Dynamic Analysis and FPGA Implementation of a New Linear Memristor-Based Hyperchaotic System with Strong Complexity

Abstract

Chaotic or hyperchaotic systems have a significant role in engineering applications such as cryptography and secure communication, serving as primary signal generators. To ensure stronger complexity, memristors with sufficient nonlinearity are commonly incorporated into the system, suffering a limitation on the physical implementation. In this paper, we propose a new four-dimensional (4D) hyperchaotic system based on the linear memristor which is the most straightforward to implement physically. Through numerical studies, we initially demonstrate that the proposed system exhibits robust hyperchaotic behaviors under typical parameter conditions. Subsequently, we theoretically prove the existence of solid hyperchaos by combining the topological horseshoe theory with computer-assisted research. Finally, we present the realization of the proposed hyperchaotic system using an FPGA platform. This proposed system possesses two key properties. Firstly, this work suggests that the simplest memristor can also induce strong nonlinear behaviors, offering a new perspective for constructing memristive systems. Secondly, compared to existing systems, our system not only has the largest Kaplan-Yorke dimension, but also has clear advantages in areas related to engineering applications, such as the parameter range and signal bandwidth, indicating promising potential in engineering applications.

1. Introduction

Hyperchaotic systems are a special kind of chaotic system with more than one positive Lyapunov Exponent (LE) number [1,2]. hyperchaotic systems with more directions have more complex dynamic behaviors than stretched ordinary hyperchaos with only a one-dimensional direction. Therefore, it is widely used in chaotic communication [3], chaotic encryption [4] and other fields [5].

In 1979, O. Rossler et al. [6] proposed the first hyperchaotic system. In the beginning, many researchers have conducted in-depth and extensive research in this field, and at the same time, many new Hyperchaotic systems have been proposed. For example, Wang et al. [7] constructed a Lorenz-like 4D Hyperchaotic system by introducing a nonlinear controller. Li et al. [8] proposed hyperchaotic memristive circuit by adding a quadratic ideal memristor. V.T.Pham et al. [9] proposed a Hyperchaotic system with no equilibrium point. Lai et al. [10] constructed a hyperchaotic system with no fixed points and an infinite number of coexisting attractors.

Among the aforementioned works, the primary focus lies in uncovering the intricate dynamical behaviors concealed within the proposed systems. These theoretical analyses and experimental observations are instrumental in comprehending the nonlinear operational mechanisms of these systems. However, while these discovered dynamic behaviors hold significance, they may not always be advantageous, and in certain cases, they can even prove detrimental to real-world applications. For instance, the transient nature of complex behaviors, such as hyperchaos, is evident in many systems. Following a period of evolution, hyperchaos tends to degrade into conventional chaos. In the context of encryption algorithms based on hyperchaos, this degradation can compromise confidentiality. For a secure communication process reliant on a chaotic system, quality communication without information errors can only be achieved when system parameters afford a wide range of chaotic behaviors. Unfortunately, system properties crucial for engineering applications, including complexity, parameter robustness, and signal bandwidth, are often overlooked.

Recently, researchers have shifted their focus to address the above-mentioned aspects related to engineering applications. For example, Mezatio et al. [11] developed a 6D hyperchaotic autonomous system by introducing a kind of two-order ideal memristor. Xiu et al. [12] adopted multimemristors to trigger the strong hyperchaotic state in a 6D system. Chen et al. [13] proposed a 4D hyperchaotic system characterized by high complexity. They further explored the system’s engineering-related properties from the aforementioned perspectives. Thanks to its favorable engineering properties, this system has demonstrated promising applications in various fields, including public-key cryptography [14] and substitution-box construction [15]. Nevertheless, systems still possess certain shortcomings, primarily stemming from the intricate coupling of multiple complex nonlinear terms, which may pose challenges during specific physical implementations of memristors or circuits. At the same time, whether the simplest linear memristor can lead to strong complexity is an interesting and open problem.

Aiming at this limitation above, this paper proposes a new 4D hyperchaotic system based on the simplest linear memristor. In order to study the properties of the system, we first discuss the basic dynamic properties of the system through theoretical analysis and dynamic analysis. Then, the properties of the system related to engineering applications are analyzed. Moreover, based on topological horseshoe theory [16,17,18], the existence of strong hyperchaotic behaviors is proved strictly. Finally, Field Programmable Gate Array (FPGA) hardware platform successfully implements the proposed 4D hyperchaotic system. Our study indicates that the linear memristor can also lead to strong and robust complexity.

To be summarized, the main contribution of this work covers three folds:

(1)

We introduce a new 4D hyperchaotic system by adding the simplest linear memristor to the Qi chaotic system, showing the potential of the linear memristor in constructing memristive systems with complex behaviors.

(2)

We show that the developed system has a simpler structure and more intricate dynamic properties relevant to real-world applications compared to previous research.

(3)

We perform in-depth dynamic analyses, including fundamental dynamic properties, computer-assisted proof for the existence of strong hyperchaos and hardware validation based on FPGA.

2. Related Work

The existing work for chaotic or hyperchaotic systems can be categorized into two groups. The first one focuses on constructing new chaotic or hyperchaotic systems. The representative works include: Wu et al. [19] proposed a hyperchaotic system with two large positive LEs and studied the Hopf bifurcation transient transition phenomenon in it. Qi et al. [20] proposed a system with strong hyperchaotic behaviors by introducing linear feedback. Kuate et al. [21] constructed a new three-dimensional chaotic system with up to eight coexisting attractors. The proposed system is multiplier-less, variable boost, and entirely based on Chua diode nonlinearity. Karawanich et al. [22] proposed a chaotic system based on the concept of third-order jerk. Zhang et al. [23]  developed a simple non-equilibrium chaotic system with only one sign function. Besides, introducing memristors is a popular construction way. For example, by connecting two discrete memristors with sine and cosine memristors in parallel, Zhang et al. [24] constructed two novel dual-memristor hyperchaotic maps. Although many impressive research studies continue to make progress, the system construction mentioned above is focused on finding new dynamics. However, it does not adequately consider the challenges of physical implementation and the robustness of the system’s behaviors, which limits its real-world applications.

The second group focuses on the complex dynamics in chaotic or hyperchaotic systems. These works are mainly in three lines. The first line focuses on the emergent mechanisms of complex behaviors. For example, in the paper by [25], the authors discussed the mechanism for generating hyperchaos from the micro-scale (intersection of homologous and heterogeneous orbits). Another paper by [26] discussed the evolutionary path from ordinary chaos to hyperchaos using the topological horse theory. The second line focuses on some interesting new dynamic phenomenons. For example, Tang et al. [27] found a new kind of topological horseshoe in the phase space of a hyperchaotic system constructed by introducing memory element. Yu et al. [28] addressed the transient behavior mechanism analysis of a new 4D chaotic system. The third line switches to systems with distinctive features. For instance, systems without equilibrium points [10,29] has attracted much attention. Within the context, the implicit dynamics, such as co-existence of multi-attractor [30,31] and hidden attractor [32,33], were found and analyzed further. From the work mentioned above, it is seen that these efforts have highlighted increasing system complexity, reported new physical phenomena and revealed the hidden working mechanism, losing focus on the system features closely attached to the real application.

In short, the study of new dynamic behaviors is an important topic in the field of nonlinear science, but it is not the only theme. The purpose of this paper is to introduce a new hyperchaotic system with excellent dynamic characteristics relevant to real engineering scenarios, such as system complexity, parameter robustness, and signal bandwidth. Therefore, the proposed system contributes to the nonlinear community by enhancing engineering applications. Additionally, to our knowledge, there are few works conducting research from this perspective, which is the main motivation behind our work.

3. The Proposed Hyperchaotic System

The mathematical model of the proposed novel 4D system is modified from the classical Qi system [34] by introducing the simplest linear memristor. According to [35], the adopted linear memristor can be defined as the following general form.

$$\dot{w}\left(t\right)=rx\left(t\right),\varphi \left(t\right)=\left(v+kw\left(t\right)\right)x\left(t\right)$$

(1)

In Equation (1), r, v and k are constants; $x\left(t\right)$ and $\varphi \left(t\right)$ are two complementary constitutive variables standing for input and output of the memristor, respectively; $w\left(t\right)$ is the internal state variable. In practice, when $x\left(t\right)$ and $\varphi \left(t\right)$ formulate the voltage and current of a two-terminal electronic element, respectively, Equation (1) essentially represents a flux-controlled memristor with linear memductance $v+kw\left(t\right)$.

Combining the linear memristor above with the Qi system, we describe the novel 4D system as Equation (2).

$$\left\{\begin{array}{c}\dot{x}=a\left(y-x\right)+eyz,\hfill \\ \dot{y}=cx-by-xz,\hfill \\ \dot{z}=xy-dz-\varphi \left(w\right)x,\hfill \\ \dot{w}=rx,\hfill \end{array}\right.$$

(2)

where $(x,y,z,w)$ are state variables, $\varphi \left(w\right)=v+kw$ and $(a,b,c,d,e,r,k,v)$ are system parameters. To present the complexity of the system  (2), we perform a comprehensive numerical study from three aspects: (1) Basic dynamic behaviors, (2) engineering application-promising characteristics, and (3) computer-assisted proof of strong complexity. Furthermore, to verify the feasibility of the proposed system, a hardware platform based on FPGA is built to verify the numerical results experimentally. These details are elaborated in the following sections.

4. Basic Dynamic Behaviors

For the proposed system  (2), when we take

$$(a,b,c,d,e,r,k,v)=(-39,39,15,41,123,-100,-35,1),$$

(3)

the LEs are $L{E}_1=14.149$, $L{E}_2=8.421$, $L{E}_3=0$, $L{E}_4=-63.475$, namely the system (2) is in a hyperchaotic state. In this paper, the LE computation uses the Jacobian method [36] with QR-decomposition. The code lec.c is available at https://drive.google.com/file/d/1sDXjSAjYKmmXSIiSN7zEvVuq58rKBmXg/view?usp=sharing. In the lec.c file, funtion ‘odesolve’ compute the trajectory of one step by local linearixed (i.e., the Jacobian method), while the QR-decomposition is achieved by the fuction ’qr’. Of note, the ‘odesolve’ is different from the ODE45 function in Matlab, which is used for discussing the engineering application-promising characteristics (Section 5). Specifically, ‘odesolve’ is a component of our LE computation based on the Jacobian method. In contrast, ODE45 is a built-in official Matlab function based on the Runge-Kutta method [37], computing the system’s trajectory under a given time range containing multi-steps.

This specific parameter setting derives from a parallel random searching algorithm for parameter identification, which is based on the LE computation and the basic parameters of the Qi system. The corresponding hyperchaotic attractor in phase space is shown in Figure 1.

In order to verify whether the attractor has 2D stretching during large 1D stretching, we transformed the original continuous system (2) into Poincaré map by selecting an appropriate hyperplane and observed the distribution characteristics of the attractor. Specifically, the same as previous work [38,39], we choose Poincaré intersection:

$$P\stackrel{\Delta }{=}\{(x,y,z,w)\left|x=0,\dot{x}\right.<0\},$$

(4)

then Poincaré map $H:P\to P$ can be defined as: for any point

$$X\stackrel{\Delta }{=}(0,y,z,w)\in P,$$

(5)

$H\left(x\right)$ stands for the intersection when a trajectory of the system (2) first return the section P under initial condition of x.

The attractor of H is shown in Figure 2 where the attractor is distributed approximately on a 2D surface (Figure 2a). It shows that the system in one direction exists strong compression, namely $L{E}_4<0$. In addition, the distribution shape of the attractor has a certain area (Figure 2b), which indicates that the system trajectory has two directions of stretching, namely, $L{E}_1>0$, $L{E}_2>0$.

It is not hard to find out that the new system (2) is easy to the conclusion:

$$\begin{array}{cc}\hfill \nabla V& =\frac{\partial \dot{x}}{\partial x}+\frac{\partial \dot{y}}{\partial y}+\frac{\partial \dot{z}}{\partial z}+\frac{\partial \dot{w}}{\partial w}\hfill \\ \hfill & =-a-b-d=-41<0.\hfill \end{array}$$

(6)

where $(\dot{x},\dot{y},\dot{z},\dot{w})\ne -(\dot{x},\dot{y},\dot{z},\dot{w})$, therefore, the new system (2) is not symmetrical and its dissipation.

The balance points of the hyperchaotic system (2) are found by solving the following algebraic system of equations:

$$\begin{array}{c}\hfill a(y-x)+eyz=0\\ \hfill cx-by-xz=0\\ \hfill xy-dz+\varphi \left(w\right)x=0\\ \hfill rx=0\end{array}$$

(7)

From the lastest sub-equation in Equation (7), we have $x=0$. Substituting $x=0$ into the second and third sub-equations, we get $y=0$ and $z=0$, respectively. Thus, w can be any real numbers, i.e., $w\in \mathbb{R}$. Namely, the system (2) has infinity equilibria as

$$s_0=\left[0,0,0,w\right].$$

(8)

To explore the nonlinear dynamics of the proposed system at a large range of parameters, we compute the spectrum of LE of the system (1) when parameter $40\le d\le 56$ with a stepsize 0.25. The numerical results are shown in Figure 3a, from which it can be seen that two positive LEs exist in the range of $40\le d\le 51.5$. The results show that our system can robustly operate in a hyperchaotic state over a wide range of parameters. At the same time, the bifurcation diagram with section hyperplane $\{x=0,\dot{x}<0\}$ is presented in Figure 3c. It seems that the bifurcation outcomes align with the Lyapunov spectrum, indicating accurate calculation of the LE. The initial condition for all computations above is $(0.5,0,0.5,0.5)$.

5. Engineering Application-Promising Characteristics

In this section, we discuss four dynamical features related to engineering applications, including (1) attractor complexity, (2) initial value sensitivity, (3) frequency spectrum and (4) robustness of system parameters.

5.1. Attractor Complexity

In chaos engineering applications, hyperchaos attractors generate random signals. The more complex the attractor means that it is more difficult to obtain all dynamic characteristics of the original system by signal analysis and signal reconstruction. Namely, the security properties of the generated signal are better. Generally, the Kaplan-Yorke dimension [40] is used to evaluate the complexity of attractors, and it is defined as follows.

$$D_{KY}=D+\sum _{i=1}^D\frac{L{E}_i}{\left|L{E}_D\right|}$$

(9)

where the constant D satisfies the following relation:

$$\sum _{i=1}^{D}L{E}_i\ge 0,\sum _{i=1}^{D+1}L{E}_i<0$$

(10)

Based on Equation (9), we can calculate the Kaplan–Yorke dimension of system (2) and compare it with six other typical systems. As shown in

Table 1. The Kaplan–Yorke dimension ($D_{KY}$) comparisons of typical hyperchaotic systems.

System	${\mathit{D}}_{\mathbf{KY}}$	${\mathbf{LE}}_1$	${\mathbf{LE}}_2$	${\mathbf{LE}}_3$	${\mathbf{LE}}_4$
System [41]	3.197	0.119	0.049	0	−0.852
System [42]	3.105	4.409	0.049	0	−0.852
System [18]	3.114	1.349	0.256	0	−14.095
System [19]	3.198	12.08	7.731	0	−97.229
System [20]	3.278	13.46	3.478	0	−61.231
System [13]	3.324	26.05	11.39	0	−115.518
System (2)	3.356	14.149	8.421	0	−63.475

, system (2) outperforms all the comparison systems. This result suggests that the attractor of system (2) is more complex than those of the other systems. Notably, even when compared to the previously best-performing system [13] with multiple complex nonlinear terms, system (2) still outperforms it despite utilizing the simplest linear memristor.

5.2. Initial Value Sensitivity

For some applications, like image encryption, the initial value sensitivity of random signals is one of the key indicators. For dynamic systems, initial sensitivity can be measured the separating time, termed $t_s$, when two tracks clearly separated from each other. Meanwhile, at the beginning, the two tracks’ initial values are very close. In the simulation, we select two different and sufficiently close points as initial values. They are respectively:

$$\begin{array}{cc}\hfill {\mathrm{p}}_1& =\left[10,10,10,10\right],\hfill \\ \hfill {\mathrm{p}}_2& ={\mathrm{p}}_2+\left[0,0,0,0.01\right].\hfill \end{array}$$

(11)

Also, we employ the ODE45 function in MATLAB to calculate the corresponding trajectory. The relative error and absolute errors were $e^{-12}$ and $e^{-9}$, respectively. At the same time, another three typical hyperchaotic systems were selected as the comparison systems. Their orbits were calculated under the same initial value conditions.

The comparison results are presented in Figure 4. The two rails of system [19], system [20] and system [13] are separated obviously at $t_s=0.9$, $t_s=0.55$ and $t_s=0.25$ respectively, whilst the separation time of the system (2) is $t_s=0.15$. These results indicate that the system (2) is more sensitive to the initial conditions change.

5.3. Frequency Spectrum

For secure communication, signal frequency bandwidth is a vital feature. According to signal theory, wider frequency bandwidth means signals comprise richer sine waves. The increase in high-frequency components makes the signal spectrum structure more complex, and the signals changes faster in the same time interval. Therefore, hyperchaotic signals with large bandwidths are more difficult to capture and extract, thus achieving secure communication purposes.

Here, we calculate the spectrum of the system [19], the system [20], the system [13], and the system (2) through simulation experiments Among them, the first three systems are used as comparison systems. In the calculation process, to eliminate the influence of variable step size in the Runge-Kutta method, the sampling time series with a sampling frequency of 400 is generated by interpolation, which was specifically realized by the DEVAL function in MATLAB.

In Figure 5, we present the one-side frequency spectra of three comparison systems and the system (2). The spectra values of y are normalized to be between 0 and 1. The same to the literature [13], the effective bandwidth is defined as spectrum frequency area of y is greater than 0.1. As shown in Figure 5d, the maximum effective frequency of system (2) is 120 Hz. Of note, There is also a 16 Hz gap width near 50 Hz (from 39 Hz to 55 Hz). Therefore, the effective bandwidth of the system (2) is 104 Hz. Compared with the previous best 83 Hz [13], the bandwidth of this paper is increased by 21 Hz, relatively increased by 25%.

5.4. Robustness of System Parameters

In many practical applications, hyperchaotic systems are usually realized by employing physical circuits, and integrating them into the specific working system. However, the circuit is inevitably affected by factors such as thermal noise, thermal accumulation, electromagnetic interference, and so on, resulting in component parameter drift. Hyperchaotic systems with poor parameter robustness often degenerate into ordinary chaos or long-period orbits, negatively affecting communication systems’ overall performance.

In this subsection, the parameter robustness of system (2) is discussed by the Lyapunov exponential spectrum method.

Specifically, for each system parameter, we select a real number range with a width of four, centring on a typical value of this parameter. Over this zone, we calculate the LEs of the system (2) and collectively present the values as a LE-varying curve. For example, for parameter $a=-39$, its corresponding changing range is $[-41,-37]$. Figure 6 shows the results of the LE-varying curve of all seven parameters.

From Figure 6, when all parameters of the system (2) have a drift of less than 2, the system still has two larger positive LEs with little value, and the system (2) continues to be hyperchaotic. Therefore, the system (2) can overcome hyperchaotic degradation caused by parameter drift to a certain extent.

6. Computer-Assisted Proof of Strong Complexity

As is known to all, LE is a numerical method, which inevitably has the computation error. Parameter selection significantly impacts the calculation results. It is not mathematically reliable to use it to judge system states. To fix this issue, in this section, we strictly prove the existence of chaos of system (2) with typical parameters utilizing computer-assisted proof.

Before proof, the symbolic dynamics and topological horseshoe are briefly introduced, which is essential for rigorous verification of chaos and estimation of topological entropy. If you want to know the detailed topological horseshoe and specific certification, you can refer to literature [16,17].

6.1. Result of Topological Horseshoes Theory

Let ${\mathrm{\Sigma }}_m$ is a sequence space that consists of all bi-infinite sequences with form as:

$$\begin{array}{cc}\hfill & s=\left\{\cdots ,s_{-m},\cdots ,s_{-1},s_0,s_1,\cdots ,s_m,\cdots \right\},\hfill \\ \hfill & s_m\in \left\{1,2,\cdots ,m-1\right\}.\hfill \end{array}$$

(12)

Definition 1. 

Let $\sigma :{\mathrm{\Sigma }}_m\to {\mathrm{\Sigma }}_m$ to be m-shift map: for every element of s, it satisfies $\sigma \left(s_i\right)=s_{i+m}$. Mathematically, it is proved that ${\mathrm{\Sigma }}_m$ is Cantor set. It satisfies three properties [43]: (1) has a countable infinity of periodic orbits containing all periods; (2) has an uncountable infinity of periodic orbits; (3) has a dense orbit. From those properties above, we know that dynamics generated by the map σ is sensitive to initial conditions, which means σ is chaotic.

Let X be a metric space, B be a compact subset of X and $f:D\to X$ be a map. Assume that there exists m mutually disjoint compact subsets $B_1,B_2,\cdots ,B_m,1\le i\le m$. For each $B_i$, $B_i^1$ and $B_i^2$ indicate two disjoint compact subsets of $B_i$ contained in the boundary $\partial B_i$.

Definition 2 

([18]). Let Γ is $B_i$ a compact subset, if for every subset $l\subset B_i$ and $\mathrm{\Gamma }\cap l\ne \varphi$, then Γ is said to completely separate $B_i^1$ and $B_i^2$, we denote it by $\mathrm{\Gamma }\updownarrow (B_i^1,B_i^2)$.

Definition 3 

([18]). Let $\mathrm{\Gamma }\subset B_i$ be a subset, we denote that $f\left(\mathrm{\Gamma }\right)$ separates $B_j$ with respect to $B_j^1$ and $B_j^2$, if Γ contains a compact subset ${\mathrm{\Gamma }}^{\prime }$ such that $f\left({\mathrm{\Gamma }}^{\prime }\right)\updownarrow (B_j^1,B_j^2)$. In this case that $f\left(\mathrm{\Gamma }\right)\mapsto B_j$ holds true for every subset $\mathrm{\Gamma }\subset B_i$ with $\mathrm{\Gamma }\updownarrow (B_i^1,B_i^2)$, we say that $f\left(B_i\right)$ separates $B_j$ with respect to two pairs $(B_i^1,B_i^2)$ and $(B_j^1,B_j^2)$, or $f\left(B_i\right)\mapsto B_j$ in case of no confusion.

Theorem 1 

([17]). If the codimension-one crossing relation $f\left(B_i\right)\mapsto B_j$, hold for $1\le i,j\le m$, then there exists a compact invariant set $\mathrm{K}\subset B$, such that $f\left|K\right.$ is semi-conjugate to the m-shift map, which is denoted by ${\mathrm{\Sigma }}_m$ and the entropy $ent\left(f\right)\ge logm$. When $ent\left(f\right)>0$, the map f is chaotic.

6.2. Horseshoe and Topological Entropy Estimation in Dynamics of Poincaré-Map

In order to apply the above Definition and Theorem, we first convert the original continuous system (2) into the corresponding Poincarémap. For the sake of simplicity, the subsequent proof is based on the Poincaré-map H defined in Section 3. Here, we find topological horseshoe geometry with 2D stretching, according to the algorithm for the 3D hyperchaotic topological horseshoes proposed in literature [44]. Firstly, through a series of attempts to find proper mapping zone containing periodic equilibrium points, the 2D topology horseshoe consisting of A and B is found on the attractor of map H, as shown in Figure 7. The four vertices of quadrilateral A in terms of $\left(y\times z\times w\right)$ are:

$$\begin{array}{c}\hfill V_1^A=\left[-1.8569,25.3930,-5.8155\right],V_2^A=\left[-1.9594,23.8266,-6.6488\right],\\ \hfill V_3^A=\left[-1.9392,23.7987,-6.6054\right],V_4^A=\left[-1.7935,24.6516,-5.9051\right].\end{array}$$

(13)

The four vertices of quadrilateral B in terms of $\left(y\times z\times w\right)$ are:

$$\begin{array}{c}\hfill V_1^B=\left[-1.8569,25.3930,-5.5985\right],V_2^B=\left[-1.9594,23.8266,-5.9157\right],\\ \hfill V_3^B=\left[-1.9392,23.7987,-5.9126\right],V_4^B=\left[-1.7935,24.6516,-5.1736\right],\end{array}$$

(14)

Following that, based on the obtained quadrilateral A, hexahedron a is constructed through the following three steps:

(1)

From the attractor of map H, the point near A is selected for surface fitting, and the surface equation is obtained.

(2)

With the aid of this obtained equation, the normal direction of the centre of A is calculated.

(3)

A is shifted along this normal line’s positive and negative directions by 0.6, and two curved surfaces are obtained. Taking the surfaces as top and bottom surfaces, respectively, we can construct the hexahedron, namely the 3D subset a.

In the same way above, we can construct 3D subset b, which is corresponding to B in the 2D topological horseshoes.

Finally, the hexahedron a and b conduct six times Poincarémap H respectively, a, b and their images $H^6\left(a\right)$, $H^6\left(b\right)$ form geometric relations as shown in Figure 8. It is easy to find that $H_a^6$ passes through both a and b clearly in the middle (Figure 8b), and there is no contact between a and b (Figure 8c). Therefore, $H_a^6$ satisfied $H^6\left(a\right)\mapsto a,b$. Similarly, by observing the geometry presented in Figure 8d, Figure 8e and Figure 8f, we can find that the $H^6\left(b\right)\mapsto a,b$ also hold.

Suppose that subsets a and b are considered as $B_1$ and $B_2$ in Theorem 1, respectively. Obviously, for any $1\le i,j\le 2$,$H\left(B_i\right)\mapsto B_j$ is always is true. According to Theorem 1, we know that on the compact invariant set $\mathrm{\Lambda }\subset a\cup b$, the map H is semi-conjugate to the 2-shift map and $ent\left(H\right)\ge log2$. Given that ${\mathrm{\Sigma }}_2$ is chaotic. Therefore, H must be chaotic. Since that $H^6$(a) and $H^6$(b) expand in two directions, the expansions along each trajectory in $\mathrm{\Lambda }$ are also in two directions. So there must be exit two positive LEs. Therefore, the system (2) is hyperchaotic.

7. Fpga Implementation

Hardware implementation of the chaotic system contains two kinds of methods. One is to build chaos circuit with basic components of circuit [45], the other is to realize the chaos circuit with the platform of FPGA. Among them, the first method has many problems, because the phase diagram of the differential equation of the dynamical system built with circuit components are displayed by analogue filter [46], which is challenging to deal with a large amount of data in practical operation. The reason is that the circuit components will be affected by temperature, illumination, voltage, and other external factors, which increases the instability of the chaotic system. It is difficult to apply this hardware to achieve the method. Another method to realize the FPGA platform [47,48]. There are two aspects of advantages. On the one hand, FPGA is a digital signal processing device with high computational accuracy, widely used in multi-tasks [49]. It can strictly match the two chaotic parameters of its sending end and receiving end, which effectively improves the fidelity of the signal demodulated at the receiving end. On the other hand, the chaotic signal generated by FPGA has good portability and high confidentiality, which makes the circuit stable and not easily affected by external factors. It facilitates the further application of the chaotic system, e.g., typical image cryptosystem [50,51].

It is well known that FPGA is a semi-custom digital integrated circuits. Thus, the chaotic system’s hardware implementation based on FPGA must first discretize the continuous system (2). Specifically, the sampling frequency $\Delta T$ must satisfy the chaotic signal cut-off frequency greater than at least twice. This makes the discrete chaotic system show dynamical characteristics closer to the system (2). The same to the other work [52,53], this paper adapts Euler’s algorithm to discretize the system (2), which is described by:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill x_1(k+1)& =\left[a{x}_2\left(k\right)-a{x}_1\left(k\right)+e{x}_2\left(k\right)x_3\left(k\right)\right]\Delta T+x_1\left(k\right)\hfill \\ \hfill x_2(k+1)& =\left[c{x}_1\left(k\right)-b{x}_2\left(k\right)-x_1\left(k\right)x_3\left(k\right)\right]\Delta T+x_2\left(k\right)\hfill \\ \hfill x_3(k+1)& =\left[x_1\left(k\right)x_2\left(k\right)-d{x}_3\left(k\right)-k{x}_4\left(k\right)x_1\left(k\right)\right]\Delta T+x_3\left(k\right)\hfill \\ \hfill x_4(k+1)& =r{x}_1\left(k\right)\Delta T+x_4\left(k\right)\hfill \end{array}\right.\end{array}$$

(15)

In Figure 9, we give the DSP-builder model of the system (2). Figure 10 shows the real FPGA hardware platform and the experimental results displayed by the digital oscilloscope. Clearly, as shown in Figure 11a–f, the chaotic attractor generated by the FPGA platform well matches the numerical simulation results of MATLAB displayed in Figure 1. These results confirm the feasibility of the proposed system.

8. Conclusions

Aiming at the core requirement of chaotic systems in engineering applications, a new 4D hyperchaotic system based on the linear memristor is proposed and analyzed comprehensively in this paper. Compared with existing systems, this new system exhibits improved properties such as strong complexity, initial value sensitivity, bandwidth, and parameter robustness. For instance, in terms of initial value sensitivity, our system relatively increased by 40% compared with the previous best system. In terms of theoretical analysis, we strictly prove the existence of the strong hyperchaos in given typical parameter condition. In addition, the new system with hyperchaotic was realized based on an FPGA digital development platform, which verifies the feasibility of the proposed system as a random number signal generator.

Our future work will focus on two main issues: First, we will explore the impact of the new dynamic characteristic of infinitely many equilibrium points. Second, we aim to expand this research to include systems with multi-linear memristors, which is significant for neuromorphic network research.