Dynamic Analysis and FPGA Implementation of a Fractional-Order Memristive Hopfield Neural Network with Hidden Chaotic Dual-Wing Attractors

Abstract

To model the response of neural networks to electromagnetic radiation in real-world environments, this study proposes a memristive dual-wing fractional-order Hopfield neural network (MDW-FOMHNN) model, utilizing a fractional-order memristor to simulate neuronal responses to electromagnetic radiation, thereby achieving complex chaotic dynamics. Analysis reveals that within specific ranges of the coupling strength, the MDW-FOMHNN lacks equilibrium points and exhibits hidden chaotic attractors. Numerical solutions are obtained using the Adomian Decomposition Method (ADM), and the system’s chaotic behavior is confirmed through Lyapunov exponent spectra, bifurcation diagrams, phase portraits, and time series. The study further demonstrates that the coupling strength and fractional order significantly modulate attractor morphologies, revealing diverse attractor structures and their coexistence. The complexity of the MDW-FOMHNN output sequence is quantified using spectral entropy, highlighting the system’s potential for applications in cryptography and related fields. Based on the polynomial form derived from ADM, a field programmable gate array (FPGA) implementation scheme is developed, and the expected chaotic attractors are successfully generated on an oscilloscope, thereby validating the consistency between theoretical analysis and numerical simulations. Finally, to link theory with practice, a simple and efficient MDW-FOMHNN-based encryption/decryption scheme is presented.

1. Introduction

With the development of new-generation information technologies such as cloud computing [1,2,3], big data [4,5,6], central networks [7,8,9], and deep learning [10,11,12], Artificial Neural Network (ANN), as a computing model that mimics the structure and function of biological neural networks, has demonstrated astonishing roles in these fields. The design of ANNs draws inspiration from the operational principles of biological neural systems, particularly the structure and functionality of the human brain. As a highly complex nonlinear system, the human brain exhibits rich chaotic behaviors. Investigating these chaotic behaviors contributes to a deeper understanding of the brain’s information processing mechanisms, holding significant exploratory value and potential applications in neuroscience [13,14,15,16]. Against this backdrop, the Hopfield neural network (HNN), characterized by its nonlinear activation functions and interacting neurons, generates diverse dynamic behaviors, making it a focal point in the fields of neural computation and nonlinear dynamics research [17,18,19]. Notably, in real-world environments, humans and their nervous systems are continuously exposed to various forms of electromagnetic radiation, which may influence neuronal activity and warrant further investigation [20,21,22,23].

The memristor, a nonlinear two-terminal dynamic device, possesses a memristance that depends on the historical profile of charge or magnetic flux passing through it [24,25,26], enabling adaptive resistance adjustments in response to external stimuli [27,28]. Consequently, memristors are regarded as ideal components for modeling the synaptic plasticity of artificial neurons. Neural networks incorporating memristors, known as memristive neural networks (MNNs), exhibit significantly richer dynamic behaviors compared to non-memristive counterparts, including neural bursting [29,30], hyperchaos [31,32], multi-scroll chaotic attractors [33,34], singular attractors [35], and hidden chaotic attractors [36,37,38]. Based on the characteristics of memristors, researchers have developed various applications and computational models, significantly advancing the development of artificial intelligence [39,40,41]. At the microscopic level, memristance variations arise from internal ion migration or physical state changes. Owing to their non-volatile nature, memristors retain their state after power-off until the next power-on. Leveraging this property, memristors effectively simulate the dynamic responses of real neurons under the influence of external electromagnetic radiation, providing a valuable tool for investigating the potential effects of electromagnetic fields on neural networks.

Research on MNNs currently centers on the exploration of novel chaotic attractors and their potential in theoretical applications and hardware implementation, including hidden attractors, multi-scroll attractors, emerging ribbon attractors and multi-wing attractors [42,43,44,45,46]. These attractors have demonstrated significant success in applications such as chaotic encryption [34,47], image classification [48], high-quality random number generation [49], and path planning [50], highlighting their substantial value in industrial contexts. Attractors with wing-shaped structures are more difficult to obtain. From a mathematical perspective, such structures typically require the introduction of cross terms and quadratic terms from the memristor’s state variables to be generated. Notably, hidden attractors, which exist in dynamic systems devoid of equilibrium points, are challenging to detect using conventional equilibrium-based analysis, resulting in limited research. A key limitation in applied studies is that complex nonlinear models of continuous MNNs, when solved using current numerical methods for differential equations, exhibit low computational efficiency, leading to slower outputs. In contrast, discrete chaotic systems offer superior computational efficiency, making them more advantageous for industrial applications [51,52,53,54]. Furthermore, optimizing designs to reduce the number of memristors while achieving comparable or more complex dynamic behaviors can effectively lower hardware costs and energy consumption, meriting further investigation.

The fractional-order system extends integer-order systems by introducing fractional calculus, showcasing non-locality and memory characteristics, thereby achieving higher accuracy in modeling complex system behaviors [55,56,57]. Compared to integer-order systems, fractional-order systems have significant advantages in nonlinear dynamics and long-term memory capture. For instance, Li et al. utilized the long-term memory of fractional calculus and complex chaotic behaviors to achieve faster convergence in solving the classical traveling salesman problem [58]. Hioual et al. revealed diverse dynamic behaviors on variable-order fractional discrete neural networks [59]. There are several methods for solving fractional-order differential equations, such as the predictor-corrector method, the Grünwald-Letnikov (G-L) method, and the Adomian decomposition method (ADM). Among these, the ADM is characterized by rapid convergence and low truncation errors [60]. The discretized results obtained through ADM require only iterative processing, significantly enhancing the feasibility of hardware implementation on field programmable gate arrays (FPGAs) [61]. It is noteworthy that the computation of decomposition terms typically entails high computational complexity. By employing co-design between hardware and software, pre-calculated coefficients can be transmitted to the hardware, significantly reducing computational burdens. The discretized fractional-order MNN is highly similar in system description to the discrete memristor neural network, enabling it to combine the rich dynamic characteristics of fractional-order MNNs with the easy hardware implementation of discrete memristor neural networks, thus holding significant potential for engineering applications.

Inspired by the aforementioned research, this study aims to construct a fractional-order MNN with unique attractor shapes and rich dynamic behaviors, proposing a memristive dual-wing fractional-order Hopfield neural network (MDW-FOMHNN) with hidden attractors. In this work, we adopted the ADM to more accurately capture the complex dynamic behaviors of the MDW-FOMHNN, such as chaos, hidden attractors, and coexisting attractors, thereby ensuring the reliability of simulation results [61]. The main contributions of this research include the following aspects:

1.

A fractional-order memristive neural network with dual-wing attractors is proposed, where the attractor morphology, modulated by internal parameters, can exhibit single-wing or dual-wing states, and such attractors are identified as rare hidden attractors.

2.

Bifurcation diagrams, Lyapunov exponent spectra, and basins of attraction are employed to reveal the influence of fractional-order and memristive coupling strength on the neural network, uncovering significant coexisting attractor behaviors.

3.

The MDW-FOMHNN is analyzed using the Adomian decomposition method, and a feasible FPGA implementation scheme is proposed, with results output to an oscilloscope accurately displaying the attractors.

4.

A simple and effective encryption/decryption algorithm based on the MDW-FOMHNN is proposed, demonstrating its strong encryption capability and robustness against noise.

The remainder of this paper is organized as follows: Section 2 presents the modeling of the MDW-FOMHNN and analyzes its equilibrium points. Section 3 introduces the numerical solution methods employed in this study and conducts a dynamical analysis. Section 4 details the FPGA implementation of MDW-FOMHNN. Section 5 presents the randomness testing of the sequence as well as a preliminary design of the encryption and decryption algorithm. Section 6 provides the conclusion.

2. Modeling and Equilibrium Analysis of the MDW-FOMHNN

2.1. Fractional-Order Memristor Model

Based on classical memristor theory, the behavior of a memristor is governed by external stimuli, internal state variables, and their corresponding state equation [62]. And memristors can be categorized into two types based on the nature of the internal state: flux-controlled and charge-controlled. By incorporating the Caputo fractional derivative operator, a fractional-order memristor model can be developed [61]. This work introduces a fractional-order flux-controlled general memristor model, with its mathematical formulation presented as follows:

$$\left\{\begin{array}{cc}{i}_m\hfill & =\mathrm{W}\left(\varphi \right)v_m\hfill \\ \mathrm{W}\left(\varphi \right)\hfill & =1-\varphi \hfill \\ \ast D_{t_0}^q\varphi \hfill & =1-\gamma \tan \mathrm{h}\left(\varphi \right)-v_m^2,\hfill \end{array}\right.$$

(1)

where, $\gamma$ is a positive real number, $i_m$ and $v_m$ represent the current flowing through the memristor and the externally applied voltage, respectively, $\varphi$ denotes the state variable of the memristor model, and $\ast D_{t0}^q$ represents the Caputo fractional derivative operator with order q, defined as

$$\ast D_{t_0}^{q}x\left(t\right)=\left\{\begin{array}{c}\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (1-q)}}{\int }_{t0}^t{\displaystyle \frac{\dot{x}\left(\tau \right)}{{(t-\tau )}^q}}d\tau ,\hfill & 0<q<1\hfill \\ \dot{x}\left(t\right),\hfill & q=1\hfill \end{array},\hfill \end{array}\right.$$

(2)

where $\Gamma (·)$ is the Gamma function.

Under periodic excitation, the memristor model exhibits a pinched hysteresis loop (PHL) with a characteristic “8”-shaped Lissajous curve. The lobe area of the PHL scales positively with the excitation amplitude and inversely with the excitation frequency. As the frequency increases, the PHL progressively collapses toward a single-valued function. In this subsection, the parameters are fixed as $\alpha =\beta =\gamma =1$, $q=0.8$, and the initial parameter $\varphi \left(0\right)=1$, with the input voltage defined as $v_m\left(t\right)=v_{0}sin\left(2\pi ft\right)$.

Figure 1a presents the PHLs obtained for three different excitation amplitudes $v_0=0.8\mathrm{V},1.0\mathrm{V},1.2\mathrm{V}$ at a constant frequency $f=1\mathrm{Hz}$, demonstrating the positive correlation between lobe area and excitation amplitude. Figure 1b shows the PHLs for a fixed $v_0=1\mathrm{V}$ at frequencies $f=0.5\mathrm{Hz},2\mathrm{Hz},10\mathrm{Hz}$, illustrating the inverse dependence on frequency and the convergence to a nearly linear response at high frequencies. Notably, all the PHLs in Figure 1 are pinched at the origin, confirming the characteristic memristive behavior described by Equation (1).

2.2. MDW-FOMHNN Model

HNN is a class of artificial neural networks exhibiting rich dynamical behaviors, which can be described by the following equation:

$$C_i{\dot{x}}_i=-{\displaystyle \frac{x_i}{R_i}}+\sum _{k=1}^n{w}_{ik}\tan \mathrm{h}\left(x_k\right)+I_i,i,k\in {\mathbb{N}}^{+},$$

(3)

where $x_i$ denotes the membrane potential of the i-th neuron, $C_i$ and $R_i$ are the corresponding membrane capacitance and membrane resistance, respectively, $I_i$ represents the external input current, $w_{ij}$ signifies the synaptic weight from neuron k to neuron i, and $tanh(·)$ is the activation function of the neuron. Typically, to simplify calculations, $C_i$ and $R_i$ are set to 1 and 1, respectively. Memristors inherently possess an inherent ability to retain their resistance state from before power-off until the subsequent power-on, providing a unique advantage for modeling the sustained effects of external electromagnetic radiation on neurons. In this subsection, the potential difference between neuron 3 and neuron 2 serves as the input voltage to drive the memristor proposed in Section 2.1. The resulting output current from the memristor, which represents the effect of electromagnetic radiation on neuron 3. By incorporating the Caputo fractional derivative, the mathematical model of the MDW-FOMHNN is formulated as follows:

$$\left\{\begin{array}{c}\ast D_{t_0}^q{x}_1\left(t\right)=-x_1+w_{11}\tan \mathrm{h}\left(x_1\right)+w_{12}\tan \mathrm{h}\left(x_2\right)+w_{13}\tan \mathrm{h}\left(x_3\right)\hfill \\ \ast D_{t_0}^q{x}_2\left(t\right)=-x_2+w_{21}\tan \mathrm{h}\left(x_1\right)+w_{22}\tan \mathrm{h}\left(x_2\right)+w_{23}\tan \mathrm{h}\left(x_3\right)\hfill \\ \ast D_{t_0}^q{x}_3\left(t\right)=-x_3+w_{31}\tan \mathrm{h}\left(x_1\right)+w_{32}\tan \mathrm{h}\left(x_2\right)+w_{33}\tan \mathrm{h}\left(x_3\right)+k(1-\varphi )(x_2-x_3)\hfill \\ \ast D_{t_0}^q\varphi \left(t\right)=1-\gamma \tan \mathrm{h}\left(\varphi \right)-{(x_3-x_2)}^2\hfill \end{array}\right.,$$

(4)

where k is the coupling strength between the memristor model and MDW-FOMHNN. To facilitate understanding, Figure 2 illustrates the topology of the MDW-FOMHNN.

For simplicity, $\gamma$ is fixed at 0.1, as it remains unchanged in subsequent analyses. Furthermore, the synaptic weight matrix in this study is configured as follows:

$$\left[\begin{array}{ccc}{w}_{11}& w_{12}& w_{13}\\ w_{21}& w_{22}& w_{23}\\ w_{31}& w_{32}& w_{33}\end{array}\right]=\left[\begin{array}{ccc}0& 0.5& -1.5\\ -0.25& -0.5& 0\\ 0& -6& 0\end{array}\right].$$

(5)

2.3. Equilibrium Analysis

Stability analysis of integer-order dynamical systems typically commences with the identification of equilibrium points, where the local stability at these points directly influences the global stability of the system. The methodology for determining equilibrium points in fractional-order systems is consistent with that employed for integer-order systems. By setting the left-hand side of Equation (4) to zero and integrating it with Equation (5), the following result is derived:

$$\left\{\begin{array}{c}0=-x_1+0.5\tan \mathrm{h}\left(x_2\right)-1.5\tan \mathrm{h}\left(x_3\right)\hfill \\ 0=-x_2-0.25\tan \mathrm{h}\left(x_1\right)-0.5\tan \mathrm{h}\left(x_2\right)\hfill \\ 0=-x_3-6\tan \mathrm{h}\left(x_2\right)+k(1-\varphi )(x_2-x_3)\hfill \\ 0=1-0.1\tan \mathrm{h}\left(\varphi \right)-{(x_2-x_3)}^2\hfill \end{array}\right..$$

(6)

Simplifying further yields two equations $E_1\mathrm{and}{E}_2$, as follows:

$$E_1:0=-\mathrm{atanh}(4x_2+2tanh\left(x_2\right))+0.5\tan \mathrm{h}\left(x_2\right)-1.5\tan \mathrm{h}\left(x_3\right).$$

(7)

$$E_2:0=-x_3-6\tan \mathrm{h}\left(x_2\right)+k(1-\mathrm{atanh}(10{(x_2-x_3)}^2-10))(x_2-x_3).$$

(8)

It is noteworthy that both equations $E_1$ and $E_2$ remain invariant under the coordinate transformation $(x_3,x_2)\to (-x_3,-x_2)$, indicating odd symmetry in the $(x_3,x_2)$ plane. For $E_1$, variable separation and derivative analysis of its inverse function demonstrate that the curve is monotonically increasing within its domain. Combined with analysis of specific function values, it is determined that the curve corresponding to $E_1$ lies in the first and third quadrants. For $E_2$, given that the domain of $\mathrm{atanh}(·)$ is $(-1,1)$, its validity requires $9<10{(x_2-x_3)}^2<11$. Under this constraint, when $0<k<2$, analysis shows that $x_2$ and $x_3$ having opposite signs is a necessary condition for $E_2$ to hold. Consequently, the curve corresponding to $E_2$ lies in the second and fourth quadrants. When $k>2$, the graph of $E_2$ can cross the line $x_3=0$, but there still exists an interval where it does not intersect the curve represented by $E_1$. In summary, when $0<k<2$, the curves of $E_1$ and $E_2$ do not intersect, implying that Equation (6) has no equilibrium points. In this case, if the MDW-FOMHNN exhibits a chaotic attractor, such an attractor would be a hidden attractor.

To verify the above analysis, the high-precision implicit function plotting function ‘ezplot’ in MATLAB R2021b was employed to plot the curves of $E_1$ and $E_2$ for $k=0.1,1,2,$ and 3. The results, shown in Figure 3, are consistent with the theoretical derivation.

3. Numerical Solution and Hidden Dynamics Analysis

From the analysis in the previous section, it is known that when $\gamma =0.1$ and $0<k<2$, the MDW-FOMHNN does not possess equilibrium points, and its dynamic behavior can be called hidden dynamics. Therefore, in this section, within this parameter range, phase portraits, time series, Lyapunov exponent (LE) spectra, bifurcation diagrams, and basins of attraction are employed as numerical tools to further investigate the hidden dynamical behaviors of the MDW-FOMHNN. It is noteworthy that the ADM is adopted in this paper to solve Equation (4), the QR-based method is utilized for computing the LEs [63], and the bifurcation diagrams are constructed by plotting the maximum values of the output sequences. The time step is set to 0.01.

3.1. Numerical Solution via the ADM

For the MDW-FOMHNN model, with $0<q<1$ and $\mathbf{X}\left(t\right)=[x_1\left(t\right);x_2\left(t\right);x_3\left(t\right);\varphi \left(t\right)]$, the system can be expressed as follows:

$$\ast D_{t0}^q\mathbf{X}\left(t\right)=L\mathbf{X}\left(t\right)+N\mathbf{X}\left(t\right)+\mathbf{C},$$

(9)

where $L\mathbf{X}\left(t\right)$, $N\mathbf{X}\left(t\right)$, and $\mathbf{C}$ represent the linear, nonlinear, and constant components of $\mathbf{X}\left(t\right)$, respectively. Coupling it with Equation (5), the linear part of Equation (4) can be expressed as $[L{\mathbf{X}}_1\left(t\right);L{\mathbf{X}}_2\left(t\right);L{\mathbf{X}}_3\left(t\right);L{\mathbf{X}}_4\left(t\right)]=[-x_1\left(t\right);-x_2\left(t\right);-x_3\left(t\right)+k(x_2\left(t\right)-x_3\left(t\right));0]$; the constant contribution in Equation (4) is given by $\mathbf{C}=[0;0;0;1]$; and the nonlinear part can be represented as follows:

$$\left[\begin{array}{c}N{\mathbf{X}}_1\left(t\right)\\ N{\mathbf{X}}_2\left(t\right)\\ N{\mathbf{X}}_3\left(t\right)\\ N{\mathbf{X}}_4\left(t\right)\end{array}\right]=\left[\begin{array}{c}0.5\tan \mathrm{h}\left(x_2\right)-1.5\tan \mathrm{h}\left(x_3\right)\\ -0.25\tan \mathrm{h}\left(x_1\right)-0.5\tan \mathrm{h}\left(x_2\right)\\ -6\tan \mathrm{h}\left(x_2\right)-k\varphi (x_2-x_3)\\ -0.1tanh\left(\varphi \right)-{(x_2-x_3)}^2\end{array}\right].$$

(10)

The nonlinear term is decomposed into Adomian polynomials,

$$\left\{\begin{array}{cc}\hfill N\mathbf{X}\left(t\right)& =\sum _{i=0}^{\infty }{\mathbf{A}}^i({\mathbf{X}}^0,{\mathbf{X}}^1,\dots ,{\mathbf{X}}^i),\hfill \\ \hfill {\mathbf{A}}^i& ={\displaystyle \frac{1}{i!}}{\left.{\displaystyle \frac{d^i}{d{\lambda }^i}}N\left(\sum _{k=0}^i{\lambda }^k{\mathbf{X}}^k\right)\right|}_{\lambda =0}.\hfill \end{array}\right.$$

(11)

where the Riemann–Liouville operator $J_{t_0}^q$ is defined as

$$J_{t_0}^{q}x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(q\right)}}{\int }_{t_0}^t{(t-\tau )}^{q-1}x\left(\tau \right)d\tau ,t>t_0,$$

(12)

and this operator has two key properties for handling Equation (9):

$$\left\{\begin{array}{cc}\hfill J_{t0}^q{(t-t_0)}^{\gamma }& ={\displaystyle \frac{\Gamma (\gamma +1)}{\Gamma (\gamma +1+q)}}{(t-t_0)}^{\gamma +q}\hfill \\ \hfill J_{t0}^{q}C& ={\displaystyle \frac{C}{\Gamma (1+q)}}{(t-t_0)}^q\hfill \end{array}.\right.$$

(13)

Therefore, the following decomposition can be performed here:

$${\mathbf{X}}^{\mathbf{0}}=\left[\begin{array}{c}{\mathbf{X}}_{\mathbf{1}}^{\mathbf{0}}\\ {\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}\\ {\mathbf{X}}_{\mathbf{3}}^{\mathbf{0}}\\ {\mathbf{X}}_{\mathbf{4}}^{\mathbf{0}}\end{array}\right]=\left[\begin{array}{c}{x}_1\left(0\right)\\ x_2\left(0\right)\\ x_3\left(0\right)\\ \varphi \left(0\right)\end{array}\right].$$

(14)

$${\mathbf{X}}^{\mathbf{1}}=\left[\begin{array}{c}{\mathbf{X}}_{\mathbf{1}}^{\mathbf{1}}\\ {\mathbf{X}}_{\mathbf{2}}^{\mathbf{1}}\\ {\mathbf{X}}_{\mathbf{3}}^{\mathbf{1}}\\ {\mathbf{X}}_{\mathbf{4}}^{\mathbf{1}}\end{array}\right]=\left[\begin{array}{c}\left({(t-t_0)}^q(0.5tanh\left({\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}\right)-{\mathbf{X}}_{\mathbf{1}}^{\mathbf{0}}-1.5tanh\left({\mathbf{X}}_{\mathbf{3}}^{\mathbf{0}}\right))\right)/\Gamma (q+1)\\ -\left({(te-t0)}^q({\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}+tanh\left({\mathbf{X}}_{\mathbf{1}}^{\mathbf{0}}\right)/4+tanh\left({\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}\right)/2)\right)/\Gamma (q+1)\\ -\left({(te-t0)}^q({\mathbf{X}}_{\mathbf{3}}^{\mathbf{0}}+6tanh\left({\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}\right)-k({\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}-{\mathbf{X}}_{\mathbf{3}}^{\mathbf{0}})+k{\mathbf{X}}_{\mathbf{4}}^{\mathbf{0}}({\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}-{\mathbf{X}}_{\mathbf{3}}^{\mathbf{0}}))\right)/\Gamma (q+1)\\ -\left({(te-t0)}^q(tanh\left({\mathbf{X}}_{\mathbf{4}}^{\mathbf{0}}\right)/10+{({\mathbf{X}}_{\mathbf{2}}^{\mathbf{0}}-{\mathbf{X}}_{\mathbf{3}}^{\mathbf{0}})}^2-1)\right)/\Gamma (q+1)\end{array}\right].$$

(15)

$${\mathbf{X}}^{\mathbf{2}}=\left[\begin{array}{c}{\mathbf{X}}_{\mathbf{1}}^{\mathbf{2}}\\ {\mathbf{X}}_{\mathbf{2}}^{\mathbf{2}}\\ {\mathbf{X}}_{\mathbf{3}}^{\mathbf{2}}\\ {\mathbf{X}}_{\mathbf{4}}^{\mathbf{2}}\end{array}\right]=\left[\begin{array}{c}{J}_{t0}^q(-{\mathbf{X}}_{\mathbf{1}}^{\mathbf{1}}+{\mathbf{A}}_{\mathbf{1}}^{\mathbf{1}})\\ J_{t0}^q(-{\mathbf{X}}_{\mathbf{2}}^{\mathbf{1}}+{\mathbf{A}}_{\mathbf{2}}^{\mathbf{1}})\\ J_{t0}^q(-{\mathbf{X}}_{\mathbf{3}}^{\mathbf{1}}+{\mathbf{A}}_{\mathbf{3}}^{\mathbf{1}})+k({\mathbf{X}}_{\mathbf{2}}^{\mathbf{1}}-{\mathbf{X}}_{\mathbf{3}}^{\mathbf{1}})\\ J_{t0}^q(1-{\mathbf{A}}_{\mathbf{4}}^{\mathbf{1}})\end{array}\right].$$

(16)

Thus, the j-th component ${\mathbf{X}}_j$ of Equation (9) is given by

$${\mathbf{X}}_j\left(t\right)=\left\{\begin{array}{cc}{\mathbf{X}}_j\left(t_0^{+}\right)+\sum _{i=1}^N{k}_j^i{\displaystyle \frac{{(t-t_0)}^q}{\Gamma (1+iq)}},\hfill & N>0\hfill \\ {\mathbf{X}}_j\left(t_0^{+}\right),\hfill & N=0\hfill \end{array}\right..$$

(17)

where N denotes the decomposition order, $(t-t_0)$ represents the decomposition step size, and $k_j^i$ is the irreducible polynomial of $(X_1^{i-1}\left(t_0\right),X_2^{i-1}\left(t_0\right),X_3^{i-1}\left(t_0\right)),X_4^{i-1}\left(t_0\right))$. As the decomposition order increases, the number of resulting terms grows exponentially, making direct presentation impractical. Here, only the first three terms are shown. In fact, all numerical simulations in this work employ fourth-order decomposition.

Based on the relationship between the memristor model and the stimulus, one can observe the relationship between $\varphi$ and the neurons $x_2$ and $x_3$. Therefore, by fixing the initial conditions (ICs) $(x_1,x_2,x_3,\varphi )=(0.1,0.1,0.1,0.1)$, with parameter $k=1.2$ and fractional order $q=0.8$, and iterating through 100,000 points, the phase portraits and time series of the attractors for the MDW-FOMHNN after ADM are obtained as shown in Figure 4. Figure 4a,b reveal the presence of a double-wing attractor, while Figure 4c exhibits characteristics of a strange attractor. Moreover, the time series depicted in Figure 4b to Figure 4e all display disordered states. Additionally, the calculation of the four LEs at this stage yields $\mathrm{LE}1=0.396$, LE1 = 0, LE1 = −1.317, and LE1 = −4.947, among which one LE is significantly positive. Synthesizing the aforementioned discussions, it is determined that the double-wing attractor is indeed chaotic and, more specifically, a hidden attractor.

3.2. Hidden Dynamics

The introduction of fractional-order derivatives and integrals into neural networks enables the description of the system’s present, past, and future states. Compared to integer-order neural networks, fractional-order models are better suited for characterizing biological neural networks and exhibit richer dynamical behaviors. To further investigate the impact of incorporating the fractional-order concept on the dynamics of neural networks, this section will explore the influence of parameter k and the evolution of fractional order q on the MDW-FOMHNN.

3.2.1. Influence of Fractional Order q and Coupling Strength k on Dynamics

In the analysis of memristive neural networks, bifurcation diagrams and LE spectra are two essential numerical tools. In this work, we fixed the initial parameter $\gamma =0.1$, $ICs=(0.1,0.1,0.1,0.1)$, and selected two different coupling strengths, $k=1.2$ and $k=1.5$, to investigate their effects on the dynamic behavior of the MDW-FOMHNN model. To ensure data convergence and accuracy, we computed only for the fractional order $q\in [0.4, 1]$, obtaining two sets of bifurcation diagrams and LE spectra, as shown in Figure 5. The results confirm that when a positive LE appears, the bifurcation diagrams exhibit corresponding dense point regions, consistent with expectations and validating the simulation’s effectiveness.

Specifically, by examining Figure 5a,c, as the fractional order q increases from $q=0.4$, the MDW-FOMHNN system transitions from a periodic state to a chaotic state. Initially, no positive Lyapunov exponents are observed, and a zero exponent is present, indicating a stable state. When q reaches approximately 0.49, a positive Lyapunov exponent emerges, marking the system’s entry into a chaotic regime through period-doubling bifurcations. Notably, around $q\approx 0.54$, the bifurcation diagram shows a distinct morphological change, indicating a transformation in the system’s attractor. A similar transition from periodic to chaotic states is also observed in Figure 5b,d.

The differences between the two sets of bifurcation diagrams suggest that the coupling strength k significantly influences the chaotic dynamics of the MDW-FOMHNN. Further analysis of different k values reveals that, as shown in Figure 5c, at $k=1.2$ and $q\approx 0.54$, the system exhibits a clear transformation in the chaotic attractor’s structure. In contrast, Figure 5d illustrates the bifurcation behavior at $k=1.5$, clearly displaying a typical period-doubling route to chaos. Comparing the results of Figure 5c,d, differences are evident in the onset points of the chaotic states under the two conditions. Moreover, at $k=1.2$, the system undergoes at least one transition from one type of chaotic attractor to another within the chaotic regime, whereas no such transition is observed at $k=1.5$. This indicates that, under the current parameter conditions, the chaotic behavior at $k=1.2$ is more complex. It also suggests that the coupling strength k not only influences the evolution of chaos but also regulates the type and complexity of the system’s attractors, playing a controlling role in the dynamic behavior of the MDW-FOMHNN system.

Bifurcation diagrams, constructed by extracting the maximum values of the system states, characterize the system’s dynamic behavior, where denser point distributions generally indicate higher chaotic complexity. As observed in Figure 5, the density of points in the bifurcation diagrams appears to decrease as the fractional order q increases. To further investigate the influence of the fractional order q on the chaotic complexity of the MDW-FOMHNN, spectral entropy (SE) analysis is employed, with results presented in Figure 6. The SE values shown in Figure 6a,b are in good agreement with the bifurcation diagrams and LE analyses in Figure 5. Taking Figure 6a as an example, within a defined chaotic regime, such as $k=1.2$ and $q\in (0.54, 1)$, the variation of SE closely follows that of the maximum LE. As q decreases, the SE values increase, indicating enhanced chaotic complexity in the MDW-FOMHNN. This trend is consistent with the denser point distribution observed in the bifurcation diagram of Figure 5c. Therefore, the incorporation of fractional-order dynamics significantly enhances the chaotic complexity of the MDW-FOMHNN.

To systematically explore the impact of the coupling strength parameter k on the dynamic behavior of the MDW-FOMHNN system, numerical simulations are performed with a fixed fractional order $q=0.8$ across the interval $k\in (0.1, 2)$. The resulting LE spectra and bifurcation diagrams are depicted in Figure 7a,b. The results indicate that within $k\in (0.9, 1.55)$, the maximum LE is positive, accompanied by intricate bifurcation patterns, signifying chaotic dynamics. Conversely, in other parameter ranges, the maximum LE is negative or zero, corresponding to periodic or stable states. Remarkably, throughout $k\in [0.1, 2]$, the MDW-FOMHNN system lacks any equilibrium points, implying that the chaotic attractors observed in $k\in (0.9, 1.55)$ are dissociated from unstable equilibrium points, characteristic of hidden chaotic attractors. This phenomenon is uncommon in memristive neural network dynamics, underscoring the special nonlinear attributes of the MDW-FOMHNN.

In conclusion, both the coupling strength k and fractional order q effectively modulate the MDW-FOMHNN system’s transitions between chaotic and periodic states. To corroborate these findings, Figure 8 presents phase portraits of the system’s attractors under six distinct $(k,q)$ parameter pairs, vividly illustrating its diverse dynamic evolution and reinforcing the conclusions on parameter effects.

3.2.2. Coexisting Attractors

By varying only the ICs, MNNs can generate attractors with distinct positions, sizes, and shapes, known as coexisting attractors. In MNNs, this property enables the simulation of diverse network responses to different neuronal stimuli. Consequently, MNN systems exhibiting coexistence behavior are highly advantageous for key design in cryptographic applications. With parameters set to $k=1.5$ and $q=0.8$, and by altering the ICs of the MDW-FOMHNN, attractors located at different positions are obtained, as shown in Figure 9a. Additionally, the corresponding basins of attraction are presented in Figure 9. Specifically, initial conditions within the green region of the basin generate the blue attractor in Figure 9a, while those in the yellow region produce the pink attractor. Furthermore, adjusting the parameters to $k=0.8$ and $k=1.8$ reveals distinct periodic coexisting attractors, with their phase portraits and time series displayed in Figure 10. Notably, as the MDW-FOMHNN undergoes a period-doubling bifurcation process, the system generates a variety of morphologically diverse periodic attractors, ultimately transitioning to a chaotic state. During this process, asymmetric periodic attractors and single-wing chaotic attractors both exhibit coexistence phenomena.

4. FPGA Hardware Implementation

Compared to analog circuits, FPGAs serve as a preferred platform for implementing nonlinear systems due to their high programmability, seamless integration, and superior electromagnetic interference resistance [64,65]. For iterative equations derived via the ADM, this work proposes an efficient FPGA-based implementation scheme to generate output attractors of the MDW-FOMHNN. The design workflow, illustrated in Figure 11, involves precomputing ADM polynomial coefficients on a software platform, which are then transferred to dedicated FPGA circuits. The current state is determined by weighted computations, relying solely on the previous state.

This design leverages Xilinx Vivado 2018.3 for development and employs the ALINX AX7020 development board, fitted with the Xilinx XC7Z020-2CLG400I chip, as the hardware platform. The board includes dual-row connectors to transmit chaotic signals to an oscilloscope. To achieve high floating-point accuracy, the floating-point IP core version 7.1 is utilized, enabling operations such as addition, multiplication, division, exponentiation, and floating-point to fixed-point conversions, compliant with the IEEE 754 single-precision format (1 sign bit, 8 exponent bits, 23 mantissa bits). Fixed-point formats are tailored to specific needs—for example, 24-bit resolution for digital-to-analog conversion and 32-bit fixed-point representation during calculations—to reduce precision errors and ensure reliable results.

To validate the output performance of the MDW-FOMHNN, parameters are set as $k=1.2$, fractional order $q=0.54$, and initial conditions $ICs=(0.1, 0.1, 0.1, 0.1)$. Dual-wing attractors were observed via an oscilloscope, as shown in Figure 12. Specifically, Figure 12b,d depict the attractor phase portraits in the $(x_3-\varphi )$ plane, consistent with the simulation results in Figure 8e, while Figure 12a,c illustrate the phase portraits in the $(x_2-\varphi )$ plane, exhibiting distinct dual-wing morphologies. These results demonstrate that the FPGA implementation successfully verified the chaotic behavior of the system and also proved the effectiveness of the proposed implementation scheme.

Furthermore, by configuring $q=0.8$ and $k=1.5$ and applying initial conditions of $\mathrm{ICs}=(0.1, 0.1, 0.1, 0.1)$ and $\mathrm{ICs}=-(0.1, 0.1, 0.1, 0.1)$, we observed coexisting chaotic attractors, illustrated in Figure 13a and Figure 13b, respectively. These results confirm the successful FPGA implementation of the condition presented in Figure 9a.

Figure 14 shows the power consumption of the system under this condition. Meanwhile,

Table 1. Resource utilization of FPGA implementation.

Resource	Utilization	Available	Utilization (%)
LUT	21,768	53,200	40.92
LUTRAM	1049	17,400	6.03
FF	29,257	106,400	27.50
DSP	118	220	53.64
IO	34	125	27.20
BUFG	1	32	3.13

lists the resources utilized by the MDW-FOMHNN implemented on the FPGA. Due to the use of the hyperbolic tangent function as the activation function in the traditional Hopfield neural network, the resource consumption during its implementation is not negligible. In this work, the verification process prioritizes high accuracy. Therefore, no simplification or optimization has been applied to computationally expensive functions such as the hyperbolic tangent.

5. Randomness and Image Encryption Results

With fixed parameters $k=1.2$, $q=0.8$, $\gamma =0.1$, and $IC=[0.1, 0.1, 0.1, 0.1]$, this work utilizes the four output sequences generated by the MDW-FOMHNN model, stored sequentially in the sequence S in the order $\{\dots , x_1\left(k\right), x_2\left(k\right), x_3\left(k\right), \varphi \left(k\right), \dots \}$. A random number generation module is designed based on this sequence to enhance the implementation of digital image encryption. The simple algorithm for generating the pseudorandom random number sequence (PRNG) is described below.

$$S_{PRNG}=\mathrm{mod}(\mathrm{floor}(S\times {10}^{10}),256),$$

(18)

where $S_{PRNG}$ will be used in the diffusion algorithms of the subsequent encryption process, and the results of the randomness test based on NIST SP 800-22 [66] are presented in

Table 2. The randomness test results of $S_{\mathrm{PRNG}}$ based on NIST SP 800-22.

No.	Statistical Tests	p-Value	Pass Rate
1	Frequency	0.275709	20/20
2	Block Frequency	0.534146	20/20
3	Cumulative Sums	0.162606	20/20
4	Runs	0.350485	20/20
5	Longest Run	0.350485	20/20
6	Rank	0.739918	20/20
7	FFT	0.370485	20/20
8	Non-Overlapping Template	0.534146	19/20
9	Overlapping Template	0.035174	20/20
10	Universal	0.911413	20/20
11	Approximate Entropy	0.350485	20/20
12	Random Excursions	0.534146	10/10
13	Random Excursions Variant	0.122325	10/10
14	Serial	0.637119	20/20
15	Linear Complexity	0.213309	20/20

. In this work, we set the significance level for randomness testing at 0.01. Based on the p-values and pass rates presented in the table, $S_{PRNG}$ exhibits exceptionally strong randomness characteristics. To further validate the potential of this PRNG in practical applications, we developed a key generation algorithm that integrates the SHA-256 hash function with the plaintext image to produce the initial key, which is then used as the input for the MDW-FOMHNN chaotic source. In the encryption process, pixel diffusion is achieved by reshaping the image data into a vector and performing an XOR operation with a high-quality PRNG sequence of equal length. This approach not only ensures effective encryption but also significantly enhances encryption efficiency. Notably, the encryption and decryption processes are symmetric in terms of operational sequence, meaning the decryption process is the reverse execution of the encryption process. The detailed steps of the encryption and decryption workflows are illustrated in Figure 15.

Figure 16 displays the images with a resolution of $512\times 512$ pixels before and after encryption, along with their corresponding histogram information. The encrypted ciphertext image exhibits a noise-like random distribution, rendering the original visual information completely indistinguishable. Moreover, the histogram of the encrypted image is almost flat, showing a very even distribution of pixel values, which effectively hides information and removes statistical patterns. Furthermore, the decrypted image and its histogram are almost identical to those of the original plaintext image, confirming the accuracy and reversibility of the decryption process. In conclusion, the encryption and decryption algorithm proposed in this study successfully achieves the expected encryption performance, demonstrating excellent feasibility and correctness.

To further validate the encryption performance, we use baboon images in Figure 16 with resolutions of $256\times 256$ and $512\times 512$ as examples for illustration and comparison. The specific evaluation metrics include information entropy, correlation coefficients after encryption, number of pixels change rate (NPCR), unified average changing intensity (UACI), encryption time, and noise resistance performance. These metrics not only reflect the effectiveness of the encryption algorithm but are also closely related to the performance of the experimental hardware. In this section, all encryption experiments are conducted on a laptop equipped with an 11th-generation Intel Core i5-1155G7 processor (base frequency 2.5 GHz) using MATLAB 2021b as the software platform.

The formula for calculating correlation coefficients is as follows:

$$\left\{\begin{array}{c}{R}_{xy}={\displaystyle \frac{\mathrm{cov}(x,y)}{\sqrt{D\left(x\right)}\sqrt{D\left(y\right)}}}\hfill \\ \mathrm{cov}(x,y)={\displaystyle \frac{1}{N}}\sum _{i=1}^N(x_i-E\left(x_i\right))(y_i-E\left(y_i\right))\hfill \end{array}\right.,$$

(19)

where $x_i$ and $y_i$ represent the pixel values of each pair of adjacent pixels, N denotes the total number of selected pixel pairs, $E(·)$ indicates the average intensity value of the pixels, and $D(·)$ represents the variance calculation. The correlation coefficient can also reflect the effect before and after encryption. The smaller the correlation coefficient of the encrypted image, the better the encryption performance.

Table 3. Comparison of correlation coefficients for encrypted images under different schemes.

Reference	Image Size	Horizontal	Vertical	Diagonal
[51]	$512\times 512$	0.007900	–	–
[49]	$256\times 256$	0.009700	−0.004700	−0.005000
[49]	$512\times 512$	−0.010700	0.006300	−0.004900
[42]	$512\times 512$	0.004000	0.002300	0.002700
Proposed	$256\times 256$	$-0.002635$	$-0.008996$	$-0.001836$
Proposed	$512\times 512$	$-0.004994$	$-0.007952$	$-0.001432$

presents a comparison of the correlation coefficients of encrypted images between the proposed method and existing studies. It can be observed that the correlation coefficients of the proposed method in the horizontal, vertical, and diagonal directions are all close to zero, indicating a high degree of decorrelation between adjacent pixels in the encrypted images. Compared to existing advanced schemes, the proposed method demonstrates superior performance in suppressing correlation in the diagonal direction, while also showing advantages in the horizontal direction compared to some studies, highlighting the effectiveness of the encryption algorithm.

The formula for calculating information entropy is as follows:

$$H\left(X\right)=-\sum _{i=1}^{n}P\left(X_i\right)\times {\log }_2\left(P\left(X_i\right)\right),$$

(20)

where $X_i$ represents the i-th pixel intensity, $p\left(X_i\right)$ denotes the probability of occurrence of that pixel intensity, and n is the total number of pixels in the image. It should be noted that the maximum pixel intensity of the images processed in this study is 256, corresponding to 8-bit grayscale images, which means the ideal information entropy value after encryption is 8. When the information entropy approaches this theoretical maximum, it indicates that the pixel distribution of the ciphertext image is highly uniform, maximizing information uncertainty and thereby enhancing resistance to statistical analysis.

The NPCR and UACI values are used to evaluate the sensitivity of the cipher image to tiny changes in the plain image, and the ideal values are NPCR = 99.6094% and UACI = 33.4635%. The formulas for calculating NPCR and UACI are as follows:

$$\left\{\begin{array}{c}\mathrm{NPCR}={\displaystyle \frac{{\sum }_{i,j}D(i,j)}{M\times N}}\times 100\%\hfill \\ \mathrm{UACI}={\displaystyle \frac{{\sum }_{i,j}|C_1(i,j)-C_2(i,j)|}{M\times N\times 255}}\times 100\%\hfill \end{array}\right.,$$

(21)

where $D(i,j)=1$ if $C_1(i,j)\ne C_2(i,j)$, otherwise $D(i,j)=0$. $C_1(i,j)$ and $C_2(i,j)$ are the pixel values at position $(i,j)$ in the two encrypted images.

Table 4. Performance comparison of encrypted images in terms of information entropy, NPCR/UACI, and encryption time.

Ref.	Image Size	Entropy	NPCR/UACI (%)	Time (s)
[51]	$512\times 512$	7.9993	99.6100/33.4500	–
[49]	$512\times 512$	–	99.6100/–	–
[42]	$512\times 512$	7.9998	99.6095/33.4632	–
[61]	$512\times 512$	7.9993	99.6080/33.4621	0.1496
[67]	$320\times 320$	7.9982	99.5900/33.4300	3.9000
[38]	$512\times 512$	7.9994	–/–	0.2810
Proposed	$256\times 256$	7.9970	99.6063/33.5505	0.0936
Proposed	$512\times 512$	7.9994	99.6227/33.4024	0.2430

compares the proposed method with existing studies in terms of information entropy, NPCR/UACI, and encryption time. The results demonstrate that the proposed encryption method is simple and effective, with a clear advantage in encryption time. Notably, for images with a resolution of $512\times 512$ pixels, the information entropy after encryption reaches 7.9994, very close to the ideal value of 8, outperforming existing studies. Moreover, the NPCR and UACI metrics closely approach their ideal values, robustly demonstrating the method’s excellent resistance to differential attacks.

Figure 17 illustrates the impact of common noise on digital image transmission, using salt-and-pepper noise as an example. Figure 17a to Figure 17d sequentially show the decrypted images after being subjected to salt-and-pepper noise of 0%, 10%, 30%, and 60%, respectively. It can be observed that even under a high noise level of 60%, the decrypted image still retains a significant amount of structural details and information of the plain image, with the overall contours still clearly discernible. This demonstrates that the proposed encryption/decryption algorithm possesses strong resistance to salt-and-pepper noise, indicating its robustness.

Based on the content of this section, it can be concluded that the encryption scheme designed using the MDW-FOMHNN with hidden dynamic behavior achieves reliable encryption performance compared to existing studies.

6. Conclusions

This paper proposes a fractional-order Hopfield neural network model with memristive electromagnetic radiation, termed MDW-FOMHNN, by leveraging the memristor’s capability to emulate electromagnetic radiation received by neurons. Through equilibrium point analysis, it is determined that the MDW-FOMHNN lacks equilibrium points within a certain range of the coupling strength parameter k. Subsequently, the MDW-FOMHNN is numerically solved using the ADM. Tools such as Lyapunov exponents, phase portraits, and time series are employed to identify the system’s current state as exhibiting hidden chaotic dynamics. Furthermore, Lyapunov exponent spectra and bifurcation diagrams reveal the influence of the coupling parameter k and the fractional order q on the MDW-FOMHNN, demonstrating that both parameters significantly affect the morphology of chaotic attractors. Several distinct attractor shapes are presented. Based on these attractor morphologies, the coexisting attractor behavior of the MDW-FOMHNN is further investigated, indicating its potential for practical applications. Then, leveraging the polynomial form of the MDW-FOMHNN derived from the ADM, an FPGA implementation scheme is proposed. The desired attractors are successfully captured and verified using an oscilloscope. Finally, to bridge the gap between theoretical analysis and practical applications, a simple and effective encryption/decryption algorithm based on the MDW-FOMHNN is designed.

Table 5. The dynamic characteristics of selected neural networks from existing studies.

Reference	Memristive Device	Attractor Shape	Hidden Attractor	Hardware	Maximum SE Value
[67]	One	Single-scroll	No	–	–
[68]	One	Multi-scroll	No	Arduino-Due	0.35
[69]	Two	Multi-wing	No	FPGA	0.55
[70]	One	Scroll	No	FPGA	0.45
[71]	Zero	Double-scroll	No	Analog Circuit	–
Proposed	One	Dual-wing	Yes	FPGA	0.62

summarizes the dynamic characteristics of some neural networks in existing studies. The proposed MDW-FOMHNN model exhibits a unique hidden double-wing attractor and employs only a single nonlinear memristive device, demonstrating advantages in both dynamic complexity and theoretical implementation efficiency. Spectral entropy analysis results indicate that by adjusting the fractional order q, MDW-FOMHNN can generate sequences with stronger randomness compared to integer-order systems, achieving the highest spectral entropy (SE) value among the studies listed in the table, reflecting superior chaotic properties.

Future research can fully leverage the excellent randomness of MDW-FOMHNN due to its hidden attractor, focusing on designing more efficient encryption schemes based on memristive neural networks (MNNs) to promote their practical applications in industrial fields such as information security. Additionally, the successful implementation of the fractional-order double-wing attractor in this work validates the feasibility of constructing fractional-order memristive neural networks with multi-wing structures, providing new insights for designing scalable complex chaotic systems. Furthermore, considering that an important direction in current research is to explore the coupling mechanisms between multiple neurons and memristor models, the dynamic behavior of such systems still offers vast potential for further in-depth investigation.