The boost signals on w direction at 300–500 s.

4.2 The Extraction of the Spectral Line in the Frequency Domain

In the weak signal detection field, to understand the trajectory of the object producing the weak signal, the line spectrum in the frequency domain is often extracted. Keeping other parameters unchanged, let the initial value w = 0.1, 0.5, 1, 1.5, and 2 respectively, the line spectrum of the test signal in the frequency domain is reflected in Figure 6. As the initial condition w increases constantly, the corresponding amplitude of the sequence is boosting constantly, and the boosting signal can be detected within the frequency range 0–1 Hz. Due to the proposed meminductive system being extremely multistable, the useful signal can be detected within different amplitudes. Even if a detector in the amplitude range fails, the weak signals can also be detected in other amplitude ranges, which increases the fault tolerance of the detection process. These boosting signals show more useful information about the running object. Based on this key information, the trajectory of the object can be predicted.

FIGURE 6

FIGURE 6. The line spectrum of the boosting weak signal with varying amplitudes.

5 Realization of the Meminductive Circuit System

Doing the circuit implementation to the system (9). Based on Eqs 8, 9, we have

$\{\begin{array}{l}\frac{dX}{dT}=\frac{1}{R_4{C}_1}(\frac{R_4}{R_1}X-\frac{R_4}{R_2}Y-\frac{R_4}{R_3}H)\hfill \\ \frac{dY}{dT}=\frac{1}{R_9{C}_2}(\frac{R_9}{R_7}X-\frac{R_9}{R_8}Y)\hfill \\ \frac{dZ}{dT}=\frac{1}{R_{11}{C}_3}(-\frac{R_{11}}{R_{19}}{W}^{2}V-\frac{R_{11}}{R_{18}}V+\frac{R_{11}}{R_{10}}H)\hfill \\ \frac{dW}{dT}=\frac{R_{17}}{R_{16}{C}_5}V\hfill \\ \frac{dV}{dT}=\frac{R_{15}}{R_{14}{C}_4}Z\hfill \end{array},(11)$

in which,

$H=X-Z+0.5\left(\right|X-Z-1|-|X-Z+1\left|\right).(12)$

Set t = KT, x = MX, y = MY, z = MZ, w = MW, v = MV, and set scale transformation factors M = 2 and K = 100, and we have

$\{\begin{array}{l}\frac{dx}{dt}=\frac{1}{K{R}_4{C}_1}(\frac{R_4}{R_1}x-\frac{R_4}{R_2}y-\frac{R_4}{R_3}H\prime )\hfill \\ \frac{dy}{dt}=\frac{1}{K{R}_9{C}_2}(\frac{R_9}{R_7}x-\frac{R_9}{R_8}y)\hfill \\ \frac{dz}{dt}=\frac{1}{K{R}_{11}{C}_3}(-\frac{R_{11}}{4R_{19}}{w}^{2}v-\frac{R_{11}}{R_{18}}v+\frac{R_{11}}{R_{10}}H\prime )\hfill \\ \frac{dw}{dt}=\frac{1}{K{R}_{17}{C}_5}\frac{R_{17}}{R_{16}}v\hfill \\ \frac{dv}{dt}=\frac{1}{K{R}_{15}{C}_4}\frac{R_{15}}{R_{14}}z\hfill \end{array},(13)$

where,

$H\prime =x-z+0.5\left(\right|x-z-0.5|-|x-z+0.5\left|\right).(14)$

In combination with the formula above, the equivalent circuit model of the system (9) can be designed as shown in Figure 7. Based on the equivalent circuit, as shown in Figure 8, the chaotic attractors can be implemented by multisim 14.0. The specific component parameters are shown in Table 1.

FIGURE 7

FIGURE 7. The equivalent circuit of the proposed meminductive circuit.

FIGURE 8

FIGURE 8. Chaotic attractors in the circuit simulation. (A) V_x-V_y plane. (B) V_x-V_z plane.

TABLE 1

TABLE 1. The circuit parameters in the circuit experiment.

6 Conclusion

This proposes a new meminductive chaotic system of extreme multistability. Due to the constant shift of equilibrium with different initial values, the proposed meminductive chaotic system can generate many infinitely coexisting attractors. By introducing the model of weak signal and noise to the meminductive chaotic system, a new weak signal detection model is established. Because this detection model inherits the extreme multistability of the original meminductive system, it can detect boosting signals of different amplitudes at the same time. The spectral line in the frequency domain can effectively reflect the motion trajectory of the object generating a weak signal. Next, we will try to apply the proposed meminductive system to other related areas.

Data Availability Statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Author Contributions

BK: Conceptualization; Methodology; Data curation; Supervision; Investigation; Project administration; Writing—original draft; Writing—review and editing. WQ: Funding acquisition; Investigation; Project administration; Resources; Validation.

Funding

This work was supported by the National Natural Science Foundation of China (No: 12171004, 12026233, and 12026221).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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