Initial-Offset-Control and Amplitude Regulation in Memristive Neural Network

Abstract

Traditional Hopfield neural networks (HNNs) suffer from limitations in generating controllable chaotic dynamics, which are essential for applications in neuromorphic computing and secure communications. Memristors, with their memory-dependent nonlinear characteristics, provide a promising approach to regulate neuronal activities, yet systematic studies on attractor offset behaviors remain scarce. In this study, we propose a fully memristive electromagnetic radiation neural network by incorporating three distinct memristors as external electromagnetic stimuli into an HNN. The parameters of the memristors were tuned to modulate chaotic oscillations, while variations in initial conditions were employed to explore multistability through bifurcation and basin stability analyses. The results demonstrate that the system enables large-scale amplitude control of chaotic signals via memristor parameter adjustments, allowing arbitrary scaling of attractor amplitudes. Various offset behaviors emerge, including parameter-driven symmetric double-scroll relocations in phase space and initial-condition-induced offset boosting that leads to extreme multistability. These dynamics were experimentally validated using an STM32-based electronic circuit, confirming precise amplitude and offset control. Furthermore, a multi-channel pseudo-random number generator (PRNG) was implemented, leveraging the initial-boosted offset to enhance security entropy. This offers a hardware-efficient chaotic solution for encrypted communication systems, demonstrating strong application potential.

1. Introduction

The human brain, with its approximately several billion neurons and trillions of synapses, is one of the most complex and efficient information processing systems known [1,2]. Neurons, the fundamental units of the brain, communicate through electrical and chemical signals, creating a vast network that underlies all cognitive functions [3,4]. The intricate dynamics of neuronal interactions, including synaptic plasticity, enable the brain to learn, adapt, and perform a multitude of tasks with remarkable efficiency and speed. Inspired by the brain’s architecture and functionality, artificial neural networks (ANNs) have been developed to emulate this biological information processing [5,6]. One specific type of neural network, the Hopfield neural network (HNN), is a recurrent network where neurons are fully connected to each other [7,8,9]. HNNs are particularly known for their ability to store and retrieve memory patterns, making them useful for associative memory tasks. To model neurons more accurately, various neuron models have been proposed, such as the Hodgkin–Huxley model, the integrate-and-fire model, and, more recently, spiking neuron models. These models aim to capture the essential characteristics of neuronal activity [10,11,12], including action potential generation, synaptic transmission, and plasticity. However, traditional HNNs face significant challenges in generating chaotic phenomena, which are essential for complex dynamic behaviors and enhanced performance in engineering applications [13,14,15].

The discovery of memristors has opened new avenues for overcoming these limitations [16,17]. Leon Chua first proposed the theoretical concept of the memristor in 1971, describing it as the fourth fundamental passive circuit element, alongside the resistor, capacitor, and inductor [18]. Wu et al. propose a core–shell silver nanowire memristor that confines conductive filament growth to a quasi-2D plane, achieving low variability, ultra-low power, and neuromorphic nociceptor emulation [19]. Tian et al. developed a 4D memristive chaotic system using magnetic-controlled memristor and fractional-order calculus, exhibiting complex dynamics like coexisting attractors and chaotic bursting [20,21]. This breakthrough confirmed the practical viability of memristors and sparked a surge of interest in their potential applications, particularly in neuromorphic engineering and neural networks. Incorporating memristors into neuronal systems has been a hot topic in recent years [22,23,24,25]. Sajad et al. explored the impact of various factors on neural system behavior and neuron membrane potential. Specifically, they examine how the conductance gain of memristive synapses, the applied current, and the speed at which electrical signals propagate along axons and dendrites influence these dynamics [26]. Alexander et al. proposed a novel ring network based on HR neurons and explored the coupling dynamic behavior [27]. Zhang et al. designed a multi-piecewise nonlinear memristor and applied it to HNN for generating any desired number of multi-scroll attractors [28]. Lin et al. extracted multi-structure attractors in neural networks by initial-boosted control [29]. Bao et al. explored scroll growing in the memristive cyclic Hopfield neural network [30]. The regulation of chaotic signal amplitude, frequency, and spatial offset can be conveniently achieved through direct manipulation of system parameters. The memristor is capable of characterizing electromagnetic radiation [31]. Some researchers have investigated the relationship between electromagnetic radiation effects and initial-offset-control [32,33,34]. However, previous works have not explored the various initial-offset behaviors. Although previous studies have proposed theoretical methods for finite-range amplitude control applicable to single-scroll [35] or double-scroll [36] regulation, none of these models demonstrate ultra-large-scale amplitude control characteristics.

Based on the above analysis, this paper proposes a novel memristive electromagnetic radiation neural network. The innovations of this work are summarized as follows:

(1) Three memristors are introduced as external electromagnetic radiation into a fully connected memristive neural network, establishing the fully memristive electromagnetic radiation neural network.

(2) Large-scale amplitude control is effectively realized through the adjustment of memristor-related parameters. By changing these parameters, the oscillation range of chaotic signals can be scaled to any desired extent. This flexibility allows for precise control over the chaotic behavior within the neural network.

(3) Various offset behaviors are found in neural networks, where parameter-dominated offset and initial-boosted are used to arrange attractors in the phase space under different locations. In addition, extreme multistability is extracted by different initial conditions.

(4) Circuit implementation is built to verify various dynamics, including amplitude control and offset control. The multi-channel pseudo-random number generator (PRNG) is designed to explore its high performance in chaos-based applications.

The rest of this paper is organized as follows: In Section 2, the model of a fully memristive electromagnetic radiation neural network is proposed. In Section 3, the basic dynamics are analyzed. In Section 4, amplitude control of neural networks is explored and analyzed. In Section 5, different offset controls are revealed by arranging attractors in different locations. In Section 6, a digital circuit implementation is built, and the application in PRNG is carried out. In Section 7, this paper is concluded.

2. Memristive-Based Hopfield Neural Network

2.1. Memritor Model

Drawing from the core principles of memristor theory, three voltage-controlled generalized memristor models are created. The generalized expression for the memristor is as follows:

$$\left\{\begin{array}{c}i=w(\phi )v\\ \dot{\phi }=F(\phi ,v)\end{array}\right.$$

(1)

where memductance is denoted by w(φ).

The memristor-1 (M1) is expressed as

$$\left\{\begin{array}{l}i=w(u_1)x=\left|u_1\right|x\\ {\dot{u}}_1=c_{1}x-d_1{u}_1\end{array}\right.$$

(2)

The memristor-2 (M2) is expressed as

$$\left\{\begin{array}{l}i=w(u_2)y=(1+u_2)y\\ {\dot{u}}_2=c_{2}y-d_{2}G(u_2)\end{array}\right.$$

(3)

G(u₂) is expressed as

$$G(u_2)=\left\{\begin{array}{l}{G}_1(u_2)=\left\{\begin{array}{l}{u}_2,M=0\\ u_2-{\displaystyle {\sum }_{i=1}^M(\mathrm{sgn}(u_2+(2i-1))+\mathrm{sgn}(u_2-(2i-1)))},M=1,2,3\dots ,\end{array}\right.\\ G_2(u_2)=\left\{\begin{array}{l}{u}_2-\mathrm{sgn}(u_2),N=0\\ u_2-\mathrm{sgn}(u_2)-{\displaystyle {\sum }_{j=1}^N(\mathrm{sgn}(u_2+2j)+\mathrm{sgn}(u_2-2j))},N=1,2,3\dots \end{array}\right.\end{array}\right.$$

(4)

The memristor-3 (M3) is expressed as

$$\left\{\begin{array}{l}i=w(u_3)z=\tanh (u_3)z\\ {\dot{u}}_3=c_{3}z\end{array}\right.$$

(5)

The excitation is achieved using a sinusoidal voltage described by V = Asin(2πft). With the initial conditions of the memristors set to φ(0) = 0 and f = 1 Hz. The numerical solution results of the pinched hysteresis loops of three memristors are shown in Figure 1.

2.2. Memristive Neural Network Model

Three memristors are introduced as external electromagnetic radiation into a fully connected Hopfield neural network to establish a new type of memristor neural network. The connection structure is shown in Figure 2.

The model of the memristive neural network is expressed as follows:

$$\left\{\begin{array}{l}\dot{x}=-x+0.8\tanh (x)-10\tanh (y)+8\tanh (z)-k_1(\left|u_1\right|)x\\ \dot{y}=-y+0.8\tanh (x)+4\tanh (y)-4\tanh (z)-k_2(1+u_2)y\\ \dot{z}=-z+1.1\tanh (x)+4\tanh (y)-0.1\tanh (z)-k_3\tanh (u_3)z\\ {\dot{u}}_1=c_{1}x-d_1{u}_1\\ {\dot{u}}_2=c_{2}y-d_{2}G(u_2)\\ {\dot{u}}_3=c_{3}z\end{array}\right.$$

(6)

where x, y, and z are the variables of neurons 1, 2, and 3. Parameters k₁, k₂, and k₃ are the coupling strengths of different memristors. Parameters c₁, c₂, c₃, d₁, and d₂, are the system parameters.

3. Basic Dynamics Analysis

The dynamics of the memristive neural network described by Equation (6) were analyzed using the fourth-order Runge–Kutta (RK4) method in MATLAB R2019b. The parameters were specified as follows: fixed time step of 0.005 and integration schemes of 500.

3.1. Equilibrium and Stability Analysis

The equilibrium points and their stability analysis play a pivotal role in unraveling the formation mechanisms of chaotic attractors. By setting the left-hand side of Equation (6) to zero, the equilibrium points can be calculated by solving the following equation:

$$\left\{\begin{array}{l}0=-x+0.8\tanh (x)-10\tanh (y)+8\tanh (z)-k_1(\left|u_1\right|)x\\ 0=-y+0.8\tanh (x)+4\tanh (y)-4\tanh (z)-k_2(1+u_2)y\\ 0=-z+1.1\tanh (x)+4\tanh (y)-0.1\tanh (z)-k_3\tanh (u_3)z\\ 0=c_{1}x-d_1{u}_1\\ 0=c_{2}y-d_{2}G(u_2)\\ 0=c_{3}z\end{array}\right.$$

(7)

Observing that z = 0, Equation (7) reduces to

$$\left\{\begin{array}{l}0=-x+0.8\tanh (x)-10\tanh (y)-k_1(\left|u_1\right|)x\\ 0=-y+0.8\tanh (x)+4\tanh (y)-k_2(1+u_2)y\\ 0=1.1\tanh (x)+4\tanh (y)\\ 0=c_{1}x-d_1{u}_1\\ 0=c_{2}y-d_{2}G(u_2)\\ 0=z\end{array}\right.$$

(8)

Given the system parameters k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 15, d₂ = 5, and c₃ = 1, Equation (7) is solved numerically using MATLAB R2019b. The equilibrium points for M = 0, N = 0, M = 1, and N = 1 are listed in

Table 1. Stability analysis of equilibrium points in the system (6).

	Equilibrium Points	Non-Zero Eigenvalues	Stabilities
M = 0	(0, 0, 0, 0, 0, 0)	−0.3593 ± 3.6242i, 2.4085, −2, −5	Unstable saddle focus
N = 0	(0, 0, 0, 0, −1, 0)	−0.3644 ± 3.6196i, 2.4087, −2, −5	Unstable saddle focus
	(0, 0, 0, 0, 1, 0)	−0.3642 ± 3.6287i, 2.4084, −2, −5	Unstable saddle focus
M = 1	(0, 0, 0, 0, −2, 0)	−0.3494 ± 3.6150i, 2.4089, −2, −5	Unstable saddle focus
	(0, 0, 0, 0, 0, 0)	−0.3593 ± 3.6242i, 2.4085, −2, −5	Unstable saddle focus
	(0, 0, 0, 0, 2, 0)	−0.3691 ± 3.6332i, 2.4082, −2, −5	Unstable saddle focus
N = 1	(0, 0, 0, 0, −3, 0)	−0.3445 ± 3.6104i, 2.409, −2, −5	Unstable saddle focus
	(0, 0, 0, 0, −1, 0)	−0.3554 ± 3.6196i, 2.4087, −2, −5	Unstable saddle focus
	(0, 0, 0, 0, 1, 0)	−0.3642 ± 3.6287i, 2.4084, −2, −5	Unstable saddle focus
	(0, 0, 0, 0, 3, 0)	−0.374 ± 3.6378i, 2.408, −2, −5	Unstable saddle focus

. To analyze the stability of these equilibrium points, we compute the Jacobian matrix at each point and determine their eigenvalues and types, as summarized in Table 1. As shown in Table 1, all of the equilibrium points are unstable saddle foci.

Furthermore, the equilibrium points and corresponding phase trajectories with the initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1) are given in Figure 3. To facilitate the observation of the relationship between the multi-scroll attractor and the nonlinear function responsible for generating multiple unstable equilibrium points, f(u₂,y) = c₂y − d₂G(u₂) = 0 is plotted as a black line in Figure 3. As shown in Figure 3, each equilibrium point (red dot) is associated with a distinct scroll of the chaotic attractor. The specific number of scrolls generated by system (6) is determined by the parameters M and N. Specifically, these parameters allow for the creation of attractors with an odd number of scrolls, given by 2M + 1, as well as those with an even number, given by 2N + 2.

3.2. Bifurcation and Lyapunov Exponents

To further illustrate the evolution process of the system’s dynamical behavior, the Lyapunov exponents and bifurcation diagram of the system are shown in Figure 4. It can be observed that when the system parameters are k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 5, c₂ = 1, d₂ = 1, c₃ = 1, M = 0, and initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1), the parameter k₁ is varied within the range of [0.1, 9], and a robust chaotic interval is exhibited by the system (6).

3.3. Symmetry and Phase Diagram

When the system parameters are k₁ = 8, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 5, c₂ = 1, d₂ = 1, c₃ = 1, M = 0, and initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1), the attractor is displayed in Figure 5. As shown in Figure 5, the phase trajectories exhibit symmetry. In Figure 5a, the phase trajectory in the plane u₁-x exhibits symmetry about the center of the origin. In Figure 5b, the phase trajectory in the plane u₁-y also exhibits symmetry about the center of the origin.

4. Amplitude Control in Memristive Neural Network

4.1. Amplitude of Single-Scroll Attractor

The topic of controlling the range of signal generation through a single parameter has become increasingly popular. In this chapter, a systematic exploration is conducted on how memristor-related parameters influence the range of chaotic oscillations. First, we investigate the control exerted by parameter M1 on the fourth-dimensional signal of system (6). The phase trajectory reveals that the attractor gradually diminishes in size with the parameter increases, as shown in Figure 6. Figure 6a,b are obtained with k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, c₂ = 1, d₂ = 1, c₃ = 1, M = 0, and initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1). d1 is set to 2, 4, and 6. As depicted in Figure 6a,b, parameter d₁ serves as the control variable for amplitude rescaling of signals along the u₁-dimension. Figure 6c,d are obtained with k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 1, c₃ = 1, M = 0, and initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1). Simulation exhibits a consistent behavior: as the parameter d₁ or d₂ increases, the range of the chaotic oscillations decreases proportionally.

4.2. Amplitude of Multi-Scroll Attractor

Memristor-related parameters are endowed with proper values to generate any desired number of multi-scroll attractors. Set k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 15, c₃ = 1, and IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1). Amplitude rescaling of multi-scroll attractors is plotted in Figure 7. In Figure 7a, N = 0 and d₂ are set to 6, 8, and 10. In Figure 7b, M = 1 and d₂ are set to 6, 8, and 10. As can be seen from Figure 7, as the value of parameter d₂ increases from 6 to 8 to 10, the amplitude in the u₂ dimension exhibits a progressive rescaling effect. Nevertheless, the amplitude control for the u₃-dimension is extraordinarily large. The details of this phenomenon will be discussed in the following sections.

4.3. Large-Scale Amplitude Control

The large-scale amplitude control of the system (6) was discovered through changes in parameter c₃. We found that the oscillation range of the attractor can be infinitely amplified, which is distinct from previously reported amplitude controls. This represents a proportional amplitude control, where the oscillation range of the system continuously expands as the parameter increases. To the best of our knowledge, such extensive control has not been reported before. Set k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 1, d₂ = 1, M = 0, and initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1). As shown in Figure 8, the amplitude of attractors is flexibly controlled. To further verify the dynamical characteristics of amplitude control, the mean value plot, Lyapunov exponents, bifurcation diagram, and waveform plot are shown in Figure 9.

To further demonstrate the characteristics of our large-scale amplitude control, parameter c₃ is set to 10⁸ and 10¹⁰ for the exploration of phase trajectory control, as shown in Figure 10.

5. Various Offset-Control and Multistability

5.1. Parameter-Offset-Control

By examining Equation (6), it was observed that when k₁ = 2.5, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 0, c₂ = 1, d₂ = 1, c₃ = 1, M = 1, and initial condition IC = (0.1, 0.1, 0.1, 0.1-p, 0.1, 0.1), the fourth-dimensional variable u₁ appears only once. According to the theory of parameter-offset-control, as illustrated in Equation (10), the parameter p can position the attractor at different locations in the phase space. As shown in Figure 10, the system’s signal graph further demonstrates the significance of offset control. Expanding on this observation, the unique occurrence of the u₁ variable in the fourth dimension underscores the potential for precise control over the system’s dynamics. Parameter-offset-control enables the strategic manipulation of system parameters, thereby allowing for the attractor’s repositioning within the phase space. Figure 11 provides a visual representation of this concept, highlighting how offset control can be employed to achieve desired signal behavior and enhance system performance.

$$\left\{\begin{array}{l}\dot{x}=-x+0.8\tanh (x)-10\tanh (y)+8\tanh (z)-k_1(\left|u_1+p\right|)x\\ \dot{y}=-y+0.8\tanh (x)+4\tanh (y)-4\tanh (z)-k_2(1+u_2)y\\ \dot{z}=-z+1.1\tanh (x)+4\tanh (y)-0.1\tanh (z)-k_3\tanh (u_3)z\\ {\dot{u}}_1=c_{1}x\\ {\dot{u}}_2=c_{2}y-d_{2}G(u_2)\\ {\dot{u}}_3=c_{3}z\end{array}\right.$$

(9)

5.2. Initial-Offset-Control

System (6) was given different initial values, revealing the multistability controlled by continuous initial values. Using the parameters k₁ = 2, k₂ = 0.05, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 1, d₂ = 1, c₃ = 0.1, M = 1, and the initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, IC-u₃), we plot the verification for initial-offset-control in Figure 12. As shown in Figure 12, the bifurcation diagram continuously climbs, reflecting the coexistence of infinitely many attractors. The system’s signal graph further verifies this characteristic.

5.3. Coexisting Attractors and Multistability

When the parameters related to the memristor are changed, set k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 5, d₂ = 5, c₃ = 1, and initial condition IC = (0.1, 0.1, 0.1, 0.1, IC-u2, 0.1), coexisting attractors are discovered, as shown in Figure 13. The number of coexisting attractors can also be arbitrarily designed, providing a stable signal source for multi-channel PRNG. The number of signal channels depends on the design of the multi-scroll-related memristor M₂. Verifications for any desired number of coexisting attractors are shown in Figure 14.

6. Circuit Implementation and Application in PRNG

In order to validate the dynamics of the proposed memristive neural network model (6), the fourth-order Runge–Kutta (RK4) method was applied to discretize the system. Digital circuits, particularly Microcontroller Units (MCUs) like the STM32 series, have gained significant popularity in the implementation of chaotic neural systems, often surpassing analog circuit approaches in performance and practicality. In this paper, we utilize STM32 technology to implement the proposed memristive neural network.

6.1. Digital Circuit Implementation

The MCU used in this experiment is the STM32F407ZGT6 (STMicroelectronics, Geneva, Switzerland), a high-performance 32-bit controller based on the Cortex-M4 core. This MCU can operate at a frequency of up to 168 MHz. It also implements a full set of DSP instructions and a Memory Protection Unit (MPU) to enhance application security. The STM32F407ZGT6 includes high-speed embedded memory (up to 1 MB of Flash memory and up to 192 KB of SRAM), up to 4 MB of backup SRAM. The notable advantage of the STM32F407ZGT6 MCU is its support for hardware floating-point operations. This capability significantly accelerates floating-point operations, which are fundamental to achieving the high computational throughput necessary for complex tasks.

To intuitively validate the output waveforms, an oscilloscope verification platform was constructed using the DAC8563 module. The circuit implementation is shown in Figure 15.

The system parameters are configured as k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 15, d₂ = 5, and c₃ = 1, with an initial condition IC = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1). Following this, the digital signals are transformed into analog signals by the DAC and subsequently observed using an oscilloscope. The results are presented in Figure 16. We can see that the results of the hardware circuit experiments are consistent with the numerical simulations of Figure 3. The hardware implementation based on STM32 successfully generates multi-scroll chaotic attractors.

6.2. Application in Pseudo-Random Number Generator

Due to many significant properties, such as sensitivity to initial conditions and unpredictability, chaotic systems are widely used in many academic and industrial fields, including pseudo-random number generators (PRNGs) [37,38,39,40]. In particular, a pseudo-random number generator (PRNG) is developed using chaotic sequences generated by the memristive neural network, which is controlled by specific initials. Let us consider a chaotic sequence. From this sequence, a series of pseudo-random numbers (PRNs) P_i with values ranging from 0 to N can be derived as follows:

$$P_i=⌊(U_i+\left|U_{\min }\right|.K)⌋\mathrm{mod}N$$

(10)

where K is a positive integer, U_min is the smallest value in U, and ⌊ ⌋ signifies the floor function, which provides the greatest integer that is less than or equal to the given number. Let K = 10⁷ and N = 256 in this paper. In this experiment, we utilized the hyperchaotic sequence generated by Equation (6) with parameters k₁ = 2, k₂ = 0.01, k₃ = 0.1, c₁ = 1, d₁ = 2, c₂ = 5, d₂ = 5, and c₃ = 1. The National Institute of Standards and Technology (NIST) test suite is used to measure its performance.

According to the multistability rising in the u₃-dimension. The results of random numbers generated by different PRNGS in the NIST test are shown in

Table 2. Multi-channel PRNG and initial condition IC = (0.1, 0.1, 0.1, 0.1, IC-u2, 0.1).

No.	Statistical Test Terms	IC-u2 = −2	IC-u2 = 0	IC-u2 = 2
01	Frequency	0.801943	0.761613	0.564310
02	Block frequency	0.491342	0.573194	0.601349
03	Cumulative sums	0.973154	0.619421	0.976104
04	Runs	0.534916	0.554913	0.651943
05	Longest run	0.546122	0.310467	0.531649
06	Rank	0.879134	0.319456	0.394614
07	FFT	0.394612	0.846120	0.761345
08	Non-overlapping template	0.808631	0.906491	0.961345
09	Overlapping template	0.804164	0.763420	0.631649
10	Universal	0.391672	0.465312	0.531946
11	Approximate entropy	0.164973	0.109712	0.549130
12	Random excursions	0.397611	0.491637	0.297613
13	Random excursions variant	0.813421	0.829460	0.531649
14	Serial	0.613401	0.651943	0.709164
15	Linear complexity	0.791342	0.319453	0.649132

. This demonstrates that the designed hyperchaotic map produces pseudo-random numbers exhibiting high randomness.

7. Conclusions and Discussion

Traditional neural networks struggle to generate chaotic behavior, and the introduction of memristors not only increases system complexity but also enhances the potential for dynamic control. In this study, three memristors were integrated into a fully connected neural network to simulate external electromagnetic radiation. Initially, limited amplitude regulation of the fourth and fifth dimensions was observed, followed by the extraction of extensive amplitude modulation in the sixth dimension, with simulation results showing an amplitude range reaching up to 10¹⁰. The unique memory properties of memristors provide new opportunities for generating multistable states. The implementation of initial-offset-control and parameter-offset-control allows attractors to be positioned differently in the phase space, offering greater flexibility for multi-channel PRNG implementation, which is promising for secure communication applications. The same attractors are induced by the offset operation. The proposed system’s effectiveness was validated through digital circuit-based simulations. Future work will focus on achieving greater controllability in simpler memristive neural networks.