Study on the Complex Dynamical Behavior of the Fractional-Order Hopfield Neural Network System and Its Implementation

Abstract

The complex dynamics analysis of fractional-order neural networks is a cutting-edge topic in the field of neural network research. In this paper, a fractional-order Hopfield neural network (FOHNN) system is proposed, which contains four neurons. Using the Adomian decomposition method, the FOHNN system is solved. The dissipative characteristics of the system are discussed, as well as the equilibrium point is resolved. The characteristics of the dynamics through the phase diagram, the bifurcation diagram, the Lyapunov exponential spectrum, and the Lyapunov dimension of the system are investigated. The circuit of the system was also designed, based on the Multisim simulation platform, and the simulation of the circuit was realized. The simulation results show that the proposed FOHNN system exhibits many interesting phenomena, which provides more basis for the study of complex brain working patterns, and more references for the design, as well as the hardware implementation of the realized fractional-order neural network circuit.

1. Introduction

Dynamics studies of artificial neurons and neural networks are important for understanding brain functions and developing neuromorphic systems. In recent years, the modeling of memory neurons and neural networks has a great potential for brain dynamics studies [1,2,3,4]. The Hopfield neural network is a type of artificial neural network, abstracted from the human brain [5], which can generate complex dynamical behaviors, due to the active firing behavior of its internal neurons [6,7,8], and these behaviors include limit loops, chaotic states, stability, multi-stability coexistence, etc. Although the construction of this neural network is simple, it is better analyzed and studied to provide a better experimental basis for understanding the behavior and working mechanism of the brain [6,9,10,11].

In recent years, Hopfield’s neural network has increasingly attracted wide attention and research, as a typical model for studying the working mechanism and information storage of the brain [12,13,14,15]. For example, Li et al. [16] constructed a new dynamic neural network model consisting of three Hopfield neurons, revealing that the coupled weighted synapses of the memory can influence the distribution of the system’s equilibrium points. Ding et al. [17] found that the Hopfield Neural Network model of the coupled local activation memristors has a multiple stability at different fractional orders and coupling coefficients. Saeed Sani et al. proposed a new algorithm for the COVID-19 detection in chest CT images, based on the Hopfield neural network analysis [18]. It has made great contributions in solving associative memory and traveler problems, by using the Hopfield neural network [19,20,21,22]. These studies suggest that studying the complex dynamical behavior of the Hopfield neural network can provide more basis for the work related to brain dynamics, and it is necessary to study this neural network model in depth.

The fractional order calculus is a theory for studying the arbitrary order differentiation and integration [23], with a wide range of applications in memory digital image processing, engineering control, and other fields [24,25,26,27,28,29,30,31,32,33,34,35,36,37]. The weighted form of the fractional-order system allows for the accumulation of global information about the function and thus a greater memorability and freedom than the integer-order system, while the fractional-order calculus is closer to reality. The use of the fractional order allows to describe the physical objects, especially some physical phenomena, with memory, more accurately than the integer order and more in line with the biological neural networks applied to reality. Therefore, fractional order calculus has been introduced into artificial neural networks in some studies to construct more accurate mathematical models [38,39,40,41,42]. The research results show that fractional-order neural networks can more accurately model the activity of brain neurons than integer-order neural network systems. Therefore, fractional-order neural networks have great application prospects and research value.

However, in the present research results, more integer-order neural network models are discussed. In the process of model building, the influence of the fractional order on the model is neglected, especially in the process of building some circuit models, the fractional order impedance that plays a decisive role in the actual electronic devices, is not taken into account. The implementation of the fractional-order neural network systems is also relatively rare in the existing studies [43,44,45]. Based on this, the fractional order theory is introduced into the Hopfield neural network system to construct a new class of the FOHNN system.

The main work of this paper is to construct a fractional-order Hopfield neural network (FOHNN) system, using the Adomian solution method algorithm, to explore the rich and complex dynamical properties present in the system, and to design a circuit for the FOHNN system. The rest of the components are as follows: Section 2 presents the FOHNN system, based on four neurons and the ADM decomposition of the system. In Section 3, some dynamical behaviors of the proposed model are analyzed. The circuit of the system is designed in Section 4 and based on the Multisim platform, the simulation is verified. In Section 4, a 4D FOHNN system is implemented, based on Multisim. In the end, some conclusions were drawn.

2. FOHNN System with Four Neurons

2.1. Principle of the Fractional Calculus

Currently, there are various definitions of the fractional order calculus, and the most commonly used ones are the Grunwald–Letnikov (G–L) derivative, the Riemann–Liouville (R–L) derivative and the Caputo derivative. These three definitions are not equivalent. However, the physical meaning of the Caputo differential definition is very clear, since only the value of the initial condition needs to be determined, when it is adopted. The Caputo definition is more conducive to the actual physical system solution and has a more practical engineering application.

Definition 1 

[46]. The Caputo fractional differentiation is defined as

$$\ast D_t^{q}f(t)=J_{t_0}^{m-q}{D}_{t_0}^{m}f(t)=\{\begin{array}{l}\frac{1}{\Gamma (m-q)}{\displaystyle {\int }_0^t\frac{f(m)(\tau )\mathrm{d}\tau }{{(t-\tau )}^{q+1-m}},m-1<q<m}\hfill \\ \frac{{\mathrm{d}}^m}{\mathrm{d}{t}^m}f(t),q=m\hfill \end{array},$$

(1)

where q $\in$ R⁺, m $\in$ N, $\ast D_t^q$ is the q-order Caputo differential operator. Γ represents the Gamma function.

Definition 2 

[47]. The Laplace transform of the fractional order differentiation is

$$L\left[{}_0{D}_{t_0}^{q}f(t)\right]={\displaystyle {\int }_0^{\infty }{e}^{-st}}{D}_{t_0}^{q}f(t)dt=s^{q}F(s)-{\displaystyle \sum _{k=0}^{m-1}{s}^k}\left[D_{t_0}^{q-k-1}f(t)\right]|{}_{t=0},m-1<q<m,$$

(2)

where F(s) = L [f(t)] denotes the Laplace transform of f(t).

Theorem 1 

[48]. The linear autonomous system

$${}^{c}D{}^{q}x=Ax,$$

(3)

where A $\in$ $R^{n\times n}$and q $\in$ (0,1) is asymptotically stable if and only if

$$\left|\arg (\lambda )\right|>\frac{q\pi }{2},\forall \lambda \in \sigma (A),$$

(4)

or, if and only if

$$\left|\mathrm{Re}(\lambda )\right|>\mathrm{Im}(\lambda )\frac{q\pi }{2},\forall \lambda \in \sigma (A),$$

(5)

where $\sigma (A)$ represents the set of all eigenvalues of the matrix A.

The proposed fractional order differential equation $\ast D_{t_0}^q$v(t) = f(v(t)), based on the Adomian solution method (ADM), can be expressed in the form,

$$\ast D_{t_0}^{q}v(t)=Lv(t)+Nv(t)+C(t)$$

(6)

where *$D_{t0}^q$ stands for the Caputo derivative operator of the order q (m−1 < q ≤ m, m ∈ N), v(t) = [v₁(t) v₂(t) … v_n(t)] ^T are the state variables. The linear and nonlinear terms are denoted by L and N, respectively. C(t) = [C₁(t) C₂(t) … C_n(t)] ^T is the constant for the autonomous system. By applying the $J_{t_0}^q$ to the left and right of Equation (1), the following equation can be obtained

$$x(t)=J_{t_0}^{q}Lv(t)+J_{t_0}^{q}Nv(t)+J_{t_0}^{q}C+v(t_{t_0}^{+}),$$

(7)

where $J_{t_0}^q$ indicates the R-L fractional integral operator of order q, the initial value is given by v($t_{t_0}^{+}$). According to ADM description, it can be known that the operational property of the integral operator $J_{t_0}^q$, are shown as follows:

$$J_{t_0}^{q}C=\frac{C}{\Gamma (q+1)}{(t-t_0)}^q,$$

(8)

$$J_{t_0}^q{(t-t_0)}^{\gamma }=\frac{\Gamma (\gamma +1)}{\Gamma (\gamma +1+q)}{(t-t_0)}^{q+\gamma },$$

(9)

$$J_{t_0}^q{J}_{t_0}^{r}v(t)=J_{t_0}^{q+r}v(t),$$

(10)

where t ∈ [t₀, t₁], q ≥ 0, γ > −1, r ≥ 0. The i-th nonlinear term can be solved using the following

$$\{\begin{array}{l}{A}_j^i=\frac{1}{i!}{[\frac{d^i}{d{\lambda }^i}N(v_j^i(\lambda ))]}_{\lambda =0}\\ v_j^i(\lambda )={\displaystyle \sum _{k=1}^i{(\lambda )}^k{v}_j^k}\end{array},$$

(11)

in which i = 0, 1, 2, 3 … ∞; j = 1, 2, 3… n. The nonlinear term N is recorded as

$$Nv={\displaystyle \sum _{i=0}^{\infty }{A}^i}(v^0,v^1,\cdots ,v^i),$$

(12)

Therefore, using the following form to represent the numerical solution of the system (1)

$$v={\displaystyle \sum _{i=1}^{\infty }{v}^i}=J_{t_0}^q{\displaystyle \sum _{i=1}^{\infty }{v}^i}+J_{t_0}^q{\displaystyle \sum _{i=1}^{\infty }{A}^i}+J_{t_0}^{q}C+v(t_0^{+}),$$

(13)

where vⁱ is calculated by

$$\begin{array}{l}{v}^0=J_{t_0}^{q}C+v(t_0),\\ v^1=J_{t_0}^{q}L{v}^0+J_{t_0}^q{A}^0(v^0),\\ v^2=J_{t_0}^{q}L{v}^1+J_{t_0}^q{A}^1(v^0,v^1),\\ \cdots \\ v^n=J_{t_0}^{q}L{v}^{i-1}+J_{t_0}^q{A}^{i-1}(v^0,v^1,\cdots v^{i-1}).\end{array}$$

(14)

2.2. Solution of the FOHNN-Based System

In 1982, J. Hopfield proposed the Hopfield neural network model

$$C_i\frac{d{v}_i}{dt}=-\frac{v_i}{R_i}+{\displaystyle \sum _{j=1}^n{T}_{ij}{g}_j}(v_j)+I_i,i=1,2,\mathrm{\dots },n,$$

(15)

where R_i, C_i, and T_i represents the resistance, capacitance, and conductance, respectively, and g_i(v_i) is the activation function generated by the operational amplifier. Normally, the selection of an as one, and when the hyperbolic tangent function is chosen as the activation function, the chaotic attractors appear in the system. The weight matrix T_ij is modified and its dimension is determined by the number of neurons. In this FOHNN system, T_ij has size 4 × 4, meaning that the Hopfield neural network has four state variables, associated with each neuron. The topology structure of the neural network is shown in Figure 1.

$$\left[\begin{array}{c}{\dot{v}}_1\\ {\dot{v}}_2\\ {\dot{v}}_3\\ {\dot{v}}_4\end{array}\right]=-\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right]+T\left[\begin{array}{c}g(v_1)\\ g(v_2)\\ g(v_3)\\ g(v_4)\end{array}\right],\left[\begin{array}{c}g(v_1)\\ g(v_2)\\ g(v_3)\\ g(v_4)\end{array}\right]=\left[\begin{array}{c}\tanh (v_1)\\ \tanh (v_2)\\ \tanh (v_3)\\ \tanh (v_4)\end{array}\right],Tij=\left[\begin{array}{cccc}{T}_{11}& T_{12}& T_{13}& T_{14}\\ T_{21}& T_{22}& T_{23}& T_{24}\\ T_{31}& T_{32}& T_{33}& T_{34}\\ T_{41}& T_{42}& T_{43}& T_{44}\end{array}\right].$$

(16)

Setting R_i = 1, I_i = 0, the Equation (16) describes the model, based on four neurons. In this case, the v is a vector containing the four elements with the four state variables. Two variable synaptic weighting parameters are set, namely β₁ and β₂. By choosing the corresponding synaptic weight coefficients, the synaptic weight matrix is represented as follows

$$T=\left[\begin{array}{cccc}0.3& 0.5& 0& 0.1\\ 2.3& {\beta }_1& 8.5& 0\\ -4& 0& 0& 1\\ 35& {\beta }_2& 0& 1\end{array}\right]$$

(17)

According to Definitions 1 and 2, a fractional order form of a fourth-FOHNN system can be written as

$$\{\begin{array}{l}\ast D_{t_0}^q{v}_1(t)=-v_1+0.3\tanh (v_1)+0.5\tanh (v_2)+0.1\tanh (v_4)\\ \ast D_{t_0}^q{v}_2(t)=-v_2+2.3\tanh (v_1)+{\beta }_1\tanh (v_2)+8.5\tanh (v_3)\\ \ast D_{t_0}^q{v}_3(t)=-v_3-4\tanh (v_1)+\tanh (v_4)\\ \ast D_{t_0}^q{v}_4(t)=-v_4+35\tanh (v_1)+{\beta }_2\tanh (v_2)+\tanh (v_4)\end{array},$$

(18)

where q represents the order of the FOHNN system. Based on the ADM algorithm, the linear and nonlinear terms of the system are represented as

$$\left[\begin{array}{c}L{v}_1\\ L{v}_2\\ L{v}_3\\ L{v}_4\end{array}\right]=\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right],\left[\begin{array}{c}N{v}_1\\ N{v}_2\\ N{v}_3\\ N{v}_4\end{array}\right]=\left[\begin{array}{c}\tanh (v_1)\\ \tanh (v_2)\\ \tanh (v_3)\\ \tanh (v_4)\end{array}\right],\left[\begin{array}{c}{c}_1\\ c_2\\ c_3\\ c_4\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],$$

(19)

For the consideration of the convergence speed of the ADM algorithm, the first five terms are intercepted here under the premise of guaranteeing the accuracy. According to Equation (6), the decomposition of A₁ = tanh(v₁) as an example, can be decomposed as follows

$$\begin{array}{ll}{A}_1^0=& \tanh (v_1^0)\\ A_1^1=& v_1^1{\mathrm{sech}}^2(v_1^0)\\ A_1^2=& v_1^2{\mathrm{sech}}^2(v_1^0)-{(v_1^1)}^2{\mathrm{sech}}^2(v_1^0)\tanh (v_1^0)\\ A_1^3=& v_1^3{\mathrm{sech}}^2(v_1^0)-2v_1^2{v}_1^1{\mathrm{sech}}^2(v_1^0)\tanh (v_1^0)+\frac{1}{3}(v_1^1{)}^3(4{\mathrm{sech}}^3(v_1^0){\tanh }^2(v_1^0)-{\mathrm{sech}}^4(v_1^0))\\ A_1^4=& x_1^4{\mathrm{sech}}^2(v_1^0)-(2v_1^3{v}_1^1+v_1^2{v}_1^2){\mathrm{sech}}^2(v_1^0)\tanh (v_1^0)+v_1^1{v}_1^1{v}_1^2(4{\mathrm{sech}}^3(v_1^0){\tanh }^2(v_1^0)\\ & -{\mathrm{sech}}^4(v_1^0))+\frac{1}{3}{(v_1^1)}^4({\mathrm{sech}}^3(v_1^0)\tanh (v_1^0)-3{\mathrm{sech}}^3(v_1^0){\tanh }^3(v_1^0))\end{array}$$

(20)

The other three nonlinear terms are similar to A₁. Setting the initial value v⁰ = [v₁($t_0^{+}$) v₂($t_0^{+}$) v₃($t_0^{+}$) v₄($t_0^{+}$)]t, the first term can be gained for

$$\{\begin{array}{l}{v}_1^0=v_1(t_0^{+})\\ v_2^0=v_2(t_0^{+})\\ v_3^0=v_3(t_0^{+})\\ v_4^0=v_4(t_0^{+})\end{array}$$

(21)

Letting $c_1^0$ = $v_1^0$, $c_2^0$ = $v_2^0$, $c_3^0$ = $v_3^0$, $c_4^0$ = $v_4^0$ therefore we obtain v⁰ = c⁰= [$c_1^0$ $c_2^0$ $c_3^0$ $c_4^0$]. v¹= $J_{t_0}^q$Lv⁰ + $J_{t_0}^q$A⁰(v⁰) is obtained by the iterative relation formula Equation (14). The second term of the state variables are expressed as

$$\{\begin{array}{l}{x}_1^1=(-c_1^0+0.3\tanh (c_1^0)+0.5\tanh (c_2^0)+0.1\tanh (c_4^0))\frac{{(t-t_0)}^q}{\Gamma (q+1)}\hfill \\ x_2^1=(-c_2^0+2.3\tanh (c_1^0)+{\beta }_1\tanh (c_2^0)+8.5\tanh (c_3^0))\frac{{(t-t_0)}^q}{\Gamma (q+1)}\hfill \\ x_3^1=(-c_3^0-4\tanh (c_1^0)+\tanh (c_4^0))\frac{{(t-t_0)}^q}{\Gamma (q+1)}\hfill \\ x_4^1=(-c_4^0+35\tanh (c_1^0)+{\beta }_2\tanh (c_2^0)+\tanh (c_4^0))\frac{{(t-t_0)}^q}{\Gamma (q+1)}\hfill \end{array}.$$

(22)

Let

$$\{\begin{array}{l}{c}_1^1=-c_1^0+0.3\tanh (c_1^0)+0.5\tanh (c_2^0)+0.1\tanh (c_4^0)\hfill \\ c_2^1=-c_2^0+2.3\tanh (c_1^0)+{\beta }_1\tanh (c_2^0)+8.5\tanh (c_3^0)\hfill \\ c_3^1=-c_3^0-4\tanh (c_1^0)+\tanh (c_4^0)\hfill \\ c_4^1=-c_4^0+35\tanh (c_1^0)+{\beta }_2\tanh (c_2^0)+\tanh (c_4^0)\hfill \end{array},$$

(23)

So v¹ will be represented as x¹ = c¹(t − t₀)^q/Γ(q + 1). By using the same method, the coefficient decompositions of the other four terms are listed in Appendix A.

Therefore, the numerical results of the five-term system, based on FOHNN are

$${\tilde{x}}_j=c_j^0+c_j^1\frac{{(t-t_0)}^q}{\Gamma (q+1)}+c_j^2\frac{{(t-t_0)}^{2q}}{\Gamma (2q+1)}+c_j^3\frac{{(t-t_0)}^{3q}}{\Gamma (3q+1)}+c_j^4\frac{{(t-t_0)}^{4q}}{\Gamma (4q+1)}+c_j^5\frac{{(t-t_0)}^{5q}}{\Gamma (5q+1)}$$

(24)

here j = 1, 2, 3, 4, 5. In the process of the calculation, the whole interval is divided into many subintervals, and the value obtained from the previous subinterval will be used as the initial value of the latter subinterval for iteration, with an iteration step of h. Setting β₁ = 0.72, β₂ = 12.5, q = 0.55, h = 0.01, and initial values x₀= (0.2, 0.2, 0.1, 0.5). The FOHNN-based system’s Lyapunov exponents are L₁ = 1.7610, L₂ = 0, L₃ = −6.8754, and L₄ = −6.8754 and the Lyapunov dimension is D_L = 2.2561. With a positive maximum Lyapunov exponent, the system has a chaotic attractor. The phase diagram of the chaotic attractor, based on the FOHNN system are displayed in Figure 2.

3. Study of the Dynamical Characteristics of the FOHNN System

3.1. Dissipation of the FOHNN System with the Existence of Attractors

The dispersion of an alternative nonlinear system to be in a chaotic state. For Equation (13), the dispersion of the FOHNN system is expressed as

$$\nabla V=\mathrm{div}V=\frac{\partial {\dot{v}}_1}{\partial v_1}+\frac{\partial {\dot{v}}_2}{\partial v_2}+\frac{\partial {\dot{v}}_3}{\partial v_3}+\frac{\partial {\dot{v}}_4}{\partial v_4}$$

(25)

according to Equations (13) and (24), the obtained

$$\nabla V=-4-0.3\sec h^2(v_1)+{\beta }_1\sec h^2(v_2)+\sec h^2(v_4)$$

(26)

due to 0 ≤ sech²(v₁) ≤ 1, 0 ≤ sech²(v₂) ≤ 1, 0 ≤ sech²(v₄) ≤ 1, therefore, $\nabla$ ≤ −4 − 0.3 + β₁ + 1 < 0. Obviously, β₁ < 3.3, the chaotic attractors will emerge from the system.

3.2. Stability Analysis of the Equilibrium Point of the FOHNN System

In order to obtain the equilibrium points, set the left of the equal sign of Equation (13) be 0 and solve the equation to obtain all of the equilibrium points of the system

$$\{\begin{array}{l}-v_1-0.3\tanh (v_1)+0.5\tanh (v_2)+\tanh (v_4)=0\\ -v_2+2.3\tanh (v_1)+{\beta }_1\tanh (v_2)+8.5\tanh (v_3)=0\\ -v_3-4\tanh (v_1)+\tanh (v_4)=0\\ -v_4+35\tanh (v_1)+{\beta }_2\tanh (v_2)+\tanh (v_4)=0\end{array}$$

(27)

It is obvious that Equation (27) is a system of the fourth order transcendental equations and all solutions of the equations need to be obtained using MATLAB. The Jacobian matrix of the system in any equilibrium point is

$$J=\left[\begin{array}{cccc}0.3 {\mathrm{sech}}^2(v_1)-1& 0.5 {\mathrm{sech}}^2(v_2)& 0& {\mathrm{sech}}^2(v_4)\\ 2.3 {\mathrm{sech}}^2(v_1)& {\beta }_1{\mathrm{sech}}^2(v_2)-1& 8.5 {\mathrm{sech}}^2(v_3)& 0\\ -4 {\mathrm{sech}}^2(v_1)& 0& -1& {\mathrm{sech}}^2(v_4)\\ 35 {\mathrm{sech}}^2(v_1)& {\beta }_2{\mathrm{sech}}^2(v_2)& 0& {\mathrm{sech}}^2(v_4)-1\end{array}\right]$$

(28)

where sech²(i) = (1 − tanh²(i)) with I = v₁, v₂, v₃, v₄. Observing Equation (27), the system always exists a zero equilibrium point E₀ = (0, 0, 0, 0), whatever the value of the parameter is taken. Setting β₁ = 0.2, β₂ = 15 the characteristic function of the equilibrium as

$${\lambda }^4+2.5{\lambda }^3-34.09{\lambda }^2+81.41\lambda +457=0$$

(29)

the corresponding eigenvalues are −2.8701 ± 1.1582i and 2.8701 ± 2.3224i, and the |larg(λ₂)| = 0.3835. According to Theorem 1, the equilibrium point is unstable and the system generates chaotic attractors, due to the existence of two conjugate complex roots with real parts greater than zero.

3.3. Stability Analysis of the Equilibrium Point of the FOHNN System

3.3.1. The Influence of the Different Orders of the FOHNN System

In order to study the influences of the different orders of the FOHNN system, the bifurcation diagram, the Lyapunov exponential spectrum, and the Lyapunov dimension of the system, were plotted in the variation of the q values from 0.2 to 1, as shown in Figure 3a–c. Setting h = 0.01, β₁ = 0.4, and β₂ = 12, the initial point v₀ = (0.2, 0.1, 0.3, 0.2). It is emphasized that in order to eliminate the effect of the transition states, the excessive number of points generated in the calculation process, is appropriately rounded off. From Figure 4a, the FOHNN system is in a periodic state when 0.3 < q < 0.426. 0.426 < q < 1, the HNN-based fractional order system appears the chaotic attractors. At q ∈ (0.426, 1), there are multiple periodic windows of the system, which are shown in

Table 1. Different orders, the periodic states of the FOHNN system.

Parameters	System Status	Parameters	System Status
q = 0.584	Period-3	q = 0.724	Period-7
q = 0.61	Period-3	q = 0.754	Period-3
q = 0.65	Period-2	q = 0.802	Period-4
q = 0.66	Period-7	q = 0.828	Period-10

. Figure 3b–c is in perfect agreement with Figure 3a. The Lyapunov exponent of the system is positive and its Lyapunov dimension is greater than 2, which implies that the system has the chaotic attractors. Different types of attractors of the FOHNN system at different orders are plotted in Figure 4.

3.3.2. The Influence of the Different Synaptic Weights of the FOHNN System

Changing the different synaptic weight values, FOHNN presents rich and complex characteristics of the dynamics. Consider the influence of the alterations in two synaptic weights, β₁ and β₂, on the FOHNN system. When the fixed parameters β₂ = 12, q = 0.8, h = 0.01, and β₁∈ (0, 0.8), the bifurcation diagram of the system, Lyapunov exponents spectrum, Lyapunov dimension diagram are drawn in Figure 5a–c. When β₁ < 0, the FOHNN system is in the periodic state; β₁ > 0, chaotic attractors are generated and return to the periodic state again after a few periodic windows with β₁ > 0.8. To better illustrate the state of the attractor, the attractor phase diagrams are presented in Figure 6 for the different β₂.

When fixing the parameters β₁ = 0.2, q = 0.8, h = 0.01, and β₂ ∈ (6, 20), the FOHNN system appears to have more complex characteristics of the dynamics. Furthermore, the Lyapunov exponential spectrum of the system, the bifurcation diagram, Lyapunov dimension diagram, are presented and plotted in Figure 7. The phase diagrams of the different types of attractors are drawn in Figure 8. When β₂ ∈ (6, 8.65), the system generates the period-2 attractor through the period-doubling bifurcation and produces the period attractor again after a short chaotic state. It enters the chaotic state again through the period-doubling bifurcation, and later produces the chaotic attractors after the period window. Compared with the integer-order HNN system, the FOHNN system appears to have more complex dynamics characteristics.

3.4. Coexistence of the Attractors with the Different Synaptic Weights

During the dynamical analysis, the multiple attractor coexistence phenomena were observed in the FOHNN system. Setting h = 0.01 and β₁ = 0.5, β₂ = 12, the coexistence of the attractors can be observed for the different initial values selected, including the coexistence of the limit loops with the limit loops and the coexistence of the chaotic attractors. The initial values v₀ = (1, 0.1, 0.1, −0.1) and v₁ = (−1, 0, 0.1, 0.1) are chosen to observe the coexistence of the attractors through the v₃–v₄ phase plane. In Figure 9, with the increase of order q, the limit loops of coexistence are moving closer and closer, and when q = 0.9, the double scroll-double coexistence appears. Setting q = 0.5, β₁ = 0.5, and changing the value of β₂, we observe the attractor coexistence through the v₁–v₄ phase plane in Figure 10. With the increase of the synaptic weight β₂, the attractors gradually intermingle together. The complex attractor coexistence phenomenon further indicates that this FOHNN system has rich dynamical properties.

3.5. Transient Chaos in the FOHNN System

The emergence of chaos with a finite lifetime, is called the transient chaos. Usually, in a system, an appropriate region is selected in which the system moves in an apparently chaotic behavior and suddenly, at some point, the system immediately reaches a steady state, which can be a periodic behavior or a point of equilibrium. If it turns to chaotic motions, but they are different from the previous chaotic state, it is called a state transition. Transient chaos appears even in very common everyday life: transient chaos can occur in any system that moves irregularly, for a period of time, and then changes to a regular behavior. Based on this, the FOHNN system was studied in relation to this system, in which the transient chaos was found.

By choosing the parameters q = 0.8, β₁ = 0.5, k = 14, and the initial value v₀ = (0.1, 0.1, −0.2, 0.1), In order to observe the phenomenon conveniently, the horizontal axis of the coordinates is set to time t, and 100 points are set to 1 s. Through the relevant experiments, it can be clearly observed that when t = 1500 s, (N = 150,000), a funny phenomenon appears in the system. The transition from the chaotic state to the periodic state. Figure 11a presents a chaotic sequence diagram of the process, where the FOHNN system is in a chaotic state at 0 s < t < 700 s, and the system generates the periodic attractors at 700 s < t < 1500 s. For a better illustration of the change of state, the Lyapunov exponents and time evolution branch diagram are plotted, as shown in Figure 11b,c. The Lyapunov exponent spectrum fits well with the bifurcation diagram. The maximum Lyapunov exponent L₁ = 0.327 at t = 100 s, and the value of LEs reaches 0 at t = 700 s as the time proceeds. In Figure 12, the phase diagrams of the periodic and chaotic attractors, when the FOHNN is at different times, and their time series are drawn. It is shown that the proposed FOHNN system has rich dynamical properties and better chaotic characteristics, and it is more appropriate for the study of the neural network dynamics.

4. Circuit Design and Simulation of the FOHNN System

To better demonstrate the realizability of the theoretical and mathematical model of the FOHNN system, the circuit of the FOHNN system is designed, based on the relevant devices of the analog circuit, as shown in Figure 13. The Tanh and F modules in the design diagram represent the tanh unit circuit and the fractional order unit circuit, and their detailed design can be shown in Figure 14. The design of the fractional order unit circuit is based on its mathematical expression, which can be described as

$$F(s)=\frac{R_1}{s{R}_1{C}_1+1}+\frac{R_2}{s{R}_2{C}_2+1}+\frac{R_3}{s{R}_3{C}_3+1}+\cdots +\frac{R_n}{s{R}_n{C}_n+1}$$

(30)

Based on this, the designed FO unit circuit is as shown in Figure 14b.

Setting β₁ = 0.35, β₂ = 19, q₁ = q₂ = q₃ = q = 0.95, and the initial value of the FOHNN system v₀= (0.2, 0.1, 0.3, 0.2), the differential equation of the FOHNN system, based on Kirchhoff’s law and the circuit design, can be expressed as

$$\{\begin{array}{l}{}^{\ast }D{}_t^q{V}_1=(\frac{1}{R_5}{V}_1+\frac{R_3}{R_2{R}_4}\tanh (V_1)+\frac{R_{15}}{R_{14}{R}_6}\tanh (V_2)+\frac{R_{38}}{R_{37}{R}_9}\tanh (V_4))\frac{R_{12}{R}_8}{R_{11}}\hfill \\ {}^{\ast }D{}_t^q{V}_2=(\frac{1}{R_{17}}{V}_2+\frac{R_3}{R_2{R}_{16}}\tanh (V_1)+\frac{R_{15}}{R_{14}{R}_6}\tanh (V_2)+\frac{R_{38}}{R_{37}{R}_9}\tanh (V_3))\frac{R_{23}{R}_{19}}{R_{22}}\hfill \\ {}^{\ast }D{}_t^q{V}_3=(\frac{1}{R_{29}}{V}_3-\frac{1}{R_{27}}\tanh (V_1)+\frac{R_{38}}{R_{37}{R}_{31}}\tanh (V_4))\frac{R_{32}{R}_{35}}{R_{34}}\hfill \\ {}^{\ast }D{}_t^q{V}_4=(\frac{1}{R_{29}}{V}_4+\frac{R_3}{R_2{R}_{39}}\tanh (V_1)+\frac{R_{15}}{R_{14}{R}_{40}}\tanh (V_2)+\frac{R_{38}}{R_{37}{R}_{41}}\tanh (V_4))\frac{R_{46}{R}_{43}}{R_{45}}\hfill \end{array}$$

(31)

According to the system equations and the designed circuit, setting the corresponding values is listed in

Table 2. Values corresponding to the components.

Components	Values	Components	Values
R2, R3, R5, R8, R10, R11, R12, R13, R17, R18, R20, R21, R22, R23, R25, R26, R27, R29, R30, R31, R32, R35, R36, R37, R39, R40	10 kΩ	Rc, Rd	1 kΩ
R4	33.333 kΩ	Rb	0.52 kΩ
R6	20 kΩ	Rf1	15.3 kΩ
R7	100 kΩ	Rf2	1.15 MΩ
R14	4.348 kΩ	Rf3	692.9 MΩ
R15	28.5714 kΩ	Cf1	3.616 μF
R16	1.176 kΩ	Cf2	4.602 μF
R24	2.5 kΩ	Cf3	1.267 μF
R33	0.286 kΩ	U1∼16, Ua, Ub	3288 RT
R34	0.526 kΩ
R9, R19, R28, R38	0.5 kΩ
Ra, Re, Rf, Rg, Rh	10 kΩ

.

The relevant parameters are set and the circuit is connected, the output of Multisim simulation of the proposed FOHNN system is shown in Figure 15, which better verifies the implement ability of the FOHNN system.

A Monte Carlo analysis of the circuit was performed for a 5% tolerance of each component value, in the fractional-order cell circuit, based on a Gaussian distribution. Since the FOHNN system is of equal order, only one unitary circuit is analyzed here, as an example. Following the 2000 experiments in the Multisim environment, Figure 16 shows the transient analysis output of the Monte Carlo circuit, with a voltage average of 0.01502 V and an output voltage standard deviation of 0.0023 V. The circuit performs the fractional order function.

5. Conclusions

In this paper, a new class of a fractional-order neural network system is designed, based on the Hopfield neural network, solved by the ADM algorithm. The dissipative and bound properties of the system are drawn. The effects of the order q and the changes of the two synaptic weights β₁ and β₂ in the system, on the dynamic characteristics of the system, were analyzed, and the results showed that the system exhibited complex dynamical behaviors with the dynamic changes of the three parameters. Interestingly, the attractor coexistence phenomenon, as well as the transient chaos phenomenon, are also found. The circuit of the system was designed and experimentally verified using the Multisim simulation platform, and the experimental results verified the correctness of the theory and provided the possibility for the application of the neural network system in the actual applications. Therefore, the complex nonlinear phenomena and rich dynamical behaviors, exhibited by the FOHNN system, provide more experimental evidence for the system in the study of the neural network dynamics and the practical applications of fractional-order systems. The next step will be to consider the implementation of the actual circuit of the system and the application of the system to the image encryption, based on the chaotic sequences, etc., to provide more reference for further research.