Hopfield neural network (HNN) can be featured by a set of circuit state equations corresponded to n neurons (Bersini and Sener, 2002). For the i-th neuron, the circuit state equation is expressed as

where x_i is a state variable standing for the voltage across the capacitor C_i, R_i is a resistor representing the membrane resistance between the inside and outside of the neuron, I_i is a input bias current, tanh(x_j) is a neuron activation function indicating the voltage input from the j-th neuron, and W = (w_ij) is an n × n synaptic weight matrix illustrating the strength of connections between the i-th and j-th neurons.

In the realization circuit of HNN, the synaptic weight w_ij is generally constructed by a resistor connecting the i-th and j-th neurons. When the connection resistor is replaced by the above hyperbolic-type memristor characterized by Equation (1), a memristor based HNN can be easily proposed.

A hyperbolic-type memristor based HNN with 3 neurons is considered in our next work. The connection topology for the hyperbolic-type memristor based HNN is displayed in Figure 3. Corresponding to this connection topology, the connection matrix considered is of the following form

Define n = 3, C_i = 1, R_i = 1, I_i = 0, and RC = 1. The autonomous ordinary differential equations describing the proposed memristive network can be derived in a dimensionless form as

where W = a – btanh(x₄) stands for the synaptic weight w₁₃ connecting the first and third neurons and k is a positive constant representing the coupling strength of the hyperbolic-type memristor. Thus, the hyperbolic-type memristor based HNN can be modeled by Equation (8), which is a four-dimensional autonomous nonlinear dynamical system.

With respect to Equation (8), the parameters of a, b, k which are related to the hyperbolic-type memristor W are positive. It can be proved that the orbits of Equation (8) including periodic and chaotic orbits are confined in a bounded region.

Referring to the approach in Nik et al. (2015), Zahedi and Nik (2015), Chen et al. (2017), and Singh and Roy (2017), a Lyapunov function is introduced as

The time derivative of Equation (9) is yield as

For convenience, denote

Equation (10) can be then rewritten as

Forx₄ ∈ R,−1 < tanh(x₄) < 1. Therefore,

and

For any x_i (i = 1, 2, 3, 4), one has|tanh(x_i)| < 1. Equation (11) can be thereby simplified as

Let D₀ > 0 be the sufficiently large region. For all (x₁, x₂, x₃, x₄) satisfying V(x₁, x₂, x₃, x₄) = D with D > D₀, there exists the following condition

where (kM + 6) is a positive constant.

Consequently, on the surface

with D > D₀, there yields

which implies that the set

is a confined region of all solutions of Equation (8), i.e., the memristive system Equation (8) is bounded.

Since the memristive system Equation (8) is invariant under the transformation (x₁, x₂, x₃, x₄, b) to (–x₁, –x₂, –x₃, –x₄, –b), the memristive system Equation (8) is symmetric about the parameter b.

By setting the left-hand side of Equation (8) to zero, the equilibrium points of the hyperbolic-type memristor based Hopfield neural network can be numerically solved and determined as a zero equilibrium point P₀ = (0, 0, 0, 0) and two nonzero equilibrium points P₁ = (δ₁, δ₂, δ₃, δ₄) and P₂ = (η₁, η₂, η₃, η₄), where the nonzero equilibrium points P₁ and P₂ can be calculated by solving the following equations

and

where the values of y₁, z₁ and y₂, z₂ are the two intersection points of the two following function curves

Therefore, the values of y₁, z₁ and y₂, z₂ can be determined through graphic analytic method. Taking k = 0.8, a = 1, and b = 0.26 as an example, the two function curves given in Equation (21) can be plotted in Figure 4, from which the solutions are gotten as y₁ = 2.1661, z₁ = −0.5971, y₂ = −1.4755, and z₂ = 0.1974. Correspondingly, the two nonzero equilibrium points P₁ and P₂ can be easily obtained from Equation (20a) and (20b).

The Jacobian matrix at the equilibrium point P = (x-1,x-2,x-3,x-4) is derived from Equation (8) as

where hi=sech2(x-i)(i=1,2,3,4) andW-=a-btanh(x-4).

For the zero equilibrium point P₀ = (0, 0, 0, 0), the characteristic polynomial equation is yielded as

which implies the eigenvalues of P₀ is only related to the memristor inner parameter a but not associated with the memristor inner parameter b. When a = 1, it is confirmed by Routh-Hurwitz criteria that P₀ is stable if k ≥ 1.4286, otherwise it is unstable if k < 1.4286. Taking k = 0.8 and a = 1 as an example, the eigenvalues of P₀ is derived as λ₁ = 0.7684, λ_{2, 3} = −0.5842 ± j1.6237, λ₄ = −1, implying that the zero equilibrium point is an unstable saddle point.

For the two nonzero equilibrium points P₁ = (δ₁, δ₂, δ₃, δ₄) and P₂ = (η₁, η₂, η₃, η₄), however, the corresponding eigenvalues should be solved by using MATLAB numerical simulations. When k = 0.8 and a = 1, for different memristor inner parameter b, two nonzero equilibrium points P₁ and P₂, their corresponding eigenvalues, and the generating attractor types are shown in Table 2. Therefore, it can be seen that the nonzero equilibrium points P₁ and P₂ are asymmetric, P₁ is a stable node-focus for b ≥ 0.44, and whereas P₂ always is an unstable saddle-focus, leading to the emergence of coexisting asymmetric attractors' behaviors. Moreover, the results in Table 2 demonstrate that when the value of the parameter b increases, the dynamical behaviors of the hyperbolic-type memristor based HNN have transitions from unstable chaotic to unstable periodic and then to stable point behaviors, i.e., the dynamical behaviors of the HNN can be stabilized by the hyperbolic-type memristor.

Furthermore, it can be amusingly found from a larger number of numerical simulations that in the region of 0.44 ≤ b < 3.06, P₁ is a stable node-focus and P₂ is an unstable saddle-focus; but in the region of b ≥ 3.06, P₁ disappears and the unique nonzero equilibrium point P₂ maintains an unstable saddle-focus.

When k = 0.8, the bifurcation diagrams of the state variable x₁ and the corresponding first two Lyapunov exponents are shown in Figures 5A,B, respectively, from which unstable chaotic, unstable periodic, and stable point behaviors as well as period doubling bifurcation routes, tangent bifurcation routes, and crisis scenarios can be found. When the parameter b is gradually increased in the region of (0, 0.4), the dynamical behaviors of system Equation (8) are basically consistent for different initial conditions and their orbits all start from chaos, enter into a large periodic window with zero largest Lyapunov exponents via reverse period doubling bifurcation routes, then break into chaos via tangent bifurcation routes, further turn into chaos with narrow bands via crisis scenarios, and last degrade into period via reverse period doubling bifurcation routes.

It is especially interesting that coexisting behaviors of asymmetric attractors can be also observed, which are mainly emerged in two regions of [0.2595, 0.2640] and [0.4, 0.6] due to the occurrences of the crisis scenarios under the initial conditions of (0, −0.1, 0, 0) and (1, 0, 0, 0). For some determined values of the parameter b, the phase portraits of coexisting asymmetric attractors in the x₁ – x₃ plane are depicted in Figure 6, where four different kinds of coexisting asymmetric attractors are exhibited. Based on the above theoretical analyses, it can be known that the generating chaotic attractors and limit cycles in Figure 6 are associated with two unstable saddle-foci and the convergent point attractor in Figure 6D is related to a stable node-focus. Consequently, the hyperbolic-type memristor based HNN is self-excited but not hidden (Danca and Kuznetsov, 2017).

When the coupling strength of the hyperbolic-type memristor is chosen as k = 0.95, the bifurcation diagrams of the state variable x₁ and the corresponding first two Lyapunov exponents are shown in Figures 7A,B, respectively. Similarly, dynamical behaviors of chaotic attractors, limit cycles, point attractors, bifurcation routes, and crisis scenarios can be found in Figure 7. Differing from the dynamical behaviors in the case 1, the globally coexisting behaviors of asymmetric attractors are existed in this case.

Several phase portraits of coexisting asymmetric attractors in the x₁ – x₃ plane are plotted in Figure 8. In special, the coexistence of three disconnected attractors at b = 0.14 are displayed in Figure 8C, where for the initial conditions (0, 0.1, 0, 0), a right-chaotic spiral attractor appears in the hyperbolic-type memristor based HNN and four Lyapunov exponents are L₁ = 0.0321, L₂ = 0, L₃ = −0.4098, and L₄ = −0.9715, respectively; whereas for the initial conditions (0, −0.1, 0, 0) and (1, 0, 0, 0), a left-period-1 limit cycle and a right-period-2 limit cycle are separately emerged in the system (8), the corresponding Lyapunov exponents are L₁ = 0, L₂ = −0.1190, L₃ = −0.1193, and L₄ = −1.0295, respectively, as well as L₁ = 0, L₂ = −0.1814, L₃ = −0.1815, and L₄ = −0.9698, respectively.

When the coupling parameter is changed as k = 1, the bifurcation diagrams of the state variable x₁ and the corresponding first two Lyapunov exponents are shown in Figures 9A,B, respectively. However, only for the initial conditions (0, 0.1, 0, 0) and (1, 0, 0, 0), complex dynamical behaviors including limit cycles with different periodicities, right-chaotic spiral attractors with different topologies, right-period-2 limit cycles with transient chaos, and stable point attractors as well as period doubling bifurcation routes, tangent bifurcation routes, and crisis scenarios can be discovered. Like as the dynamical behaviors in the case 2, the globally coexisting behaviors of asymmetric attractors can be also found in this case.

Some phase portraits of coexisting asymmetric attractors in the x₁ – x₃ plane are demonstrated, as shown in Figure 10. It should be mentioned that two types of coexisting asymmetric multiple attractors' behaviors are generated, as shown in Figures 10C,D, in which a strikingly coexisting phenomenon of right-period-2 limit cycle with transient chaos, left-period-1 limit cycle, and right-period-2 limit cycle at b = 0.34 is presented in Figure 10D.

But above all, different kinds of coexisting asymmetric attractors can be easily found in the hyperbolic-type memristor based HNN. It is significant to stress that the coexistence of two and three disconnected attractors revealed in the above three cases are all asymmetric, which are different from the symmetric coexistence of multiple attractors encountered in many nonlinear dynamical systems (Kengne et al., 2015; Bao et al., 2016a; Xu et al., 2016). Therefore, the hyperbolic-type memristor based HNN is a specific nonlinear dynamical system.

Additionally, according to the above three cases, it can be thereby concluded that the coupling strength and inner parameters of the hyperbolic-type memristor have significant effects on the dynamics of the hyperbolic-type memristor based HNN, i.e., the newly introduced hyperbolic-type memristor can cause the HNN initially in chaotic state to be stabilized to the stable state or can be used to generate chaotic signals in the HNN. In other words, the memristor can be regarded as a controller to realize the dynamical control of the HNN.

With the mathematical model of Equation (8), the hyperbolic-type memristor based HNN can be physically implemented by some operational amplifiers linked with resistors and/or capacitors (Bao et al., 2017), a nonlinear memristor, and three nonlinear function circuits. The circuit scheme for the memristive HNN is drawn in Figure 11, where the memristor W represents the proposed hyperbolic-type memristor emulator in Figure 1A and the three circuit modules T_a, T_b, and T_c marked by –tanh with solid box are the inverting hyperbolic tangent function circuit units drawn in Figure 1B.

The desired circuit in Figure 11 has four dynamic elements of three capacitors and a hyperbolic-type memristor W, corresponding to four state variables of v₁, v₂, v₃, and v₀, respectively. Therefore, the circuit state equations for Figure 11 are established as

Assuming that the integrating time constant RC = 1 ms, the resistance and the capacitance can be chosen as R = 10 kΩ and C = 100 nF, respectively. Based on the element values of the weight matrix (7), other resistances are calculated as R₁ = R/1.4 = 7.143 kΩ, R₂ = R/1.2 = 8.333 kΩ, R₃ = R/7 = 1.429 kΩ, R₄ = R/1.1 = 9.091 kΩ, R₅ = R/2.8 = 3.571 kΩ, R₆ = R/2 = 5 kΩ, R₇ = R/4 = 2.5 kΩ. It is significant to illuminate that two adjustable parameters of the coupling strength k and the inner parameter b are achieved by adjusting precision potentiometers R_a and R_b of the hyperbolic-type memristor emulator in Figure 1A, whose calculating relations between the resistances and the system parameters can be determined as R_a = R/ka kΩ and R_b = gR/kb kΩ, respectively.

With the circuit schemes displayed in Figures 1, 11, a circuit simulation model plotted by using PSPICE electronic circuit simulator and a hardware experimental circuit fabricated by commercial electronic components are gotten ready for validating the coexisting behaviors of asymmetric attractors in the hyperbolic-type memristor based HNN. The pictures of several circuit breadboards are photographed as depicted in Figure 12, where Figure 12A is the hyperbolic-type memristor emulator, Figure 12B is the three inverting hyperbolic tangent function circuit units, and Figure 12C is main circuit of the hyperbolic-type memristor based HNN. Additionally, the experimentally captured phase portraits are obtained by 4 channel digital oscilloscope.

Consider k = 0.95 as an example to perform PSPICE circuit simulations and hardware experiments of the memristive HNN. When a = 1, the resistance R_a is fixed as 10.53 kO and only the resistance R_b is adjustable. By adjusting the resistance R_b of the hyperbolic-type memristor emulator, some phase portraits in the v₁ – v₃ plane are simulated by PSPICE simulator with time step 10⁻⁵ and captured by a 4 channel digital oscilloscope in XY mode, as shown in Figure 13, where the orbits marked with red, blue, and black colors represent those triggered by three different initial values. It should be stressed that, in PSPICE circuit simulations the different initial values are assigned by setting different initial capacitor voltages, whereas during hardware circuit experiments the desired initial values are randomly achieved by the induced voltages of four capacitors through switching on and off the hardware circuit power supplies.

The simulated and captured results given in Figure 13 are consistent with numerical simulations given in Figure 8, which availably illustrate that the circuit realization form of the hyperbolic-type memristor based HNN can also display coexisting behaviors of asymmetric attractors. However, corresponding to Figures 13A–D, the values of the resistor R_b in PSPICE circuit simulations are 105.82 kΩ, 26.13 kΩ, 7.45 kΩ, and 2.21 kΩ, respectively, while the values of the resistor R_b in hardware circuit experiments are equal to 105.2 kΩ, 25.45 kΩ, 7.2 kΩ, and 2.13 kΩ, respectively. Due to the computational errors in PSPICE simulation (Kuznetsov et al., 2017) and the existence of parasitic parameters in the practical hardware circuit, the differences of the values of the resistor R_b indeed appear in PSPICE circuit simulations and hardware circuit experiments.

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