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

: 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.


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 * D q t f (t) = J m−q t 0 where q ∈ R + , m ∈ N, * D q t is the q-order Caputo differential operator. Γ represents the Gamma function. Definition 2 [47]. The Laplace transform of the fractional order differentiation is (2) where F(s) = L [f(t)] denotes the Laplace transform of f(t).
The proposed fractional order differential equation * D q t 0 v(t) = f (v(t)), based on the Adomian solution method (ADM), can be expressed in the form, * D q t 0 v(t) = Lv(t) + Nv(t) + C(t) (6) where *D q t0 stands for the Caputo derivative operator of the order q (m−1 < q ≤ m, m ∈ N), v(t) = [v 1 (t) v 2 (t) . . . v n (t)] T are the state variables. The linear and nonlinear terms are denoted by L and N, respectively. C(t) = [C 1 (t) C 2 (t) . . . C n (t)] T is the constant for the autonomous system. By applying the J q t 0 to the left and right of Equation (1), the following equation can be obtained where J q t 0 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 q t 0 , are shown as follows: where t ∈ [t 0 , t 1 ], q ≥ 0, γ > −1, r ≥ 0. The i-th nonlinear term can be solved using the following      in which i = 0, 1, 2, 3 . . . ∞; j = 1, 2, 3 . . . n. The nonlinear term N is recorded as Therefore, using the following form to represent the numerical solution of the sys- where v i is calculated by

Solution of the FOHNN-Based System
In 1982, J. Hopfield proposed the Hopfield neural network model 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.
in which i = 0, 1, 2, 3 … ∞; j = 1, 2, 3… n. The nonlinear term N is recorded as Therefore, using the following form to represent the numerical solution of th system (1) where v i is calculated by

Solution of the FOHNN-Based System
In 1982, J. Hopfield proposed the Hopfield neural network model where Ri, Ci, and Ti represents the resistance, capacitance, and conductance, respectively and gi(vi) is the activation function generated by the operational amplifier. Normally, th selection of an as one, and when the hyperbolic tangent function is chosen as th activation function, the chaotic attractors appear in the system. The weight matrix Tij i modified and its dimension is determined by the number of neurons. In this FOHNN system, Tij has size 4 × 4, meaning that the Hopfield neural network has four stat variables, associated with each neuron. The topology structure of the neural network i shown in Figure 1.  Setting R i = 1, I i = 0, the Equation (16)  corresponding synaptic weight coefficients, the synaptic weight matrix is represented as follows According to Definitions 1 and 2, a fractional order form of a fourth-FOHNN system can be written as 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 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 A 1 = tanh(v 1 ) as an example, can be decomposed as follows The other three nonlinear terms are similar to A 1 . Setting the initial value the first term can be gained for Letting c 0 is obtained by the iterative relation formula Equation (14). The second term of the state variables are expressed as Fractal Fract. 2022, 6, 637 So v 1 will be represented as x 1 = c 1 (t − t 0 ) 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 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 β 1 = 0.72,  ( ) ( 4 tanh( ) tanh( )) ( 1) So v 1 will be represented as x 1 = c 1 (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 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

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 according to Equations (13) and (24), the obtained 3, the chaotic attractors will emerge from the system.

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 according to Equations (13) and (24), the obtained Obviously, β 1 < 3.3, the chaotic attractors will emerge from the system.

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 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 where sech 2 Observing Equation (27), the system always exists a zero equilibrium point E 0 = (0, 0, 0, 0), whatever the value of the parameter is taken. Setting β 1 = 0.2, β 2 = 15 the characteristic function of the equilibrium as the corresponding eigenvalues are −2.8701 ± 1.1582i and 2.8701 ± 2.3224i, and the |larg(λ 2 )| = 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.

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 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. 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. 1), there are multiple periodic windows of the system, which are shown in Tabl Figure 3b-c is in perfect agreement with Figure 3a. The Lyapunov exponent of system is positive and its Lyapunov dimension is greater than 2, which implies that system has the chaotic attractors. Different types of attractors of the FOHNN system different orders are plotted in Figure 4.   Changing the different synaptic weight values, FOHNN presents rich and com characteristics of the dynamics. Consider the influence of the alterations in two syna weights, β1 and β2, on the FOHNN system. When the fixed parameters β2 = 12, q = 0.8 1), there are multiple periodic windows of the system, which are shown in Table  Figure 3b-c is in perfect agreement with Figure 3a. The Lyapunov exponent of t system is positive and its Lyapunov dimension is greater than 2, which implies that t system has the chaotic attractors. Different types of attractors of the FOHNN system different orders are plotted in Figure 4.   Changing the different synaptic weight values, FOHNN presents rich and compl characteristics of the dynamics. Consider the influence of the alterations in two synap weights, β1 and β2, on the FOHNN system. When the fixed parameters β2 = 12, q = 0.8, h  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, β 1 and β 2 , on the FOHNN system. When the fixed parameters β 2 = 12, q = 0.8, h = 0.01, and β 1 ∈ (0, 0.8), the bifurcation diagram of the system, Lyapunov exponents spectrum, Lyapunov dimension diagram are drawn in Figure 5a-c. When β 1 < 0, the FOHNN system is in the periodic state; β 1 > 0, chaotic attractors are generated and return to the periodic state again after a few periodic windows with β 1 > 0.8. To better illustrate the state of the attractor, the attractor phase diagrams are presented in Figure 6 for the different β 2 . 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.  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.  When fixing the parameters β 1 = 0.2, q = 0.8, h = 0.01, and β 2 ∈ (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 β 2 ∈ (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.
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.

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.

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 β 1 = 0.5, β 2 = 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 v 0 = (1, 0.1, 0.1, −0.1) and v 1 = (−1, 0, 0.1, 0.1) are chosen to observe the coexistence of the attractors through the v 3 -v 4 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, β 1 = 0.5, and changing the value of β 2 , we observe the attractor coexistence through the v 1 -v 4 phase plane in Figure 10. With the increase of the synaptic weight β 2 , the attractors gradually intermingle together. The complex attractor coexistence phenomenon further indicates that this FOHNN system has rich dynamical properties.

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

Transient Chaos in the FOHNN System
The emergence of chaos with a finite lifetime, is called the transient chaos. Usuall in a system, an appropriate region is selected in which the system moves in a apparently chaotic behavior and suddenly, at some point, the system immediate reaches a steady state, which can be a periodic behavior or a point of equilibrium. If turns to chaotic motions, but they are different from the previous chaotic state, it called a state transition. Transient chaos appears even in very common everyday lif transient chaos can occur in any system that moves irregularly, for a period of time, an then changes to a regular behavior. Based on this, the FOHNN system was studied 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. −0.2, 0.1), In order to observe the phenomenon conveniently, the horizontal axis of th coordinates is set to time t, and 100 points are set to 1 s. Through the releva experiments, it can be clearly observed that when t = 1500 s, (N = 150,000), a funn phenomenon appears in the system. The transition from the chaotic state to the period state. Figure 11a presents a chaotic sequence diagram of the process, where the FOHN system is in a chaotic state at 0 s < t < 700 s, and the system generates the period attractors at 700 s < t < 1500 s. For a better illustration of the change of state, the Lyapuno exponents and time evolution branch diagram are plotted, as shown in Figure 11b,c. Th Lyapunov exponent spectrum fits well with the bifurcation diagram. The maximu Lyapunov exponent L1 = 0.327 at t = 100 s, and the value of LEs reaches 0 at t = 700 s as th time proceeds. In Figure 12, the phase diagrams of the periodic and chaotic attractor when the FOHNN is at different times, and their time series are drawn. It is shown th the proposed FOHNN system has rich dynamical properties and better chaot characteristics, and it is more appropriate for the study of the neural network dynamics

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, β 1 = 0.5, k = 14, and the initial value v 0 = (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 L 1 = 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.

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 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

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 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

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          Based on this, the designed FO unit circuit is as shown in Figure 14b. Setting β 1 = 0.35, β 2 = 19, q 1 = q 2 = q 3 = q = 0.95, and the initial value of the FOHNN system v 0 = (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 According to the system equations and the designed circuit, setting the corresponding values is listed in Table 2. 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. According to the system equations and the designed circuit, setting the corresponding values is listed in Table 2  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.

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 β 1 and β 2 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.