Application of Two General Memristor Models in Chaotic Systems

Abstract

The memristor has attracted more and more attention due to its broad application prospect. The theory of memristors is being gradually improved. The first is to derive a general voltage-controlled discrete memristor (DM) model from another perspective-circuit, which proves that it conforms to the definition of Memristor, and a Simulink model of the discrete memristor is built to implement the simulation. The other one is the conventional magnetron titanium dioxide (i.e., D) memristor model. In order to explore the adaptability of these two types of memristor models, this paper constructs two types of memristor chaotic systems based on them and performs dynamics analysis to verify the applicability of the above memristor models, which lays the foundation for the application of memristor chaotic systems in the field of communication security.

1. Introduction

The concept of memristor was proposed by Prof. Chua from the University of California, Berkeley in 1971 based on the completeness of circuit theory, which represents the relationship between magnetic flux and electric charge. The memristor is also considered to be the fourth basic circuit element in addition to resistance, capacitance, and inductance [1, 2]. In 2008, Strukov et al. from Hewlett-Packard Laboratories first developed a nano-memristor device when doing TiO₂ cross-array experiments [2], and the experiments confirmed that the nano-memristor has a unique switching mechanism, automatic memory function, continuous input and output characteristics, and the emergence of the memristor opens up a new space for the development of the design and application of electronic circuits. Because of the memory function of the memristor, its potential application value has attracted extensive attention from academia and industry. Memristors will show attractive application prospects in fields such as neural networks [3], electrical engineering [4], and secure communications [5, 6]. In addition, memristor can mimic the associative learning and forgetting properties of the human brain and thus is used in neuromorphic circuits and artificial intelligence [7].

Since the development of the nano-memristor in HP Labs, various models of the memristor have been proposed. For example, the nonlinear drift model of HP memristor in literature [8], and the flux-controlled memristor model focusing on threshold characteristics in literature [9]. These models are based on ion migration, and although they are consistent with the basic properties of the memristor, many practical behavioral properties are still not represented. In particular, the ion mobility model can simulate the nonlinear transmission properties of synapses when the memristor acts as a synapse, but it is unable to explain the phenomena of long-term plasticity (LTP), short-term plasticity (STP), and the memory fading phenomena in biological systems, which is a common shortcoming of the current ion mobility models. This is a common shortcoming of current ion mobility models. Therefore, it is necessary to find a mathematical model that can fit the above phenomena.

It is worth noting that references [10, 11] replace the Chua diode in the Chua oscillator with a segmented linear memristor, and propose several memristor chaotic oscillation circuits based on Chua circuits. Driven by this groundbreaking work, many scholars have devoted themselves to the study of various memristor chaotic systems [12, 13, 14, 15]. The chaotic signal generator is the foundation of chaotic secure communication. Generally speaking, the nonlinearity of the circuit is a necessary condition for chaos generation, so it is necessary to make the system have more complex chaotic behavior. As a controllable nonlinear device, memristors, coupled with their small size and low power consumption, are very suitable for application in high-frequency chaotic circuits. High-frequency chaotic signals have broad application prospects in image encryption [16] and chaotic secure communication [17]. Therefore, the use of memristors to construct chaotic systems has received close attention from researchers, and research on the design and dynamic behavior analysis of memristor chaotic systems has become a hot topic [12, 14].

Since chaos theory has been proposed, the research results on the design of chaotic systems, dynamics analysis, chaos-based nonlinear circuits, and their applications have been quite mature. With the continuous deepening of the application aspects, the requirements for the system complexity are getting higher and higher. The memristor, as a kind of nonlinear element with memory function and nanometer size, also the nonlinear part of the chaotic system, not only can greatly improve the chaotic system’s signal randomness and complexity but also reduce the physical size of the system. On the one hand, some scholars have tried to enhance the chaotic properties of chaotic systems by incorporating memristor into them [15, 18, 19], such as fractional-order chaotic systems [18, 20, 21], conservative chaotic systems [22, 23], multiscroll chaotic systems [24, 25, 26], and fractal chaotic systems [27, 28]. On the other hand, despite the wide range of applications of memristor in nonlinear systems, most of the studies are based on continuous memristor models. However, continuous memristor is difficult to be applied to discrete systems or digital circuits. Therefore, researchers have started to focus on discrete memristors and have achieved some remarkable results in this area. Several researchers have derived discrete memristor models using continuous memristors [29, 30, 31]. Literature [30] modeled a discrete memristor using the forward Eulerian difference method and obtained a new two-dimensional hyperchaotic mapping from sampled memristor-capacitor circuits. Literature [31] proposed a discrete memristor model based on the difference theory, proved the three-fingerprint characterization of the model according to the definition of generalized memristor, and applied it to the Hénon mapping. In addition, the discrete HP memristor derived from the continuous HP memristor is the most popular research topic [29, 32, 33]. Literature [29] constructed a multidimensional closed-loop coupling model and then introduced HP discrete memristors into the coupling model. It also constructed a discrete memristor coupling model based on Logistic mapping and sinusoidal mapping, which can produce an arbitrary high-dimensional hyperchaotic system. Literature [32] designed two discrete memristors with cosine band amplitude memristors and established Simulink models of two discrete memristors which verified that they conformed to the definition of memristor [33] so that the dynamics of classical chaotic mapping can be improved. Several scholars have derived general generic models for discrete memristor. Literature [34] proposed a general discrete memristor model and a unified discrete memristor mapping model, also proposed four 2D-discrete memristor hyperchaotic mappings with infinite immovable points, and studied their dynamical behaviors. Literature [35] proposed a generalized voltage-controlled discrete memristor model, improved three typical discrete chaotic mappings, and studied their dynamical behaviors. In summary, memristor is of great significance in enhancing nonlinear systems, especially in the research of combining with chaotic fields, due to its good nonlinear characteristics, and the study and control of memristor chaotic systems is a new issue that needs to be paid attention to in the engineering applications of nonlinear dynamical systems.

Therefore, there is a need to further develop and improve the theory of memristor electrical networks, to explore the nonlinear characteristics of memristor and its role in the complex dynamics and formation mechanism of chaotic systems, to reveal the plasticity of memristor is of great significance in the evolution and function of neural networks [36, 37, 38] and neuromorphic circuits [39]. The combination of memristor and chaos can be mainly divided into two aspects: on the one hand, memristor has become one of the hotspots of research due to its inherent nonlinear properties and complex dynamical behaviors [30, 31, 33, 34, 40, 41]. On the other hand, memristor is often used as one of the applications to introduce into other chaotic systems to enhance the chaotic properties due to its memory function and nonlinearity [15, 18, 24, 26, 29, 31, 32, 35].

In the above studies, the memristor model is usually established by defining the memristor’s memristance or memductance, or the model is established directly from a specific memristor. However, the ideal memristor model cannot completely simulate the characteristics of the actual device, so the establishment of a general-purpose memristor model is more practical significance, but there has been less exploration of the adaptability of the universal memristor model. In this context, through the relevant theoretical derivation of memristor, this paper proposes two generalized memristor models, gives the specific construction methods and application rules, and applies them to two classical chaotic systems respectively which verifies the correctness and applicability of the proposed models from the experimental results of dynamics analysis.

The rest of the paper is organized as follows. Section 2 mainly proposes two generalized memristor models, namely, the generalized voltage-controlled discrete memristor model and the generalized magnetic-controlled titanium dioxide memristor model, constructs the corresponding memristor chaotic system, analyzes the dynamics of the system, and verifies the correctness and applicability of the proposed memristor models. Section 3 summarizes and discusses the whole paper.

2. Design and application of generalized memristor modeling

In this section, two types of generalized memristor mathematical models are proposed, namely, the generalized voltage-controlled discrete memristor model and the generalized magnetic-controlled titanium dioxide memristor model. By studying the mechanism of action and nonlinear properties inside of the memristor, the above two memristor models are applied to two types of classical chaotic systems: discrete Hénon chaotic mapping and Lorenz chaotic systems, respectively, to verify the universality of the proposed generalized memristor mathematical model.

2.1 Generalized voltage-controlled discrete memristor model and its application to chaotic mapping

2.1.1 Generalized voltage-controlled discrete memristor model

In general, the ideal mathematical model of a magnetic-controlled memristor is defined as:

$$q=\widehat{q}\left(\phi \right),$$E1

or

$$\left\{\begin{array}{l}i\left(t\right)=W\left(\phi \right)\cdot v\left(t\right),\\ \mathit{d\phi }\left(t\right)/\mathit{dt}=v\left(t\right),\end{array}\right.$$E2

where $\phi$ and $q$ denote the input magnetic flux and the charge flowing through the device, respectively, $v$ is the input voltage, and $i$ is the output current. $W\left(\phi \right)$ is the memductance, which is a nonlinear function of the magnetic flux $\phi \left(t\right)$.

Chua [42] gave the steps of an ideal voltage-controlled memristor through the nonlinear intrinsic relationship between the device terminal voltage $v$ and the terminal current $i$, moreover, Bao et al. [30] constructed a model of a magnetic-controlled memristor with nonlinear memductance, which is defined as:

$$\left\{\begin{array}{l}i\left(t\right)=cos\phi \left(t\right)\cdot v\left(t\right),\\ \mathit{d\phi }\left(t\right)/\mathit{dt}=v\left(t\right).\end{array}\right.$$E3

Using the above model of the memristor, a sampling memristor-capacitor circuit with a sample interval switch is further proposed to connect the memristor to the capacitor, which is shown in Figure 1.

Figure 1.

The sampling memristor-capacitor circuit.

The charge-controlled discrete memristor is derived by introducing the forward difference method into the charge-controlled continuous memristor [43], and similarly, the voltage-controlled discrete memristor can be derived using the difference method [44].

Consider an ideal memristor as shown in Eq. (2). Denote $v_n,{\phi }_n$ and $i_n$ as the capacitor voltage value, the memristor magnetic flux, and the current value at the time $n$, respectively. In the sampling memristor-capacitor circuit of Figure 1, the intrinsic relationship between the memristor and the capacitor can be expressed by the Euler method as follows:

$$\left\{\begin{array}{l}{i}_n=W\left({\phi }_n\right)\cdot v_n,\\ {\phi }_{n+1}-{\phi }_n=\mathit{\Delta T}\cdot v_n,\end{array}\right.$$E4

and

$$C\left(v_{n+1}-v_n\right)=-\mathit{\Delta T}\cdot i_n,$$E5

where $C$ is the capacitance.

This can be obtained by applying Kirchhoff’s current law to the circuit in Figure 1.

$$\left\{\begin{array}{l}{v}_{n+1}=v_n-k\cdot W\left({\phi }_n\right)\cdot v_n,\\ {\phi }_{n+1}={\phi }_n+\epsilon v_n,\end{array}\right.$$E6

where $k=\mathit{\Delta T}/C,\epsilon =\mathit{\Delta T}$ are two positive parameters, and the relation between $v_n$ and $W\left({\phi }_n\right)$ can be obtained from the above equation:

$$\begin{array}{l}W\left({\phi }_n\right)=W\left({\phi }_{n-1}+\epsilon \cdot v_{n-1}\right)\\ =W\left({\phi }_0+\epsilon \cdot \sum _{j=0}^{n-1}{v}_j\right)\\ =W\left(q_0+\epsilon \cdot \sum _{j=0}^{n-1}{v}_j\right),\end{array}$$E7

where $q_0$ can be regarded as the initial charge, so that the generalized voltage-controlled discrete memristor model can be obtained as:

$$i_n=v_n\cdot W\left(q_0+e\cdot \sum _{j=0}^{n-1}{v}_j\right),$$E8

where $e$ is a parameter related to the sampling interval, which can also be viewed as a parameter of the discrete memristor.

In the generalized voltage-controlled discrete memristor model, a new discrete memristor model can be constructed by defining memductance, which can satisfy the essential characteristics of the memristor while satisfying the diversity of the memristor. It is worth noting that the steps involved in constructing a new discrete memristor based on the generalized voltage-controlled discrete memristor model are as follows:

Step 1: Select a suitable continuous function as the memductance $W\left(\phi \right)$, to obtain:

$$i_n=v_n\cdot W\left(q_0+e\cdot \sum _{j=0}^{n-1}{v}_j\right),$$E9

Step 2: Apply a sine voltage signal with different frequencies to the above equation, and the volt-ampere characteristic curve can be obtained.

Step 3: The volt-ampere characteristic curve is used to determine whether the constructed discrete memristor satisfies the three basic characteristics of a memristor [45]:

1. Have pinched hysteresis loops at the origin;

2. Frequency dependence of hysteresis loop lobe areas, hysteresis loop lobe areas decrease monotonically with increasing frequency of the periodic excitation signal;

3. When the frequency is infinite, the hysteresis loops contract to a single-valued function.

To verify the applicability of the proposed generalized voltage-controlled discrete memristor model, in this paper, the memductance is defined as a common primitive function, as shown in Table 1.

Function categories	$W\left(\phi \right)$
Trigonometric functions	$W\left(\phi \right)=sin\phi ,W\left(\phi \right)=cos\phi$
Inverse trigonometric function	$W\left(\phi \right)=arccos\phi$
Power function	$W\left(\phi \right)={\phi }^{\pi }$
Exponential function	$W\left(\phi \right)=e^{\phi }$
Logarithmic function	$W\left(\phi \right)=log\phi$

Table 1.

Defining functions of memductance in discrete memristors.

A sine voltage signal $v_n=A_{0}sin\left(\mathit{\omega n}\right)$ is used as the input to the discrete memristor model. The volt-ampere characteristic curves at different frequencies are shown in Table 2, where it can be observed that the trajectory in the $v-i$ plane is a pinched hysteresis loop at the origin, and the hysteresis loop lobe areas decrease monotonically with increasing frequency of the sinusoidal voltage signal. When the frequency is infinite, the hysteresis loops shrink as a single-valued function. Therefore, the constructed discrete memristor models all have the three fingerprints characteristics of the memristor [31].

Table 2.

Hysteresis loops of discrete memristors.

Based on the mathematical model of a generalized voltage-controlled discrete memristor, its Simulink model is constructed as shown in Figure 2. The model consists of a unit delay block, a summation block, a memory block, an addition block, a gain block, a product block, a constant block, and a $W\left(\phi \right)$ function block. The summing block and the memory block form an accumulator. The $W\left(\phi \right)$ function block represents the function that defines the memductance. Different discrete memristor Simulink models can be created by setting the $W\left(\phi \right)$ function. XY Graph represents the relationship between the input voltage $v_n$ and the memristor current $i_n$.

Figure 2.

Simulink model of the generic voltage-controlled discrete memristor model.

To verify the correctness of the Simulink model, the similarity between the Simulink model of the generalized voltage-controlled discrete memristor and the experimental data from the numerical simulation has been measured. The $v_n$ and $i_n$ obtained from the Simulink model will be recorded by using the To Workspace module. Due to the difference in sampling time between the Simulink model and the numerical simulation, the Simulink model sine module is used in the numerical simulation as the input sine voltage signal $v_n$ . The amplitude of the sine voltage is $A=3$ and the several frequency are set as $\omega =2\times {10}^{-4},5\times {10}^{-4},3\times {10}^{-3},2\times {10}^{-2},$respectively. Here, the $W\left(\phi \right)$ function is set as a logarithmic function. The relationship between the Simulink model and the numerical simulation of the volt-ampere characteristic curve is shown in Figure 3. Each volt-ampere characteristic curve corresponds to 400,000 points. The results show that the Simulink model is consistent with the change of hysteresis loop trend of the numerical simulation experiments, thus verifying the correctness of the generalized voltage-controlled discrete memristor model.

Figure 3.

The pinched hysteresis loops obtained from (a) the Simulink model; (b) the numerical simulation.

2.1.2 Hénon mapping model based on discrete memristor

The Hénon mapping [46] is a common two-dimensional discrete chaotic mapping with the system equation:

$$\left\{\begin{array}{l}{x}_n=1+y_{n-1}-{\mathit{ax}}_{n-1}^2,\\ y_n=\mathit{bx},\end{array}\right.$$E10

where $a$ and $b$ are the control parameters of the system.

For introducing a discrete memristor into the Hénon mapping, there are two choices of the discrete memristor model’s inputs, $x_n$ or $y_n$. Here, $x_n$ is chosen as the input to the voltage-controlled discrete memristor, and the $y_{n-1}$ variable in $x_n$ is chosen to multiply the output of the memristor, let $W\left(\phi \right)=cos\phi$. Thus, the Hénon mapping model based on the discrete memristor is as follows:

$$\left\{\begin{array}{l}{x}_n=\left\{\begin{array}{l}1-{\mathit{ax}}_{n-1}^2+y_{n-1}{x}_{n-1}W\left(q_0+e\sum _{i=1}^{n-1}{x}_i\right),2\le n\le 100\\ 1-{\mathit{ax}}_{n-1}^2+y_{n-1}{x}_{n-1}W\left(q_0+e\sum _{i=n-99}^{n-1}{x}_i\right),n\ge 101\end{array}\right.\\ y_n={\mathit{bx}}_{n-1},\\ \\ W\left(\phi \right)=cosx.\end{array}\right.$$E11

Where $q_0$ and $e$ are respectively the internal initial values and parameters of the discrete memristor. Inspired by the literature [31], the “short-term memory” method is used to limit the memory of the discrete memristor, i.e., for the first 100 items of the sequence, the memristor remembers all the items, and after the first 100 items of the sequence, the memristor only remembers the first 99 items.

2.1.3 Dynamics analysis of Hénon mapping based on discrete memristor

The dynamical behavior of DM-based Hénon mappings, including attractor trajectories, Lyapunov exponential spectrum, bifurcation maps, attractor basins, and sequence complexity, are studied and analyzed to compare and contrast the specific effects of voltage-controlled discrete memristors on Hénon chaotic mappings.

The chaotic attractor is shown in Figure 4, setting the initial value of the system Eq. (11) to $\left(x_0,y_0\right)=\left(0.1,0.1\right)$ and the internal parameter $q_0=0.1,e=0.0001$ of the memristor model (the same as below), unlike the original chaotic mapping, the trajectory of the attractor is an arc. Figure 4 shows the attractors of DM-based Hénon chaotic mapping for different parameters. In Figure 4(a), when the fixed parameter $b=0.3$, the gap of the arcs disappears and the length of the attractor arcs increases as the parameter $a$ increases. When $a=1.6$, the attractor arc length is the longest. In addition, in Figure 4(b), the folding degree of the attractor decreases with the increase of the parameter $b$ when the fixed parameter $a=1.5$. It can be seen that when $b=0.2$, there is a large gap between the arcs, and the attractor consists of three arcs, and there are two arcs in all the upper part of the attractor, and the two arcs intersect at a point, and the distance between these two arcs increases as the parameter $b$ increases. It can be summarized that the parameter $a$ controls the length of the arcs and the gap between the arcs, while the parameter $b$ controls the number of arcs, the gap between the arcs, the distance between the arcs, and the folding state of the attractor.

Figure 4.

Chaotic attractor maps for DM-based Hénon mapping. (a) $b=0.3$. (b) $a=1.5$.

The Lyapunov exponential spectrum and its corresponding bifurcation under different parameters control for the Hénon mapping and the DM-based Hénon mapping are shown in Figures 5 and 6, respectively. Figure 5 shows the Lyapunov exponential spectrum and bifurcation maps of the corresponding systems under the control of the parameter $a$. The Hénon mapping has the system in a chaotic state at $a\in \left[1.15,1.607\right]$, while the DM-based Hénon mapping has the system in a chaotic state at $a\in \left[1.354,1.953\right]$. The chaotic region of the DM-based Hénon mapping is wider than that of the original Hénon mapping, a situation that is more evident in the parameter $b$. Figure 6 shows the Lyapunov exponential spectrum and the bifurcation diagram for the corresponding system under the control of the parameter $b$. It can be observed that the LE value of the DM-based Hénon mapping is almost always greater than zero within $b\in \left[0,0.9\right]$, and the Hénon mapping has a LE value greater than zero only within $b\in \left[0,0.313\right]$. Therefore, it can be concluded that the voltage-controlled discrete memristor effectively expands the chaotic parameter space of the Hénon mapping.

Figure 5.

Lyapunov exponential spectrum and bifurcation diagrams of Hénon mapping and DM-based Hénon mapping $x_0=0.0001,y_0=0.0001$. (a) Lyapunov exponential spectrum of Hénon mapping $a\in \left[0,1.7\right],b=0.2$. (b) Lyapunov exponential spectrum of DM-based Hénon mapping $a\in \left[0,2\right],b=0.2$. (c) Bifurcation diagram of Hénon mapping $a\in \left[0,1.7\right],b=0.2$. (d) Bifurcation diagram of DM-based Hénon mapping $a\in \left[0,2\right],b=0.2$.

Figure 6.

Lyapunov exponential spectrum and bifurcation diagrams of Hénon mapping and DM-based Hénon mapping $x_0=0.0001,y_0=0.0001$. (a) Lyapunov exponential spectrum of Hénon mapping $b\in \left[-0.3,0.4\right],a=1.4$. (b) Lyapunov exponential spectrum of DM-based Hénon mapping $b\in \left[-0.3,0.9\right],a=1.4$. (c) Bifurcation diagram of Hénon mapping $b\in \left[-0.3,0.4\right],a=1.4$. (d) Bifurcation diagram of DM-based Hénon mapping $b\in \left[-0.3,0.9\right],a=1.4$.

To further investigate the effect of the initial state values, an attraction basin consisting of the $x_0-y_0$ plane is shown in Figure 7, where the white region is the divergence region, i.e., the sequence generated by the system is divergent at the corresponding initial value of the point, and the non-white region is the chaotic region. It is worth noting that the voltage-controlled L effectively extends the valid region of the initial values $x_0$ and $y_0$ of the original system, and the valid region of the initial values of the DM-based Hénon mapping is larger than that of the original Hénon mapping, which suggests that the discrete memristor can change and extend the range of the initial values of the chaotic region of the Hénon mapping.

Figure 7.

Attraction basin in the $x_0-y_0$ plane. (a) Hénon mapping $a=1.4$, $b=0.2$. (b) DM-based Hénon mapping $a=1.4,b=0.2$.

To investigate the effect of the internal parameters of the memristor on the system’s degree of chaos, the parameters and initial values of the system are fixed, and the internal parameters of the memristor, $e\in \left[0,1\right],q_0\in \left[0,5\right],$ are chosen to be 600 × 600 samples. Figure 8 shows the Lyapunov exponential chaotic map based on the Hénon mapping of the DM. It can be observed that with the changes of $e$ and $q_0$, the chaotic region (dark red) and the non-chaotic region (yellow) are distributed in parallel lines, and the system’s degree of chaos will change with the selection of the internal parameters of the memristor, which indicates that the nonlinear characteristics of the discrete memristor will have a more obvious effect on the degree of chaos of the Hénon mapping.

Figure 8.

Chaotic map of $e-q_0$ plane based on Lyapunov exponent $e\in \left[0,1\right],q_0\in \left[0,5\right],$$a=1.4,b=0.2,x_0=0.1,y_0=0.1$.

Figure 9 shows the Permutation Entropy (PE) of the system under two-parameter control, i.e., from the point of view of the entropy value of the system sequence to determine the system’s degree of chaos, the red and yellow regions indicate that the system’s PE value is larger, i.e., the system parameters can obtain a high-complexity sequence by selecting the corresponding region, and at this time, the system’s degree of chaos is larger. It can be noted that in Figure 9(a), the Hénon mapping is only in the interval of $a\in \left[0,2\right],b\in \left[0,1\right]$ when the PE complexity of the system is greater than 0. The PE value in the dark blue region is equal to 0 when the system is in the divergence state. In Figure 9(b), on the contrary, the complexity region of the system Eq. (11) has increased significantly compared to system Eq. (10), which is consistent with the results of the multiple dynamics analysis methods described above. The results show that based on the voltage-controlled discrete memristor proposed in this paper, the parameter range and initial value range of the chaotic mapping as well as the robustness of the strengthened chaotic system can be effectively enhanced, which is more applicable to the field of communication encryption.

Figure 9.

Permutation entropy with parameters $a,b$ and $x_0=0.0001,y_0=0.0001$. (a) Hénon map with $a\in \left[0,2\right],b\in \left[-0.3,1.5\right]$. (b) DM-based Hénon map with $a\in \left[0,2\right],b\in \left[-0.3,3\right]$.

A comparison of the main metrics of the DM-based Hénon mapping discussed in this section with the classical Hénon mapping is given in Table 3, highlighting the fact that the DM-based Hénon mapping exhibits significant advantages in all the main metrics, with a larger chaotic range.

	Chaos range of a	Chaos range of b	Initial value chaos ratio	High PE radio
Hénon mapping	0.457	0.313	20.14%	22.43%
DM-based Hénon mapping	0.599	0.9	30.36%	36.08%

Table 3.

Comparison of DM-based Hénon mapping and Hénon mapping key metrics.

2.2 Generalized HP memristor model and its application in chaotic systems

2.2.1 Generalized HP memristor model

Firstly, reviewing the structure and definition of the HP memristor model, Strukov et al. found that a layer of titanium dioxide (${\mathit{TiO}}_2$) film was sandwiched between two platinum electrodes, and that the ${\mathit{TiO}}_2$ film is divided into two parts, one being a pure ${\mathit{TiO}}_2$ thin film layer, which has high resistivity, and the other being a ${\mathit{TiO}}_{2-X}$ thin film layer. Due to the “doping” of pure ${\mathit{TiO}}_2$, some of the oxygen atoms are missing, so the ${\mathit{TiO}}_{2-X}$ film layer is positively charged and has a lower resistance to the passage of current. When current flows through the device in a counterclockwise direction, the boundary between the two materials shifts, thereby increasing the proportion of ${\mathit{TiO}}_{2-X}$ film layers, at which point the overall resistance decreases. When the current is reversed, the boundary moves in the opposite direction and the proportion of pure ${\mathit{TiO}}_2$ layers increases, increasing the overall resistance. When the current stops, the boundary between the two film layers stops moving and maintains its final resistance value. In other words, the memristor “remembers” the current flowing through it. The physical model of a memristor [2] is shown in Figure 10. The corresponding mathematical model is shown as:

Figure 10.

Schematic of HP memristor.

$$\begin{array}{l}M\left(t\right)=R_{\mathrm{ON}}\frac{w\left(t\right)}{D}+R_{\mathrm{OFF}}\left(1-\frac{w\left(t\right)}{D}\right),\\ =R_{\mathrm{OFF}}+\left(R_{\mathrm{ON}}-R_{\mathrm{OFF}}\right)\frac{w\left(0\right)}{D},\end{array}$$E12

where $M\left(t\right)$ is the memristor value, $R_{\mathrm{OFF}}$ and $R_{\mathrm{ON}}$ are the two limiting values of the memristor resistance, $w\left(t\right)$ is the time-dependent thickness of the ${\mathit{TiO}}_{2-X}$ film layer, and $D$ is the thickness of the $\mathrm{Pt}-{\mathrm{TiO}}_{2-\mathrm{X}}-{\mathrm{TiO}}_2-\mathrm{Pt}$ device film. The $R_{\mathrm{OFF}}$ value corresponds to the resistance of the device at $w\left(t\right)=0$, and the $R_{\mathrm{ON}}$ value corresponds to the device resistance at $w\left(t\right)=D$. At this time, the linear drift model of the device can be represented as:

$$\frac{\mathit{dw}\left(t\right)}{\mathit{dt}}={\mu }_{\mathrm{V}}\frac{R_{\mathrm{ON}}}{D}i\left(t\right),$$E13

Where ${\mu }_{\mathrm{V}}$ is the average mobility of oxygen vacancy.

To derive the complete mathematical model of the HP memristor, it is necessary to consider the interdependence between memristor, charge, and magnetic flux as well as the boundary and initial conditions of the operation. Usually, the initial value of the memristor is not 0, i.e., ${\left.w\left(t\right)\right|}_{t=0}\ne 0$, and $w\left(0\right)$ is set as the initial thickness of $w\left(t\right)$, when the initial state value of the memristor is:

$$M\left(0\right)=R_{\mathrm{OFF}}+\left(R_{\mathrm{ON}}-R_{\mathrm{OFF}}\right)\frac{w\left(0\right)}{D},$$E14

which is obtained by integrating Eq. (13):

$$w\left(t\right)={\mu }_{\mathrm{V}}\frac{R_{\mathrm{ON}}}{D}q\left(t\right)+w\left(0\right),$$E15

thus Eq. (12) can be rewritten as:

$$M\left(t\right)=M\left(0\right)+\frac{\left(R_{\mathrm{ON}}-R_{\mathrm{OFF}}\right){\mu }_{\mathrm{V}}{R}_{\mathrm{ON}}}{D^2}q\left(t\right),$$E16

i.e.:

$$M\left(t\right)=M\left(0\right)+k\cdot q\left(t\right),$$E17

where $k=\frac{\left(R_{\mathrm{ON}}-R_{\mathrm{OFF}}\right){\mu }_{\mathrm{V}}{R}_{\mathrm{ON}}}{D^2},$ and is a constant.

Consider that there exists a boundary condition $w\left(t\right)\in \left[0,D\right]$ for the memristor, i.e., $R_{\mathrm{ON}}\le M\left(t\right)\le R_{\mathrm{OFF}}$. From Eq. (17), the linear relationship between $q\left(t\right)$ and $M\left(t\right)$ is given by:

$$\frac{R_{\mathrm{OFF}}-M\left(0\right)}{k}\le q\left(t\right)\le \frac{R_{\mathrm{ON}}-M\left(0\right)}{k}.$$E18

let

$$\frac{R_{\mathrm{OFF}}-M\left(0\right)}{k}=c_1,\frac{R_{\mathrm{ON}}-M\left(0\right)}{k}=c_2.$$E19

Since $R_{\mathrm{ON}}\ll R_{\mathrm{OFF}}$, thus $k<0$, Eq. (17) is a monotonically decreasing function, the charge control model of the HP memristor is a continuous function with the following mathematical expression:

$$M\left(t\right)=\left\{\begin{array}{l}{R}_{\mathrm{OFF}},q\left(t\right)<c_1\\ M\left(0\right)+kq\left(t\right),c_1\le q\left(t\right)<c_2\\ R_{\mathrm{ON}},q\left(t\right)\ge c_2.\end{array}\right.$$E20

Assuming that an input signal, i.e., $q\left(0\right)=0$, is applied to the device at the moment $t=0$, the relationship between charge and magnetic flux can be expressed according to the definition of magnetic flux in a memristor:

$$\phi \left(t\right)={\int }_0^{t}M\left(t\right)i\left(t\right)\mathit{dt}={\int }_{q\left(0\right)}^{q\left(t\right)}M\left(t\right)\mathit{dq}\left(t\right)$$E21

then

$$\phi \left(t\right)=\left\{\begin{array}{l}{\int }_{q\left(0\right)}^{c_1}\left[M\left(0\right)+kq\left(t\right)\right]\mathit{dq}\left(t\right)+{\int }_{c_1}^{q\left(t\right)}{R}_{\mathrm{OFF}}\mathit{dq}\left(t\right)=R_{\mathrm{OFF}}q\left(t\right)+c_3,q\left(t\right)<c_1\\ {\int }_{q\left(0\right)}^{q\left(t\right)}\left[M\left(0\right)+kq\left(t\right)\right]\mathit{dq}\left(t\right)=\frac{k}{2}{q}^2\left(t\right)+M\left(0\right)q\left(t\right),c_1\le q\left(t\right)<c_2\\ {\int }_{q\left(0\right)}^{c_2}\left[M\left(0\right)+kq\left(t\right)\right]\mathit{dq}\left(t\right)+{\int }_{c_2}^{q\left(t\right)}{R}_{\mathrm{ON}}\mathit{dq}\left(t\right)=R_{\mathrm{ON}}q\left(t\right)+c_4,q\left(t\right)\ge c_2\end{array}\right.$$E22

where

$$\left\{\begin{array}{l}{c}_3=\frac{k}{2}{c_1}^2+\left[M\left(0\right)-R_{\mathrm{OFF}}\right]c_1=-\frac{{\left[R_{\mathrm{OFF}}-M\left(0\right)\right]}^2}{2k},\\ c_4=\frac{k}{2}{c_2}^2+\left[M\left(0\right)-R_{\mathrm{ON}}\right]c_2=-\frac{{\left[R_{\mathrm{ON}}-M\left(0\right)\right]}^2}{2k}.\end{array}\right.$$E23

Therefore, from Eq. (22), the charge as a function of magnetic flux is given by:

$$q\left(t\right)=\left\{\begin{array}{l}\frac{\phi \left(t\right)-c_3}{R_{\mathrm{OFF}}},\phi \left(t\right)<c_5\\ \frac{\sqrt{2\mathit{k\phi }\left(t\right)+M^2\left(0\right)}-M\left(0\right)}{k},c_5\le \phi \left(t\right)<c_6\\ \frac{\phi \left(t\right)-c_4}{R_{\mathrm{ON}}},\phi \left(t\right)\ge c_6\end{array}\right.$$E24

where

$$\left\{\begin{array}{l}{c}_5=c_1{R}_{\mathrm{OFF}}+c_3=\frac{R_{\mathrm{OFF}}^2-M^2\left(0\right)}{2k},\\ c_6=c_2{R}_{\mathrm{ON}}+c_4=\frac{R_{\mathrm{ON}}^2-M^2\left(0\right)}{2k}.\end{array}\right.$$E25

Bringing Eq. (24) into Eq. (20) yields a generalized magnetic control model for HP memristors:

$$M\left(t\right)=\left\{\begin{array}{l}{R}_{\mathrm{OFF}},\phi \left(t\right)<c_5\\ \sqrt{2\mathit{k\phi }\left(t\right)+M^2\left(0\right)},c_5\le \phi \left(t\right)<c_6\\ R_{\mathrm{ON}},\phi \left(t\right)\ge c_6.\end{array}\right.$$E26

2.2.2 Modeling of Lorenz chaotic systems based on HP memristor

It is well known that the classical Lorenz chaotic system is a continuous nonlinear system with state equations:

$$\left\{\begin{array}{l}\frac{\mathit{dx}}{\mathit{dt}}=\sigma \left(y-x\right),\\ \frac{\mathit{dy}}{\mathit{dt}}=\mathit{\rho x}-y-\mathit{xz},\\ \frac{\mathit{dz}}{\mathit{dt}}=\mathit{xy}-\mathit{\gamma z},\end{array}\right.$$E27

where $\sigma ,\rho ,\gamma$ are the system parameters, and when $\sigma =10,\rho =28,\gamma =\frac{8}{3}$ (the same as below), the system is in a chaotic state.

Without changing the structure of the original system equation of state, the input flux of the HP memristor is used to replace “some single variable” of the Lorenz system equation, i.e., the relationship between charge and magnetic flux in Eq. (24) is utilized to generate complex chaotic signals. It is worth noting that the introduction of a generalized HP memristor model into common nonlinear chaotic systems (which include high-dimensional nonlinear chaotic systems of arbitrary dimensions) requires the following two rules to be followed simultaneously:

Rule 1: Pick any single variable in the system that has a negative coefficient;

Rule 2: In the other two nonlinear differential equations, one differential equation has one and only one selected single variable, and the other differential equation has one and only one cross-product term containing the selected single variable.

Based on the above approach, a Lorenz chaotic system based on HP memristor, i.e., Lorenz memristor chaotic system, is proposed in this paper, and the state equations of the constructed new system are as follow:

$$\left\{\begin{array}{l}\frac{\mathit{dx}}{\mathit{dt}}=\sigma \left(y-x\right),\\ \frac{\mathit{dy}}{\mathit{dt}}=\mathit{\rho x}-f\left(\left|y\right|\right)-\mathit{xz},\\ \frac{\mathit{dz}}{\mathit{dt}}=\mathit{xy}-\mathit{\gamma z},\end{array}\right.$$E28

where $f\left(\cdot \right)$ is the nonlinear term of the system that satisfies the relationship between magnetic flux and charge with the expression:

$$f\left(y\right)=\left\{\begin{array}{l}\frac{y-c_3}{R_{\mathrm{OFF}}},y<c_5\\ \frac{\sqrt{2\mathit{ky}+M^2\left(0\right)}-M\left(0\right)}{k},c_5\le y<c_6\\ \frac{y-c_4}{R_{\mathrm{ON}}},y\ge c_6\end{array}\right.$$E29

where the $y$ variable in the second differential equation is the input magnetic flux $\phi$ of the memristor, and the expressions for $c_i\left(i=3,4,5,6\right)$ are in Section 2.2.1 and will not be repeated here.

2.2.3 Dynamics analysis of Lorenz chaotic system based on HP memristor

The dynamical behavior of the Lorenz memristor chaotic system is studied and analyzed, including the attractor phase diagram, largest Lyapunov exponents (LLE), bifurcation diagram, sequence complexity, etc., to compare and analyze the specific effect of HP memristor on Lorenz chaotic system.

$R_{\mathrm{OFF}},R_{\mathrm{ON}}$ are the two limit values of the memristor, set $R_{\mathrm{OFF}}=20\mathrm{k}\Omega$, $R_{\mathrm{ON}}=100\Omega$, $M\left(0\right)=1.6\times {10}^4\mathrm{k}\Omega$ is the initial value of the state of the memristor, ${\mu }_{\mathrm{V}}$ is the average mobility of oxygen deficiencies, $D=10\mathrm{nm}$ is the thickness of the film of the memristor device, the initial value of the system $\left(x_0,y_0,z_0\right)=\left(0.3,0.1,0.2\right)$ (the same as below), the variable step-size fourth-order Runge-Kutta algorithm is used to simulate the above two systems. Their attractor phase diagram is shown in Figure 11.

Figure 11.

Chaotic attractors in the $x-y$ plane. (a) Lorenz chaotic system. (b) Lorenz memristor chaotic system.

Both system Eq. (27) and system Eq. (28) present a chaotic attractor with two scrolls in the plane, one scroll trajectory of the Lorenz chaotic system is relatively concentrated, and its surrounding trajectories are attracted in the region and spread around, while the distribution law of the two scroll trajectories of the Lorenz memristor chaotic system is not very much different from each other, and maintains the basic structure of the Lorenz chaotic attractor. This is consistent with the rules in the above-proposed method of constructing memristor chaotic systems, which also shows the correctness of the method of constructing memristor chaotic systems proposed in the text, i.e., without changing the structure of the equation of state of the original system, it can also basically maintain the structure of the attractor phase diagram of the original system, whereas the degree of chaos and the robustness of the system need to be determined from several performance indicators.

To investigate the specific effect of the HP memristor in the original system, the control parameter $\gamma \in \left[0,5\right]$ is set, and the bifurcation diagrams of the Lorenz chaotic system and the Lorenz memristor chaotic system are plotted respectively, as shown in Figure 12(a). The bifurcation diagram of Lorenz chaotic system is divided into three parts: periodic state, chaotic state, and non-chaotic state, when $\gamma =3.779$, the system is not chaotic, while in the bifurcation diagram of Lorenz memristor chaotic system, although there is no significant change in the region of the periodic state, but the region of the chaotic state is increased in Figure 12(b). This shows that the HP memristor can effectively extend the parameter interval of the original Lorenz chaotic system.

Figure 12.

Bifurcation of the system with $\gamma \in \left[0,5\right]$. (a) Lorenz chaotic system. (b) Lorenz memristor chaotic system.

To study and compare the degree of chaos of the two systems, the LLE method and Multi-scale permutation entropy (MPE) [47] are used to examine the degree of chaos and the complexity of the sequences of the two systems, respectively. As shown in Figure 13, the LLE of the Lorenz memristor chaotic system is significantly larger compared to the Lorenz chaotic system, when $\gamma =3.5$, the LLE of the Lorenz chaotic system rapidly decreases to 0, and in the subsequent parameter control, the LLE of the system is constantly less than 0, at which time the system is in a non-chaotic state, while the LLE of the Lorenz memristor chaotic system is continuously maintained at around 1, which remains consistent with the phenomenon of the bifurcation diagram. In the same case, the MPE of the system exhibits the same trend as the LLE, which demonstrates that the generalized HP memristor model proposed in this paper can effectively expand the parameter range of the system and strengthen the degree of chaos of the system without changing the structure of the original system equations and the attractor structure.

Figure 13.

LLE and MPE of two systems (a) LLE; (b) MPE.

In Table 4, the chaos range, larger LLE ratio and larger MPE ratio under the parameter $\gamma$ for the Lorenz memristor chaotic system discussed in this section and the classical Lorenz chaotic system are given, demonstrating that the Lorenz memristor chaotic system has a larger chaos range and a stronger degree of chaos compared to the classical system.

	Chaos range of $\gamma$	Larger LLE ratio	Larger MPE ratio
Lorenz	3.249	13%	5%
Lorenz memristor	4.470	87%	95%

Table 4.

Comparison of Lorenz chaotic system and Lorenz memristor chaotic system.

3. Conclusion

Memristor is a kind of component with great development possibility, which has the characteristics of small size, low power consumption, and memory function, and has high application value in chaotic circuits. In this paper, the nonlinear characteristics and the mechanism of action inside the memristor are investigated through theoretical analysis. Firstly, two types of generalized memristor models are proposed. On the one hand, based on sampling memristor-capacitor circuits, a generalized voltage-controlled discrete memristor model is derived through the intrinsic relationship between the memristor and the capacitor. On the other hand, under the definition of charge-flux in the HP memristor model, considering the operating initial conditions of the memristor device and the film boundary movement conditions, the interdependence between charge, magnetic flux, and memristor is analyzed in detail, and a generalized magnetically controlled titanium dioxide memristor model is deduced. Secondly, the above two types of memristor models are applied to two classical chaotic systems, and the construction methods and application rules of the new systems are elaborated in detail. Then, the dynamical behaviors of the original system and the memristor system are compared and analyzed by various dynamical analysis methods to verify the correctness and applicability of the proposed generalized memristor model. Finally, the experimental results show that the generalized memristor model proposed in this paper can effectively strengthen the chaotic characteristics and robustness of the system, which provides an effective theoretical basis and experimental foundation for applications in the field of confidential communication.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61971109, in part by the GF Science and Technology Special Innovation Zone Project, and in part by the Fundamental Research Funds of Central Universities under Grant 2672018ZYGX2018J009.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.