Star Memristive Neural Network: Dynamics Analysis, Circuit Implementation, and Application in a Color Cryptosystem

At present, memristive neural networks with various topological structures have been widely studied. However, the memristive neural network with a star structure has not been investigated yet. In order to investigate the dynamic characteristics of neural networks with a star structure, a star memristive neural network (SMNN) model is proposed in this paper. Firstly, an SMNN model is proposed based on a Hopfield neural network and a flux-controlled memristor. Then, its chaotic dynamics are analyzed by using numerical analysis methods including bifurcation diagrams, Lyapunov exponents, phase plots, Poincaré maps, and basins of attraction. The results show that the SMNN can generate complex dynamical behaviors such as chaos, multi-scroll attractors, and initial boosting behavior. The number of multi-scroll attractors can be changed by adjusting the memristor’s control parameters. And the position of the coexisting chaotic attractors can be changed by switching the memristor’s initial values. Meanwhile, the analog circuit of the SMNN is designed and implemented. The theoretical and numerical results are verified through MULTISIM simulation results. Finally, a color image encryption scheme is designed based on the SMNN. Security performance analysis shows that the designed cryptosystem has good security.


Introduction
Because of its rich chaotic dynamics, Hopfield neural networks (HNNs) have attracted much attention from chaos scholars [1][2][3]. It is well known that the human brain nervous system has abundant chaotic discharge behaviors [4], which are closely related to advanced intelligence. HNNs are considered to be the best example of studying the dynamics of the brain's nervous system [5,6]. At the same time, many researchers have found that neural networks with chaotic behavior have a wide range of applications in many fields including associative memory, pattern recognition, and combinatorial optimization [7,8]. Therefore, the study of the chaotic behavior of HNNs has important theoretical and practical significance for brain science and artificial intelligence.
Since the HNN was proposed, various HNNs have been proposed and their chaotic behaviors studied. For example, normal chaos [9][10][11] and hyperchaos [12][13][14] have been detected in some small HNNs with several neurons. Especially in recent years, memristive neural networks have received much attention from many scholars [15,16]. For example, Pham et al. [17] proposed a three-neuron memristive neural network and revealed its hidden chaotic behavior. Bao et al. [18] found that a memristive neural network consisting of three neurons can produce coexisting asymmetric attractors. A memristive neural network with complex chaotic attractors was proposed by Yu et al. in [19]. Furthermore, plenty of similar works have been reported [20][21][22]. To sum up, these studies are mainly focused on the chaotic dynamics of neural networks with mixed structures. Very recently,
(sgn(ϕ + 2j) + sgn(ϕ − 2j)) M = 1, 2, 3, . . . (2) where i, v, and ϕ are the current, voltage, and state variables of the memristor, respectively. The function h(ϕ) is the state function of the memristor, where N and M are two control parameters. The memristor model can exhibit typical memristive characteristics, as shown in Figure 1. As can be seen from Figure 1a, the memristor model can generate three tight hysteresis loops under different voltage amplitudes (A = 1, 2, 3). Figure 1b shows that with the increase in voltage frequency (F = 0.05, 0.1, 0.5), the area of the tight hysteresis loop of the memristor decreases gradually.
where i, v, and φ are the current, voltage, and state variables of the tively. The function h(φ) is the state function of the memristor, where control parameters. The memristor model can exhibit typical memris as shown in Figure 1. As can be seen from Figure 1a, the memristor m three tight hysteresis loops under different voltage amplitudes (A = shows that with the increase in voltage frequency (F = 0.05, 0.1, 0.5), t hysteresis loop of the memristor decreases gradually.

Memristive Star Neural Network Modeling
A Hopfield neural network with a brain-like structure can be used behaviors of the biological nervous system. A Hopfield neural netwo can be described by where Ci, Ri, and xi are the membrane capacitance, membrane resistan potential of the neuron i, respectively. wij is the synaptic weight coeffi ron j and neuron i. In addition, tanh(.) denotes the neuron activation fu external input current. Based on the original HNN (3), setting Ci = 1, Ri HNN with four neurons is proposed as follows:

Memristive Star Neural Network Modeling
A Hopfield neural network with a brain-like structure can be used to emulate chaotic behaviors of the biological nervous system. A Hopfield neural network with n neurons can be described by [1] where C i , R i , and x i are the membrane capacitance, membrane resistance, and membrane potential of the neuron i, respectively. w ij is the synaptic weight coefficient between neuron j and neuron i. In addition, tanh(.) denotes the neuron activation function, and I i is an external input current. Based on the original HNN (3), setting C i = 1, R i = 1, and I i = 0, a new HNN with four neurons is proposed as follows: where x i is the membrane potentials of neuron i. As shown in Figure 2, the four neurons are made up of a star neural network. A large number of research results show that the electromagnetic induction effect can be described by a flux-controlled memristor. According to the principle, the effect of external stimuli induced by electromagnetic radiation can be regarded as an additional forcing current IEMR. Consequently, when we consider that neuron 1 is stimulated by external electromagnetic radiation, the memristive star neural network can be modeled and described as follows: where ϕ is the magnetic flux across the membrane of neuron 1, µ is the feedback intensity of external stimuli induced by electromagnetic radiation, and µw(ϕ)x 1 is the electromagnetic induction current.
where φ is the magnetic flux across the membrane of neuron 1, µ is of external stimuli induced by electromagnetic radiation, and µw(φ netic induction current.

Equilibrium Points and Their Stability
The equilibrium points of the star memristive neural network a analyzed in this subsection. Setting the left side of Equation (5) to 0, are solved by

Equilibrium Points and Their Stability
The equilibrium points of the star memristive neural network and their stabilities are analyzed in this subsection. Setting the left side of Equation (5) to 0, the equilibrium points are solved by Solving Equation (6), one can obtain Obviously, there are infinite discrete equilibrium points. The Jacobian matrix of each equilibrium point can be calculated by where s i = sech 2 (x i ). When setting a = 1, b = 0.01, c = 0.7, and d = 1.3, the corresponding eigenvalues and stabilities are given in Table 1. From Table 1, all the equilibrium points are of unstable saddle type. According to the Shilnikov theorem, the star memristive neural network may have a chaotic phenomenon.

Dynamic Analysis
In this section, chaotic dynamics of the proposed star memristive neural network are revealed by using nonlinear dynamics methods including bifurcation diagrams, Lyapunov exponents, phase plots, Poincaré maps, and basins of attraction. It should be noted that the numerical simulations are completed in MATLAB R2017a with the ODE45 algorithm. Meanwhile, the start time, the time step, and the time length are set as 500, 0.01, and 4000, respectively.

µ-Related Chaotic Dynamics
In this subsection, the chaotic dynamics related to the memristive coupling strength µ is analyzed. Setting the parameters a = 1, b = 0.01, c = 0.7, d = 1.3, N = 0, and initial conditions (x 10 , x 20 , x 30 , x 40 , ϕ 0 ) = (0.1, 0.1, 0.1, 0.1, 0.1), the parameter µ increases from −0.2 to 0, and the µ-based bifurcation diagram is plotted in Figure 3a, where x 1 max is the maxima of the membrane potential x 1 . Moreover, the corresponding Lyapunov exponents are shown in Figure 3b. It can be seen from Figure 3 that the star memristive neural network can generate complex dynamical behaviors including period, transient chaos, and chaos. For example, with µ increasing from −0.2 to 0, the dynamical trajectory of the star memristive neural network starts from the period of entering into transient chaos at µ = −0.1, and then the transient chaotic behavior degrades into periodic behavior at u =−0.08. Interestingly, with the µ increasing to −0.055, the star memristive neural network enters chaos. It can be seen that the star memristive neural network exhibits a wide range of chaos until µ = 0. The phase portraits of the star memristive neural network with different values of u are given to illustrate its dynamical evolution with the parameter µ, as shown in Figure 4. It is obvious that the star memristive neural network successively produces periodic attractors, transient chaotic attractors, and chaotic attractors with the increase in µ.

N/M-Related Multi-Scroll Attractors
In this subsection, the chaotic dynamics related to memristor control param are analyzed. The numerical simulation shows that the SMNN can generate an number of multi-scroll chaotic attractors related to N/M. When the parameter 0.01, d = 1.3, µ = −0.01, N = 3, and initial states are kept unchanged, c is selec control parameter. A bifurcation diagram and the corresponding Lyapunov exp lated to parameter c∈(0, 2) are shown in Figure 5a and Figure 5b, respectively ingly, it can be seen from the bifurcation diagram that the bifurcation diagram i of a dense patch of points with seven steps. This means that the SMNN not only chaotic attractors but also produces seven-scroll attractors. From the Lyapunov e it shows one positive value, namely, it is chaotic behavior. Further research show SMNN can generate an arbitrary number of multi-scroll attractors by selecting control parameters N/M. To better understand the multi-scroll attractors, Figu the phase portraits of multi-scroll attractors under different numbers of scrolls. O the number of scrolls generated by the star memristive neural network can be by 2N+1 and 2M+2.

N/M-Related Multi-Scroll Attractors
In this subsection, the chaotic dynamics related to memristor control par are analyzed. The numerical simulation shows that the SMNN can generat number of multi-scroll chaotic attractors related to N/M. When the parame 0.01, d = 1.3, µ = −0.01, N = 3, and initial states are kept unchanged, c is se control parameter. A bifurcation diagram and the corresponding Lyapunov lated to parameter c∈(0, 2) are shown in Figure 5a and Figure 5b, respectiv ingly, it can be seen from the bifurcation diagram that the bifurcation diagra of a dense patch of points with seven steps. This means that the SMNN not o chaotic attractors but also produces seven-scroll attractors. From the Lyapun it shows one positive value, namely, it is chaotic behavior. Further research s SMNN can generate an arbitrary number of multi-scroll attractors by selec control parameters N/M. To better understand the multi-scroll attractors, F the phase portraits of multi-scroll attractors under different numbers of scrol the number of scrolls generated by the star memristive neural network can by 2N+1 and 2M+2.

N/M-Related Multi-Scroll Attractors
In this subsection, the chaotic dynamics related to memristor control parameters N/M are analyzed. The numerical simulation shows that the SMNN can generate an arbitrary number of multi-scroll chaotic attractors related to N/M. When the parameters a = 1, b = 0.01, d = 1.3, µ = −0.01, N = 3, and initial states are kept unchanged, c is selected as the control parameter. A bifurcation diagram and the corresponding Lyapunov exponents related to parameter c∈(0, 2) are shown in Figures 5a and 5b, respectively. Interestingly, it can be seen from the bifurcation diagram that the bifurcation diagram is made up of a dense patch of points with seven steps. This means that the SMNN not only generates chaotic attractors but also produces seven-scroll attractors. From the Lyapunov exponents, it shows one positive value, namely, it is chaotic behavior. Further research shows that the SMNN can generate an arbitrary number of multi-scroll attractors by selecting different control parameters N/M. To better understand the multi-scroll attractors, Figure 6 gives the phase portraits of multi-scroll attractors under different numbers of scrolls. Obviously, the number of scrolls generated by the star memristive neural network can be controlled by 2N + 1 and 2M + 2.  In addition, the Poincaré maps on φ-x3 and φ-x4 phase planes for the 7-scro with x1 = 0 are depicted in Figure 7a and Figure 7b, respectively. Clearly, the Poin have approximately 7-scroll maps, implying that the star memristive neural ne erates extremely complex multi-scroll attractors.   In addition, the Poincaré maps on φ-x3 and φ-x4 phase planes for the 7-scroll attractor with x1 = 0 are depicted in Figure 7a and Figure 7b, respectively. Clearly, the Poincaré maps have approximately 7-scroll maps, implying that the star memristive neural network generates extremely complex multi-scroll attractors.

φ0-Related Initial Boosting Behavior
In this subsection, the initial boosting chaotic phenomenon is analyzed. It is wonderful that the presented SMNN can generate initial boosting coexisting chaos. For instance, we plot the bifurcation diagram of the φ0 under a = 1, b = 0.01, c = 0.5, d = 1.3, µ = −0.01, M = 4, and x10 = x20 = x30 = x40 = 0.1, as shown in Figure 8a. As can be seen, the SMNN displays a complicated initial boosting behavior. Meanwhile, the corresponding constant Lyapunov exponents in the whole range of the φ0 variation are given in Figure 8b. Obviously, In addition, the Poincaré maps on ϕ-x 3 and ϕ-x 4 phase planes for the 7-scroll attractor with x 1 = 0 are depicted in Figures 7a and 7b, respectively. Clearly, the Poincaré maps have approximately 7-scroll maps, implying that the star memristive neural network generates extremely complex multi-scroll attractors.  In addition, the Poincaré maps on φ-x3 and φ-x4 phase planes for the 7-scro with x1 = 0 are depicted in Figure 7a and Figure 7b, respectively. Clearly, the Poin have approximately 7-scroll maps, implying that the star memristive neural net erates extremely complex multi-scroll attractors.

φ0-Related Initial Boosting Behavior
In this subsection, the initial boosting chaotic phenomenon is analyzed. It i ful that the presented SMNN can generate initial boosting coexisting chaos. Fo we plot the bifurcation diagram of the φ0 under a = 1, b = 0.01, c = 0.5, d = 1.3, µ = 4, and x10 = x20 = x30 = x40 = 0.1, as shown in Figure 8a. As can be seen, the SMNN a complicated initial boosting behavior. Meanwhile, the corresponding con complicated initial boosting behavior. Meanwhile, the corresponding constant Lyapunov exponents in the whole range of the ϕ 0 variation are given in Figure 8b. Obviously, the SMNN has an infinite wide chaotic range along the ϕ 0 -axis. That is to say, the SMNN enjoys complex dynamics of initial boosting coexisting chaos, which means that it has excellent robustness. Moreover, to further verify initial boosting dynamics, the local attraction basin in the ϕ 0 -x 10 plane is given in Figure 9. From Figure 10, the local attraction basin has complex manifold structures and basin boundaries, and the specified initial value regions are composed of different colored zones marked by 1-13, among which the colored zones marked by 1-13 correspond to the attractors with different positions. Therefore, the star memristive neural network has complex initial boosting behavior.
Entropy 2023, 25, x FOR PEER REVIEW enjoys complex dynamics of initial boosting coexisting chaos, which means that it cellent robustness. Moreover, to further verify initial boosting dynamics, the local tion basin in the φ0-x10 plane is given in Figure 9. From Figure 10, the local attractio has complex manifold structures and basin boundaries, and the specified initial v gions are composed of different colored zones marked by 1-13, among which the zones marked by 1-13 correspond to the attractors with different positions. Theref star memristive neural network has complex initial boosting behavior.   enjoys complex dynamics of initial boosting coexisting chaos, which means that it has excellent robustness. Moreover, to further verify initial boosting dynamics, the local attraction basin in the φ0-x10 plane is given in Figure 9. From Figure 10, the local attraction basin has complex manifold structures and basin boundaries, and the specified initial value regions are composed of different colored zones marked by 1-13, among which the colored zones marked by 1-13 correspond to the attractors with different positions. Therefore, the star memristive neural network has complex initial boosting behavior.

Circuit Validation
With the rapid development of artificial intelligence, the physical realization of neural network models is very important for developing neuromorphic hardware systems [42,43]. It is a reliable way to realize neural network models through analog circuits. This is because the analog neural network circuit can not only achieve real-time calculation but enjoys complex dynamics of initial boosting coexisting chaos, which means cellent robustness. Moreover, to further verify initial boosting dynamics, th tion basin in the φ0-x10 plane is given in Figure 9. From Figure 10, the local att has complex manifold structures and basin boundaries, and the specified in gions are composed of different colored zones marked by 1-13, among whic zones marked by 1-13 correspond to the attractors with different positions. T star memristive neural network has complex initial boosting behavior.

Circuit Validation
With the rapid development of artificial intelligence, the physical realiz ral network models is very important for developing neuromorphic hardw

Circuit Validation
With the rapid development of artificial intelligence, the physical realization of neural network models is very important for developing neuromorphic hardware systems [42,43]. It is a reliable way to realize neural network models through analog circuits. This is because the analog neural network circuit can not only achieve real-time calculation but also reproduce the behavior of the real nervous system. In this section, the proposed star memristive neural network model is realized by using basic electronic circuit elements such as resistors, capacitors, operational amplifiers, and analog multipliers. We first design the circuit of the star memristive neural network model and verify its complex behavior in Multisim.

Design of the SMNN Circuit
Before implementing the neural network circuit, we first introduce a hyperbolic tangent excitation function circuit [23] and a memristor circuit [15], respectively. Based on the two circuit units, a memristive Hopfield neural network circuit can be designed. According to the star memristive neural network model (5), the circuit structure is designed in Figure 11. Four membrane potentials x 1 , x 2 , x 3 , and x 4 are emulated by four out voltages v 1 , v 2 , v 3 , and v 4 , respectively. All synaptic weight coefficients are simulated by the resistors R 1 -R 10 . Based on the circuit in Figure 11, the circuit state equations can be described by Entropy 2023, 25, x FOR PEER REVIEW 9 memristive neural network model is realized by using basic electronic circuit elem such as resistors, capacitors, operational amplifiers, and analog multipliers. We firs sign the circuit of the star memristive neural network model and verify its complex havior in Multisim.

Design of the SMNN Circuit
Before implementing the neural network circuit, we first introduce a hyperbolic gent excitation function circuit [23] and a memristor circuit [15], respectively. Base the two circuit units, a memristive Hopfield neural network circuit can be designed cording to the star memristive neural network model (5), the circuit structure is desig in Figure 11. Four membrane potentials x1, x2, x3, and x4 are emulated by four out volt v1, v2, v3, and v4, respectively. All synaptic weight coefficients are simulated by the resi R1-R10. Based on the circuit in Figure 11, the circuit state equations can be described b Figure 11. Circuit implementation. Figure 11. Circuit implementation.

Measurement of the SMNN Circuit
The designed star memristive neural network circuit is verified by using the MULTI-SIM platform. When a = 1, b = 0.01, µ = 0.01, c = 0.7, d = 1.3, and initial values (0.1, 0.1, 0.1,  0.1, 0.1), and setting Ra = 1 MΩ, Rb = 10 MΩ, Rc = 13.9 kΩ, Rd = 8 kΩ, and initial capacitor voltages to 0.1 V, multi-scroll attractors can be generated by selecting different switches. For example, closing W 1 , W 2 , and W 3 and setting e 1 = 2 V and e 2 = 4 V, a six-scroll chaotic attractor can be generated from the designed circuit, as shown in Figure 12a. But when changing Rc = 20 kΩ (c = 0.5), the initial boosting coexisting attractors can be realized by adjusting different initial capacitor voltages v ϕ0 . As shown in Figure 12b, six coexisting chaotic attractors are obtained from the neural network circuit. Similarly, keeping Rc = 13.9 kΩ; closing W 2 , W 3 , and W 4 ; and setting e 1 = 1 V, e 2 = 3 V, and e 3 = 5 V, a seven-scroll chaotic attractor can be obtained, as shown in Figure 13a. Keeping Rc = 20 kΩ, seven coexisting chaotic attractors can be obtained as shown in Figure 13b. It should be noted that the circuit simulation results are slightly different from the numerical results because of the computing errors between two different tools.

Measurement of the SMNN Circuit
The designed star memristive neural network circuit is verified by usi SIM platform . When a = 1, b = 0.01, µ = 0.01, c = 0.7, d = 1.3, and initial valu  0.1, 0.1), and setting Ra = 1 MΩ, Rb = 10 MΩ, Rc = 13.9 kΩ, Rd = 8 kΩ, and i voltages to 0.1 V, multi-scroll attractors can be generated by selecting diffe For example, closing W1, W2, and W3 and setting e1 = 2 V and e2 = 4 V, a six attractor can be generated from the designed circuit, as shown in Figure  changing Rc = 20 kΩ (c = 0.5), the initial boosting coexisting attractors can adjusting different initial capacitor voltages vφ0. As shown in Figure 12b, chaotic attractors are obtained from the neural network circuit. Similarly 13.9 kΩ; closing W2, W3, and W4; and setting e1 = 1 V, e2 = 3 V, and e3 = 5 V, chaotic attractor can be obtained, as shown in Figure 13a. Keeping Rc = 20 existing chaotic attractors can be obtained as shown in Figure 13b. It should the circuit simulation results are slightly different from the numerical resu the computing errors between two different tools.

Application in a Color Cryptosystem
Chaos can be used for information encryption due to its high randomn tivity [44,45]. Chaotic neural networks with complex dynamic behavior hav ising applications for information encryption [46][47][48]. In this section, a ne encryption scheme is designed based on the proposed star memristive n with multi-scroll attractors.
Step 2: To obtain a pseudo-random sequence, the generated sequenc cessed as Step 4: K(i) is used to encrypt the original image using the XOR operati To demonstrate the efficiency of the designed image encryption crypt

Application in a Color Cryptosystem
Chaos can be used for information encryption due to its high randomness and sensitivity [44,45]. Chaotic neural networks with complex dynamic behavior have more promising applications for information encryption [46][47][48]. In this section, a new color image encryption scheme is designed based on the proposed star memristive neural network with multi-scroll attractors.
The encryption and decryption process is described in the following steps.
Step 2: To obtain a pseudo-random sequence, the generated sequences are preprocessed as where mod(x) is the modulo operation and floor(x) is the flooring operation.
Step 3: A chaotic sequence is generated, which can be described as Step 4: K(i) is used to encrypt the original image using the XOR operation, as follows: To demonstrate the efficiency of the designed image encryption cryptosystem, three color images "Lena", "virus", and "chameleon" with a size of 512 × 512 are chosen as the encryption object. The experimental results and the security performance analyses including the histogram, correlation coefficient, information entropy, key sensitivity, data loss, and noise attacks are given in the following.
(1) Histogram analysis: Histograms are used to evaluate the distribution of pixel intensity values in an image. In theory, a good image encryption system should produce a uniform histogram. Figure 14 gives the original images, the encrypted images, and their respective histograms. Obviously, the encrypted images in Figure 14(c1) look cluttered and completely lose their original information. The histograms of the encrypted images in Figure 14(d1-d3) are almost uniform, which means that it is difficult to obtain any useful statistical information from the encrypted image. Therefore, the proposed image encryption scheme is enough to resist statistical attacks.
Entropy 2023, 25, x FOR PEER REVIEW original information. The histograms of the encrypted images in Figure 14(d1-d3 most uniform, which means that it is difficult to obtain any useful statistical info from the encrypted image. Therefore, the proposed image encryption scheme is to resist statistical attacks. (2) Correlation analysis: The relationship between adjacent pixels in an imag described using correlation. In general, a stronger correlation indicates a more image and a weaker correlation indicates a more chaotic image. The correlation co is computed by [45] Table 2, after the original images were encrypted, their relevance was reduced. Therefore, the designed cryptosystem has a strong ability to resist stati tacks.  (2) Correlation analysis: The relationship between adjacent pixels in an image can be described using correlation. In general, a stronger correlation indicates a more regular image and a weaker correlation indicates a more chaotic image. The correlation coefficient is computed by [45] where x and y are the intensity values of two adjacent pixels. Here, 10,000 pairs of pixels were randomly chosen in horizontal, vertical, and diagonal directions from the original image and corresponding encrypted image to evaluate the correlation coefficient. As shown in Table 2, after the original images were encrypted, their relevance was greatly reduced. Therefore, the designed cryptosystem has a strong ability to resist statistical attacks. (3) Entropy analysis: Information entropy can be used to evaluate the randomness of image information. The information entropy is calculated as [45] where P(x i ) denotes the probability of x i and 2N represents the number of the information source. The maximum theoretical information entropy is 8. Table 3 gives the calculation results of information entropy under different channels. From the results in Table 3, it can be seen that compared with other similar schemes, the information entropy of this scheme is closer to the theoretical value. Table 3. Information entropy in different signal channels and different encryption schemes.

RGB Red Green Blue
(4) Sensitivity analysis: Key sensitivity is an important indicator for measuring the security of encryption algorithms. A good image encryption scheme should be key-sensitive.
The initial values are used as secret keys in this encryption algorithm. The decrypted images are shown in Figure 15(a1-b3) with a slight change of the secret key. Despite the fact that the secret key has been changed a little (10)(11)(12)(13)(14)(15)(16), the decrypted images are completely different from the original image. Figure 15(c1-c3) shows the decrypted image with the correct secret key. As shown in Table 4, compared with other similar image encryption schemes, the image encryption scheme proposed in this paper has a higher sensitivity to the key. (5) Data loss and noise attacks Data loss and noise attacks can seriously affect the decryption effect of encrypted images. To evaluate the ability of data loss and noise attacks, we cut off some parts of the encrypted image and then decrypt it. As shown in Figure 16(a1-b4), data loss attacks for the different lost areas are successfully decrypted for the original image to be recovered. In order to test the resistance of the algorithm to noise attack, salt and pepper noise with different concentrations is added to the encrypted images. It can be seen from Figure 16(c1-c4) that some pixel values in the decrypted images have been changed; however, the approximate information of the original image can still be recovered successfully. Furthermore, when different concentrations of Gaussian noise are added to the encrypted images, these images are still able to be decrypted, as shown in Figure 16(d1-d4). Consequently, we can conclude that the proposed color image encryption scheme is able to withstand data loss and noise attacks and has high security. (3) Sensitivity analysis: Key sensitivity is an important indicator for measuring the security of encryption algorithms. A good image encryption scheme should be key-sensitive. The initial values are used as secret keys in this encryption algorithm. The decrypted images are shown in Figure 15(a1-b3) with a slight change of the secret key. Despite the fact that the secret key has been changed a little (10)(11)(12)(13)(14)(15)(16), the decrypted images are completely different from the original image. Figure 15(c1-c3) shows the decrypted image with the correct secret key. As shown in Table 4, compared with other similar image encryption schemes, the image encryption scheme proposed in this paper has a higher sensitivity to the key.   (4) Data loss and noise attacks Data loss and noise attacks can seriously affect the decryption effect of encrypted images. To evaluate the ability of data loss and noise attacks, we cut off some parts of the encrypted image and then decrypt it. As shown in Figure 16(a1-b4), data loss attacks for the different lost areas are successfully decrypted for the original image to be recovered. In order to test the resistance of the algorithm to noise attack, salt and pepper noise with different concentrations is added to the encrypted images. It can be seen from Figure  16(c1-c4) that some pixel values in the decrypted images have been changed; however, the approximate information of the original image can still be recovered successfully. Furthermore, when different concentrations of Gaussian noise are added to the encrypted images, these images are still able to be decrypted, as shown in Figure 16(d1-d4). Consequently, we can conclude that the proposed color image encryption scheme is able to withstand data loss and noise attacks and has high security.

Conclusions
In this article, the chaotic dynamics of a star memristive neural network are studied. First, based on a Hopfield neural network with four neurons and a flux-controlled memristor, a star memristive neural network model is constructed. Then, its chaotic behaviors are revealed by using various numerical methods. Analysis results show that the star memristive neural network can generate abundant chaotic dynamics including chaos,

Conclusions
In this article, the chaotic dynamics of a star memristive neural network are studied. First, based on a Hopfield neural network with four neurons and a flux-controlled memristor, a star memristive neural network model is constructed. Then, its chaotic behaviors are revealed by using various numerical methods. Analysis results show that the star memristive neural network can generate abundant chaotic dynamics including chaos, multi-scroll attractors, and initial boosting coexisting behavior. Especially, the number of scrolls for the multi-scroll attractors can be changed by adjusting the memristor's control parameters. In addition, the position and number of coexisting attractors can be changed by switching the memristor's initial value. To further verify these results, an analog neural network circuit is designed and implemented. All numerical results are experimentally verified by MULTISIM circuit simulation. Finally, a color image encryption scheme based on the proposed star memristive neural network is designed. Simulation results such as histograms, correlation, information entropy, key sensitivity, and data loss and noise attacks demonstrate that the designed image encryption scheme has good security.
With the rapid development of memristors, memristor-based neural networks have been widely used in various fields including memristive neurodynamics [51,52], memristive neuromorphic computation [53,54], and so on [55]. In future work, we will devote ourselves to studying the chaotic dynamics of the memristive neural networks with different topology structures. We will also explore the practical applications of the memristive neural network developed here.